Classical Algebraic Geometry: a modern view · Preface The main purpose of the present treatise is...

Classical Algebraic Geometry: a modernview

IGOR V. DOLGACHEV

Preface

The main purpose of the present treatise is to give an account of some of thetopics in algebraic geometry which while having occupied the minds of manymathematicians in previous generations have fallen out of fashion in moderntimes. Often in the history of mathematics new ideas and techniques make thework of previous generations of researchers obsolete, especially this appliesto the foundations of the subject and the fundamental general theoretical factsused heavily in research. Even the greatest achievements of the past genera-tions which can be found for example in the work of F. Severi on algebraiccycles or in the work of O. Zariskis in the theory of algebraic surfaces havebeen greatly generalized and clarified so that they now remain only of histor-ical interest. In contrast, the fact that a nonsingular cubic surface has 27 linesor that a plane quartic has 28 bitangents is something that cannot be improvedupon and continues to fascinate modern geometers. One of the goals of thispresent work is then to save from oblivion the work of many mathematicianswho discovered these classic tenets and many other beautiful results.

In writing this book the greatest challenge the author has faced was distillingthe material down to what should be covered. The number of concrete facts,examples of special varieties and beautiful geometric constructions that haveaccumulated during the classical period of development of algebraic geometryis enormous and what the reader is going to find in the book is really onlythe tip of the iceberg; a work that is like a taste sampler of classical algebraicgeometry. It avoids most of the material found in other modern books on thesubject, such as, for example, [10] where one can find many of the classicalresults on algebraic curves. Instead, it tries to assemble or, in other words, tocreate a compendium of material that either cannot be found, is too dispersed tobe found easily, or is simply not treated adequately by contemporary researchpapers. On the other hand, while most of the material treated in the book existsin classical treatises in algebraic geometry, their somewhat archaic terminology

iv Preface

and what is by now completely forgotten background knowledge makes thesebooks useful to but a handful of experts in the classical literature. Lastly, onemust admit that the personal taste of the author also has much sway in thechoice of material.

The reader should be warned that the book is by no means an introductionto algebraic geometry. Although some of the exposition can be followed withonly a minimum background in algebraic geometry, for example, based onShafarevichs book [530], it often relies on current cohomological techniques,such as those found in Hartshornes book [282]. The idea was to reconstructa result by using modern techniques but not necessarily its original proof. Forone, the ingenious geometric constructions in those proofs were often beyondthe authors abilities to follow them completely. Understandably, the price ofthis was often to replace a beautiful geometric argument with a dull cohomo-logical one. For those looking for a less demanding sample of some of thetopics covered in the book, the recent beautiful book [39] may be of great use.

No attempt has been made to give a complete bibliography. To give an ideaof such an enormous task one could mention that the report on the status oftopics in algebraic geometry submitted to the National Research Council inWashington in 1928 [535] contains more than 500 items of bibliography by130 different authors only in the subject of planar Cremona transformations(covered in one of the chapters of the present book.) Another example is thebibliography on cubic surfaces compiled by J. E. Hill [295] in 1896 whichalone contains 205 titles. Meyers article [385] cites around 130 papers pub-lished 1896-1928. The title search in MathSciNet reveals more than 200 papersrefereed since 1940, many of them published only in the past 20 years. Howsad it is when one considers the impossibility of saving from oblivion so manynames of researchers of the past who have contributed so much to our subject.

A word about exercises: some of them are easy and follow from the defi-nitions, some of them are hard and are meant to provide additional facts notcovered in the main text. In this case we indicate the sources for the statementsand solutions.

I am very grateful to many people for their comments and corrections tomany previous versions of the manuscript. I am especially thankful to SergeyTikhomirov whose help in the mathematical editing of the book was essentialfor getting rid of many mistakes in the previous versions. For all the errors stillfound in the book the author bears sole responsibility.

Contents

1 Polarity page 11.1 Polar hypersurfaces 1

1.1.1 The polar pairing 11.1.2 First polars 71.1.3 Polar quadrics 131.1.4 The Hessian hypersurface 151.1.5 Parabolic points 181.1.6 The Steinerian hypersurface 211.1.7 The Jacobian hypersurface 25

1.2 The dual hypersurface 321.2.1 The polar map 321.2.2 Dual varieties 331.2.3 Plucker formulas 37

1.3 Polar s-hedra 401.3.1 Apolar schemes 401.3.2 Sums of powers 421.3.3 Generalized polar s-hedra 441.3.4 Secant varieties and sums of powers 451.3.5 The Waring problems 52

1.4 Dual homogeneous forms 541.4.1 Catalecticant matrices 541.4.2 Dual homogeneous forms 571.4.3 The Waring rank of a homogeneous form 581.4.4 Mukais skew-symmetric form 591.4.5 Harmonic polynomials 62

1.5 First examples 671.5.1 Binary forms 67

vi Contents

1.5.2 Quadrics 70Exercises 72Historical Notes 74

2 Conics and quadric surfaces 772.1 Self-polar triangles 77

2.1.1 Veronese quartic surfaces 772.1.2 Polar lines 792.1.3 The variety of self-polar triangles 812.1.4 Conjugate triangles 85

2.2 Poncelet relation 912.2.1 Darbouxs Theorem 912.2.2 Poncelet curves and vector bundles 962.2.3 Complex circles 99

2.3 Quadric surfaces 1022.3.1 Polar properties of quadrics 1022.3.2 Invariants of a pair of quadrics 1082.3.3 Invariants of a pair of conics 1122.3.4 The Salmon conic 117

Exercises 121Historical Notes 125

3 Plane cubics 1273.1 Equations 127

3.1.1 Elliptic curves 1273.1.2 The Hesse equation 1313.1.3 The Hesse pencil 1333.1.4 The Hesse group 134

3.2 Polars of a plane cubic 1383.2.1 The Hessian of a cubic hypersurface 1383.2.2 The Hessian of a plane cubic 1393.2.3 The dual curve 1433.2.4 Polar s-gons 144

3.3 Projective generation of cubic curves 1493.3.1 Projective generation 1493.3.2 Projective generation of a plane cubic 151

3.4 Invariant theory of plane cubics 1523.4.1 Mixed concomitants 1523.4.2 Clebschs transfer principle 1533.4.3 Invariants of plane cubics 155

Exercises 157

Contents vii

Historical Notes 160

4 Determinantal equations 1624.1 Plane curves 162

4.1.1 The problem 1624.1.2 Plane curves 1634.1.3 The symmetric case 1684.1.4 Contact curves 1704.1.5 First examples 1744.1.6 The moduli space 176

4.2 Determinantal equations for hypersurfaces 1784.2.1 Determinantal varieties 1784.2.2 Arithmetically Cohen-Macaulay sheaves 1824.2.3 Symmetric and skew-symmetric aCM sheaves 1874.2.4 Singular plane curves 1894.2.5 Linear determinantal representations of surfaces 1974.2.6 Symmetroid surfaces 201


5 Theta characteristics 2095.1 Odd and even theta characteristics 209

5.1.1 First definitions and examples 2095.1.2 Quadratic forms over a field of characteristic 2 210

5.2 Hyperelliptic curves 2135.2.1 Equations of hyperelliptic curves 2135.2.2 2-torsion points on a hyperelliptic curve 2145.2.3 Theta characteristics on a hyperelliptic curve 2165.2.4 Families of curves with odd or even theta

characteristic 2185.3 Theta functions 219

5.3.1 Jacobian variety 2195.3.2 Theta functions 2225.3.3 Hyperelliptic curves again 224

5.4 Odd theta characteristics 2265.4.1 Syzygetic triads 2265.4.2 Steiner complexes 2295.4.3 Fundamental sets 233

5.5 Scorza correspondence 2365.5.1 Correspondences on an algebraic curve 2365.5.2 Scorza correspondence 240

viii Contents

5.5.3 Scorza quartic hypersurfaces 2435.5.4 Contact hyperplanes of canonical curves 246


6 Plane Quartics 2516.1 Bitangents 251

6.1.1 28 bitangents 2516.1.2 Aronhold sets 2536.1.3 Riemanns equations for bitangents 256

6.2 Determinant equations of a plane quartic 2616.2.1 Quadratic determinantal representations 2616.2.2 Symmetric quadratic determinants 265

6.3 Even theta characteristics 2706.3.1 Contact cubics 2706.3.2 Cayley octads 2716.3.3 Seven points in the plane 2756.3.4 The Clebsch covariant quartic 2796.3.5 Clebsch and Luroth quartics 2836.3.6 A Fano model of VSP(f, 6) 291

6.4 Invariant theory of plane quartics 2946.5 Automorphisms of plane quartic curves 296

6.5.1 Automorphisms of finite order 2966.5.2 Automorphism groups 2996.5.3 The Klein quartic 302


7 Cremona transformations 3117.1 Homaloidal linear systems 311

7.1.1 Linear systems and their base schemes 3117.1.2 Resolution of a rational map 3137.1.3 The graph of a Cremona transformation 3167.1.4 F-locus and P-locus 3187.1.5 Computation of the multidegree 323

7.2 First examples 3277.2.1 Quadro-quadratic transformations 3277.2.2 Bilinear Cremona transformations 3297.2.3 de Jonquieres transformations 334

7.3 Planar Cremona transformations 3377.3.1 Exceptional configurations 337

Contents ix

7.3.2 The bubble space of a surface 3417.3.3 Nets of isologues and fixed points 3447.3.4 Quadratic transformations 3497.3.5 Symmetric Cremona transformations 3517.3.6 de Jonquieres transformations and hyperellip-

tic curves 3537.4 Elementary transformations 356

7.4.1 Minimal rational ruled surfaces 3567.4.2 Elementary transformations 3597.4.3 Birational automorphisms of P1 P1 361

7.5 Noethers Factorization Theorem 3667.5.1 Characteristic matrices 3667.5.2 The Weyl groups 3727.5.3 Noether-Fano inequality 3767.5.4 Noethers Factorization Theorem 378


8 Del Pezzo surfaces 3868.1 First properties 386

8.1.1 Surfaces of degree d in Pd 3868.1.2 Rational double points 3908.1.3 A blow-up model of a del Pezzo surface 392

8.2 The EN -lattice 3988.2.1 Quadratic lattices 3988.2.2 The EN -lattice 4018.2.3 Roots 4038.2.4 Fundamental weights 4088.2.5 Gosset polytopes 4108.2.6 (1)-curves on del Pezzo surfaces 4128.2.7 Effective roots 4158.2.8 Cremona isometries 418

8.3 Anticanonical models 4228.3.1 Anticanonical linear systems 4228.3.2 Anticanonical model 427

8.4 Del Pezzo surfaces of degree 6 4298.4.1 Del Pezzo surfaces of degree 7, 8, 9 4298.4.2 Del Pezzo surfaces of degree 6 430

8.5 Del Pezzo surfaces of degree 5 4338.5.1 Lines and singularities 433

x Contents

8.5.2 Equations 4348.5.3 OADP varieties 4368.5.4 Automorphism group 437

8.6 Quartic del Pezzo surfaces 4418.6.1 Equations 4418.6.2 Cyclid quartics 4438.6.3 Lines and singularities 4468.6.4 Automorphisms 447

8.7 Del Pezzo surfaces of degree 2 4518.7.1 Singularities 4518.7.2 Geiser involution 4548.7.3 Automorphisms of del Pezzo surfaces of

degree 2 4568.8 Del Pezzo surfaces of degree 1 457

8.8.1 Singularities 4578.8.2 Bertini involution 4598.8.3 Rational elliptic surfaces 4618.8.4 Automorphisms of del Pezzo surfaces of

degree 1 462Exercises 470Historical Notes 471

9 Cubic surfaces 4759.1 Lines on a nonsingular cubic surface 475

9.1.1 More about the E6-lattice 4759.1.2 Lines and tritangent planes 4829.1.3 Schurs quadrics 4869.1.4 Eckardt points 491

9.2 Singularities 4949.2.1 Non-normal cubic surfaces 4949.2.2 Lines and singularities 495

9.3 Determinantal equations 5019.3.1 Cayley-Salmon equation 5019.3.2 Hilbert-Burch Theorem 5049.3.3 Cubic symmetroids 509

9.4 Representations as sums of cubes 5129.4.1 Sylvesters pentahedron 5129.4.2 The Hessian surface 5159.4.3 Cremonas hexahedral equations 5179.4.4 The Segre cubic primal 520

Contents xi

9.4.5 Moduli spaces of cubic surfaces 5349.5 Automorphisms of cubic surfaces 538

9.5.1 Cyclic groups of automorphisms 5389.5.2 Maximal subgroups of W (E6) 5469.5.3 Groups of automorphisms 5499.5.4 The Clebsch diagonal cubic 555


10 Geometry of Lines 56610.1 Grassmannians of lines 566

10.1.1 Generalities about Grassmannians 56610.1.2 Schubert varieties 56910.1.3 Secant varieties of Grassmannians of lines 572

10.2 Linear line complexes 57710.2.1 Linear line complexes and apolarity 57710.2.2 Six lines 58410.2.3 Linear systems of linear line complexes 589

10.3 Quadratic line complexes 59210.3.1 Generalities 59210.3.2 Intersection of two quadrics 59610.3.3 Kummer surfaces 59810.3.4 Harmonic complex 61010.3.5 The tangential line complex 61510.3.6 Tetrahedral line complex 617

10.4 Ruled surfaces 62110.4.1 Scrolls 62110.4.2 Cayley-Zeuthen formulas 62510.4.3 Developable ruled surfaces 63410.4.4 Quartic ruled surfaces in P3 64110.4.5 Ruled surfaces in P3 and the tetrahedral line

complex 653Exercises 655Historical Notes 657

Bibliography 660References 661Index 691

1Polarity

1.1 Polar hypersurfaces

1.1.1 The polar pairing

We will take C as the base field, although many constructions in this bookwork over an arbitrary algebraically closed field.

We will usually denote by E a vector space of dimension n + 1. Its dualvector space will be denoted by E.

Let S(E) be the symmetric algebra of E, the quotient of the tensor algebraT (E) = d0Ed by the two-sided ideal generated by tensors of the formv w w v, v, w E. The symmetric algebra is a graded commutativealgebra, its graded components Sd(E) are the images of Ed in the quotient.The vector space Sd(E) is called the d-th symmetric power ofE. Its dimensionis equal to

(d+nn

). The image of a tensor v1 vd in Sd(E) is denoted by

v1 vd.The permutation group Sd has a natural linear representation in Ed via

permuting the factors. The symmetrization operator Sd is a projectionoperator onto the subspace of symmetric tensors Sd(E) = (Ed)Sd multi-plied by d!. It factors through Sd(E) and defines a natural isomorphism

Sd(E) Sd(E).

Replacing E by its dual space E, we obtain a natural isomorphism

pd : Sd(E) Sd(E). (1.1)

Under the identification of (E)d with the space (Ed), we will be ableto identify Sd(E) with the space Hom(Ed,C)Sd of symmetric d-multilinearfunctions Ed C. The isomorphism pd is classically known as the totalpolarization map.

Next we use that the quotient map Ed Sd(E) is a universal symmetric

2 Polarity

d-multilinear map, i.e. any linear map Ed F with values in some vectorspace F factors through a linear map Sd(E) F . If F = C, this gives anatural isomorphism

(Ed) = Sd(E) Sd(E).

Composing it with pd, we get a natural isomorphism

Sd(E) Sd(E). (1.2)

It can be viewed as a perfect bilinear pairing, the polar pairing

, : Sd(E) Sd(E) C. (1.3)

This pairing extends the natural pairing between E and E to the symmetricpowers. Explicitly,

l1 ld, w1 wd =Sd

l1(1)(w1) l1(d)(wd).

One can extend the total polarization isomorphism to a partial polarizationmap

, : Sd(E) Sk(E) Sdk(E), k d, (1.4)

l1 ld, w1 wk =

1i1...ikn

li1 lik , w1 wk

j 6=i1,...,ik

lj .

In coordinates, if we choose a basis (0, . . . , n) in E and its dual basist0, . . . , tn in E, then we can identify S(E) with the polynomial algebraC[t0, . . . , tn] and Sd(E) with the space C[t0, . . . , tn]d of homogeneous poly-nomials of degree d. Similarly, we identify Sd(E) with C[0, . . . , n]. The po-larization isomorphism extends by linearity of the pairing on monomials

ti00 tinn , j00 jnn =

{i0! in! if (i0, . . . , in) = (j0, . . . , jn),0 otherwise.

One can give an explicit formula for pairing (1.4) in terms of differentialoperators. Since ti, j = ij , it is convenient to view a basis vector j asthe partial derivative operator j = tj . Hence any element S

k(E) =

C[0, . . . , n]k can be viewed as a differential operator

D = (0, . . . , n).

The pairing (1.4) becomes

(0, . . . , n), f(t0, . . . , tn) = D(f).

1.1 Polar hypersurfaces 3

For any monomial i = i00 inn and any monomial tj = tj00 tjnn , we

have

i(tj) =

{j!

(ji)!tji if j i 0,

0 otherwise.(1.5)

Here and later we use the vector notation:

i! = i0! in!,(k

i

)=k!

i!, |i| = i0 + + in.

The total polarization f of a polynomial f is given explicitly by the followingformula:

f(v1, . . . , vd) = Dv1vd(f) = (Dv1 . . . Dvd)(f).

Taking v1 = . . . = vd = v, we get

f(v, . . . , v) = d!f(v) = Dvd(f) =|i|=d

(di

)aiif. (1.6)

Remark 1.1.1 The polarization isomorphism was known in the classical liter-ature as the symbolic method. Suppose f = ld is a d-th power of a linear form.Then Dv(f) = dl(v)d1 and

Dv1 . . . Dvk(f) = d(d 1) (d k + 1)l(v1) l(vk)ldk.

In classical notation, a linear formaixi on Cn+1 is denoted by ax and the

dot-product of two vectors a, b is denoted by (ab). Symbolically, one denotesany homogeneous form by adx and the right-hand side of the previous formulareads as d(d 1) (d k + 1)(ab)kadkx .

Let us take E = Sm(U) for some vector space U and consider the linearspace Sd(Sm(U)). Using the polarization isomorphism, we can identifySm(U) with Sm(U). Let (0, . . . , r) be a basis in U and (t0, . . . , tr+1) bethe dual basis in U. Then we can take for a basis of Sm(U) the monomialsj. The dual basis in Sm(U) is formed by the monomials 1i!x

i. Thus, for anyf Sm(U), we can write

m!f =|i|=m

(mi

)aix

i. (1.7)

In symbolic form, m!f = (ax)m. Consider the matrix

=

(1)0 . . .

(d)0

......

...

(1)r . . .

(d)r

,

4 Polarity

where ((k)0 , . . . , (k)r ) is a copy of a basis in U . Then the space Sd(Sm(U))

is equal to the subspace of the polynomial algebra C[((i)j )] in d(r + 1) vari-ables (i)j of polynomials which are homogeneous of degree m in each columnof the matrix and symmetric with respect to permutations of the columns. LetJ {1, . . . , d}with #J = r+1 and (J) be the corresponding maximal minorof the matrix . Assume r+1 divides dm. Consider a product of k = dmr+1 suchminors in which each column participates exactlym times. Then a sum of suchproducts which is invariant with respect to permutations of columns representsan element from Sd(Sm(U)) which has an additional property that it is invari-ant with respect to the group SL(U) = SL(r + 1,C) which acts on U by theleft multiplication with a vector (0, . . . , r). The First Fundamental Theoremof invariant theory states that any element in Sd(Sm(U))SL(U) is obtained inthis way (see [182]). We can interpret elements of Sd(Sm(U)) as polyno-mials in coefficients of ai of a homogeneous form of degree d in r + 1 vari-ables written in the form (1.7). We write symbolically an invariant in the form(J1) (Jk) meaning that it is obtained as sum of such products with somecoefficients. If the number d is small, we can use letters, say a, b, c, . . . , in-stead of numbers 1, . . . , d. For example, (12)2(13)2(23)2 = (ab)2(bc)2(ac)2

represents an element in S3(S4(C2)).In a similar way, one considers the matrix

(1)0 . . .

(d)0 t

(1)0 . . . t

(s)0

......

......

......

(1)r . . .

(d)r t

(1)r . . . t

(s)r

.The product of k maximal minors such that each of the first d columns occursexactly k times and each of the last s columns occurs exactly p times representsa covariant of degree p and order k. For example, (ab)2axbx represents theHessian determinant

He(f) = det

(2fx21

2fx1x2

2fx2x1

2fx22

)of a ternary cubic form f .

The projective space of lines in E will be denoted by |E|. The space |E|will be denoted by P(E) (following Grothendiecks notation). We call P(E)the dual projective space of |E|. We will often denote it by |E|.

A basis 0, . . . , n in E defines an isomorphism E = Cn+1 and identi-fies |E| with the projective space Pn := |Cn+1|. For any nonzero vectorv E we denote by [v] the corresponding point in |E|. If E = Cn+1 and


v = (a0, . . . , an) Cn+1 we set [v] = [a0, . . . , an]. We call [a0, . . . , an]the projective coordinates of a point [a] Pn. Other common notation for theprojective coordinates of [a] is (a0 : a1 : . . . : an), or simply (a0, . . . , an), ifno confusion arises.

The projective space comes with the tautological invertible sheaf O|E|(1)whose space of global sections is identified with the dual space E. Its d-thtensor power is denoted by O|E|(d). Its space of global sections is identifiedwith the symmetric d-th power Sd(E).

For any f Sd(E), d > 0, we denote by V (f) the corresponding ef-fective divisor from |O|E|(d)|, considered as a closed subscheme of |E|, notnecessarily reduced. We call V (f) a hypersurface of degree d in |E| definedby equation f = 01 A hypersurface of degree 1 is a hyperplane. By definition,V (0) = |E| and V (1) = . The projective space |Sd(E)| can be viewedas the projective space of hypersurfaces in |E|. It is equal to the complete lin-ear system |O|E|(d)|. Using isomorphism (1.2), we may identify the projectivespace |Sd(E)| of hypersurfaces of degree d in |E| with the dual of the pro-jective space |SdE|. A hypersurface of degree d in |E| is classically knownas an envelope of class d.

The natural isomorphisms

(E)d = H0(|E|d,O|E|(1)d), Sd(E) = H0(|E|d,O|E|(1)d)Sd

allow one to give the following geometric interpretation of the polarizationisomorphism. Consider the diagonal embedding d : |E| |E|d. Then thetotal polarization map is the inverse of the isomorphism

d : H0(|E|d,O|E|(1)d)Sd H0(|E|,O|E|(d)).

We view a00 + + ann 6= 0 as a point a |E| with projective coordi-nates [a0, . . . , an].

Definition 1.1.2 Let X = V (f) be a hypersurface of degree d in |E| andx = [v] be a point in |E|. The hypersurface

Pak(X) := V (Dvk(f))

of degree d k is called the k-th polar hypersurface of the point a with respectto the hypersurface V (f) (or of the hypersurface with respect to the point).

1 This notation should not be confused with the notation of the closed subset in Zariski topologydefined by the ideal (f). It is equal to V (f)red.

6 Polarity

Example 1.1.3 Let d = 2, i.e.

f =

ni=0

iit2i + 2

0i


Note that

Dakbm(f) = Dak(Dbm(f)) = Dbm(a) = Dbm(Dak(f)) = Dak(f)(b).

(1.11)This gives the symmetry property of polars

b Pak(X) a Pbdk(X). (1.12)

Since we are in characteristic 0, if m d, Dam(f) cannot be zero for all a. Tosee this we use the Euler formula:

df =

ni=0

tif

ti.

Applying this formula to the partial derivatives, we obtain

d(d 1) (d k + 1)f =|i|=k

(ki

)tiif (1.13)

(also called the Euler formula). It follows from this formula that, for all k d,

a Pak(X) a X. (1.14)

This is known as the reciprocity theorem.

Example 1.1.4 Let Md be the vector space of complex square matrices ofsize d with coordinates tij . We view the determinant function det : Md Cas an element of Sd(Md ), i.e. a polynomial of degree d in the variables tij .Let Cij = dettij . For any point A = (aij) in Md the value of Cij at A is equalto the ij-th cofactor of A. Applying (1.6), for any B = (bij) Md, we obtain

DAd1B(det) = Dd1A (DB(det)) = D

d1A (

bijCij) = (d 1)!

bijCij(A).

Thus Dd1A (det) is a linear functiontijCij on Md. The linear map

Sd1(Mn)Md , A 71

(d 1)!Dd1A (det),

can be identified with the function A 7 adj(A), where adj(A) is the cofactormatrix (classically called the adjugate matrix of A, but not the adjoint matrixas it is often called in modern text-books).

1.1.2 First polars

Let us consider some special cases. LetX = V (f) be a hypersurface of degreed. Obviously, any 0-th polar of X is equal to X and, by (1.12), the d-th polar

8 Polarity

Pad(X) is empty if a 6 X . and equals Pn if a X . Now take k = 1, d 1.By using (1.6), we obtain

Da(f) =

ni=0

aif

ti,

1

(d 1)!Dad1(f) =

ni=0

f

ti(a)ti.

Together with (1.12) this implies the following.

Theorem 1.1.5 For any smooth point x X , we have

Pxd1(X) = Tx(X).

If x is a singular point of X , Pxd1(X) = Pn. Moreover, for any a Pn,

X Pa(X) = {x X : a Tx(X)}.

Here and later on we denote by Tx(X) the embedded tangent space of aprojective subvariety X Pn at its point x. It is a linear subspace of Pn equalto the projective closure of the affine Zariski tangent space Tx(X) of X at x(see [278], p. 181).

In classical terminology, the intersection X Pa(X) is called the apparentboundary of X from the point a. If one projects X to Pn1 from the point a,then the apparent boundary is the ramification divisor of the projection map.

The following picture makes an attempt to show what happens in the casewhen X is a conic.

a

Pa(X)

X

Figure 1.1 Polar line of a conic

The set of first polars Pa(X) defines a linear system contained in the com-plete linear system

OPn(d1). The dimension of this linear system n. Wewill be freely using the language of linear systems and divisors on algebraicvarieties (see [282]).


Proposition 1.1.6 The dimension of the linear system of first polars r ifand only if, after a linear change of variables, the polynomial f becomes apolynomial in r + 1 variables.

Proof LetX = V (f). It is obvious that the dimension of the linear system offirst polars r if and only if the linear map E Sd1(E), v 7 Dv(f) haskernel of dimension n r. Choosing an appropriate basis, we may assumethat the kernel is generated by vectors (1, 0, . . . , 0), etc. Now, it is obvious thatf does not depend on the variables t0, . . . , tnr1.

It follows from Theorem 1.1.5 that the first polar Pa(X) of a point a withrespect to a hypersurface X passes through all singular points of X . One cansay more.

Proposition 1.1.7 Let a be a singular point of X of multiplicity m. For eachr degX m, Par (X) has a singular point at a of multiplicity m and thetangent cone of Par (X) at a coincides with the tangent cone TCa(X) of X ata. For any point b 6= a, the r-th polar Pbr (X) has multiplicity m r at aand its tangent cone at a is equal to the r-th polar of TCa(X) with respect tob.

Proof Let us prove the first assertion. Without loss of generality, we mayassume that a = [1, 0, . . . , 0]. Then X = V (f), where

f = tdm0 fm(t1, . . . , tn) + tdm10 fm+1(t1, . . . , tn) + + fd(t1, . . . , tn).

(1.15)The equation fm(t1, . . . , tn) = 0 defines the tangent cone of X at b. Theequation of Par (X) is

rf

tr0= r!

dmri=0

(dmi

r

)tdmri0 fm+i(t1, . . . , tn) = 0.

It is clear that [1, 0, . . . , 0] is a singular point of Par (X) of multiplicity m withthe tangent cone V (fm(t1, . . . , tn)).

Now we prove the second assertion. Without loss of generality, we mayassume that a = [1, 0, . . . , 0] and b = [0, 1, 0, . . . , 0]. Then the equation ofPbr (X) is

rf

tr1= tdm0

rfmtr1

+ + rfdtr1

= 0.

The point a is a singular point of multiplicity m r. The tangent cone ofPbr (X) at the point a is equal to V (

rfmtr1

) and this coincides with the r-thpolar of TCa(X) = V (fm) with respect to b.

10 Polarity

We leave it to the reader to see what happens if r > dm.Keeping the notation from the previous proposition, consider a line ` through

the point a such that it intersectsX at some point x 6= awith multiplicity largerthan one. The closure ECa(X) of the union of such lines is called the envelop-ing cone of X at the point a. If X is not a cone with vertex at a, the branchdivisor of the projection p : X \ {a} Pn1 from a is equal to the projectionof the enveloping cone. Let us find the equation of the enveloping cone.

As above, we assume that a = [1, 0, . . . , 0]. LetH be the hyperplane t0 = 0.Write ` in a parametric form ua + vx for some x H . Plugging in Equation(1.15), we get

P (t) = tdmfm(x1, . . . , xn)+tdm1fm+1(x1, . . . , xm)+ +fd(x1, . . . , xn) = 0,

where t = u/v.We assume that X 6= TCa(X), i.e. X is not a cone with vertex at a (oth-

erwise, by definition, ECa(X) = TCa(X)). The image of the tangent coneunder the projection p : X \ {a} H = Pn1 is a proper closed subset ofH . If fm(x1, . . . , xn) 6= 0, then a multiple root of P (t) defines a line in theenveloping cone. Let Dk(A0, . . . , Ak) be the discriminant of a general poly-nomial P = A0T k + +Ak of degree k. Recall that

A0Dk(A0, . . . , Ak) = (1)k(k1)/2Res(P, P )(A0, . . . , Ak),

where Res(P, P ) is the resultant of P and its derivative P . It follows fromthe known determinant expression of the resultant that

Dk(0, A1, . . . , Ak) = (1)k2k+2

2 A20Dk1(A1, . . . , Ak).

The equation P (t) = 0 has a multiple zero with t 6= 0 if and only if

Ddm(fm(x), . . . , fd(x)) = 0.

So, we see that

ECa(X) V (Ddm(fm(x), . . . , fd(x))), (1.16)ECa(X) TCa(X) V (Ddm1(fm+1(x), . . . , fd(x))).

It follows from the computation of rftr0

in the proof of the previous Propositionthat the hypersurface V (Ddm(fm(x), . . . , fd(x))) is equal to the projectionof Pa(X) X to H .

Suppose V (Ddm1(fm+1(x), . . . , fd(x))) and TCa(X) do not share anirreducible component. Then

V (Ddm(fm(x), . . . , fd(x))) \ TCa(X) V (Ddm(fm(x), . . . , fd(x)))


= V (Ddm(fm(x), . . . , fd(x))) \ V (Ddm1(fm+1(x), . . . , fd(x))) ECa(X),

gives the opposite inclusion of (1.16), and we get

ECa(X) = V (Ddm(fm(x), . . . , fd(x))). (1.17)

Note that the discriminant Ddm(A0, . . . , Ak) is an invariant of the groupSL(2) in its natural representation on degree k binary forms. Taking the diago-nal subtorus, we immediately infer that any monomial Ai00 A

ikk entering in

the discriminant polynomial satisfies

k

ks=0

is = 2

ks=0

sis.

It is also known that the discriminant is a homogeneous polynomial of degree2k 2 . Thus, we get

k(k 1) =ks=0

sis.

In our case k = dm, we obtain that

deg V (Ddm(fm(x), . . . , fd(x))) =dms=0

(m+ s)is

= m(2d 2m 2) + (dm)(dm 1) = (d+m)(dm 1).

This is the expected degree of the enveloping cone.

Example 1.1.8 Assume m = d 2, then

D2(fd2(x), fd1(x), fd(x)) = fd1(x)2 4fd2(x)fd(x),

D2(0, fd1(x), fd(x)) = fd2(x) = 0.

Suppose fd2(x) and fd1 are coprime. Then our assumption is satisfied, andwe obtain

ECa(X) = V (fd1(x)2 4fd2(x)fd(x)).

Observe that the hypersurfaces V (fd2(x)) and V (fd(x)) are everywhere tan-gent to the enveloping cone. In particular, the quadric tangent cone TCa(X) iseverywhere tangent to the enveloping cone along the intersection of V (fd2(x))with V (fd1(x)).

12 Polarity

For any nonsingular quadric Q, the map x 7 Px(Q) defines a projectiveisomorphism from the projective space to the dual projective space. This is aspecial case of a correlation.

According to classical terminology, a projective automorphism of Pn iscalled a collineation. An isomorphism from |E| to its dual space P(E) is calleda correlation. A correlation c : |E| P(E) is given by an invertible linear map : E E defined uniquely up to proportionality. A correlation transformspoints in |E| to hyperplanes in |E|. A point x |E| is called conjugate to apoint y |E| with respect to the correlation c if y c(x). The transpose of theinverse map t1 : E E transforms hyperplanes in |E| to points in |E|. Itcan be considered as a correlation between the dual spaces P(E) and |E|. It isdenoted by c and is called the dual correlation. It is clear that (c) = c. IfH is a hyperplane in |E| and x is a point in H , then point y |E| conjugateto x under c belongs to any hyperplane H in |E| conjugate to H under c.

A correlation can be considered as a line in (E E) spanned by a nonde-generate bilinear form, or, in other words as a nonsingular correspondence oftype (1, 1) in |E| |E|. The dual correlation is the image of the divisor underthe switch of the factors. A pair (x, y) |E| |E| of conjugate points is justa point on this divisor.

We can define the composition of correlations c c. Collineations andcorrelations form a group PGL(E) isomorphic to the group of outer auto-morphisms of PGL(E). The subgroup of collineations is of index 2.

A correlation c of order 2 in the group PGL(E) is called a polarity. Inlinear representative, this means that t = for some nonzero scalar . Aftertransposing, we obtain = 1. The case = 1 corresponds to the (quadric)polarity with respect to a nonsingular quadric in |E|which we discussed in thissection. The case = 1 corresponds to a null-system (or null polarity) whichwe will discuss in Chapters 2 and 10. In terms of bilinear forms, a correlationis a quadric polarity (resp. null polarity) if it can be represented by a symmetric(skew-symmetric) bilinear form.

Theorem 1.1.9 Any projective automorphism is equal to the product of twoquadric polarities.

Proof Choose a basis in E to represent the automorphism by a Jordan matrix


J . Let Jk() be its block of size k with at the diagonal. Let

Bk =

0 0 . . . 0 1

0 0 . . . 1 0...

......

......

0 1 . . . 0 0

1 0 . . . 0 0

.

Then

Ck() = BkJk() =

0 0 . . . 0

0 0 . . . 1...

......

......

0 . . . 0 0

1 . . . 0 0

.

Observe that the matrices B1k and Ck() are symmetric. Thus each Jordanblock of J can be written as the product of symmetric matrices, hence J is theproduct of two symmetric matrices. It follows from the definition of composi-tion in the group PGL(E) that the product of the matrices representing thebilinear forms associated to correlations coincides with the matrix representingthe projective transformation equal to the composition of the correlations.

1.1.3 Polar quadrics

A (d 2)-polar of X = V (f) is a quadric, called the polar quadric of X withrespect to a = [a0, . . . , an]. It is defined by the quadratic form

q = Dad2(f) =|i|=d2

(d2i

)aiif.

Using Equation (1.9), we obtain

q =|i|=2

(2

i

)tiif(a).

By (1.14), each a X belongs to the polar quadric Pad2(X). Also, byTheorem 1.1.5,

Ta(Pad2(X)) = Pa(Pad2(X)) = Pad1(X) = Ta(X). (1.18)

This shows that the polar quadric is tangent to the hypersurface at the point a.Consider the line ` = ab through two points a, b. Let : P1 Pn be

14 Polarity

its parametric equation, i.e. a closed embedding with the image equal to `. Itfollows from (1.8) and (1.9) that

i(X, ab)a s+ 1 b Padk(X), k s. (1.19)

For s = 0, the condition means that a X . For s = 1, by Theorem 1.1.5,this condition implies that b, and hence `, belongs to the tangent plane Ta(X).For s = 2, this condition implies that b Pad2(X). Since ` is tangent to Xat a, and Pad2(X) is tangent to X at a, this is equivalent to that ` belongs toPad2(X).

It follows from (1.19) that a is a singular point of X of multiplicity s+ 1if and only if Padk(X) = Pn for k s. In particular, the quadric polarPad2(X) = Pn if and only if a is a singular point of X of multiplicity 3.

Definition 1.1.10 A line is called an inflection tangent to X at a point a if

i(X, `)a > 2.

Proposition 1.1.11 Let ` be a line through a point a. Then ` is an inflectiontangent toX at a if and only if it is contained in the intersection of Ta(X) withthe polar quadric Pad2(X).

Note that the intersection of an irreducible quadric hypersurface Q = V (q)with its tangent hyperplane H at a point a Q is a cone in H over the quadricQ in the image H of H in |E/[a]|.

Corollary 1.1.12 Assume n 3. For any a X , there exists an inflectiontangent line. The union of the inflection tangents containing the point a is thecone Ta(X) Pad2(X) in Ta(X).

Example 1.1.13 Assume a is a singular point of X . By Theorem 1.1.5, thisis equivalent to that Pad1(X) = Pn. By (1.18), the polar quadric Q is alsosingular at a and therefore it must be a cone over its image under the projectionfrom a. The union of inflection tangents is equal to Q.

Example 1.1.14 Assume a is a nonsingular point of an irreducible surface Xin P3. A tangent hyperplane Ta(X) cuts out in X a curve C with a singularpoint a. If a is an ordinary double point of C, there are two inflection tangentscorresponding to the two branches of C at a. The polar quadric Q is nonsingu-lar at a. The tangent cone of C at the point a is a cone over a quadric Q in P1.If Q consists of two points, there are two inflection tangents corresponding tothe two branches of C at a. If Q consists of one point (corresponding to non-reduced hypersurface in P1), then we have one branch. The latter case happensonly if Q is singular at some point b 6= a.


1.1.4 The Hessian hypersurface

LetQ(a) be the polar quadric ofX = V (f) with respect to some point a Pn.The symmetric matrix defining the corresponding quadratic form is equal tothe Hessian matrix of second partial derivatives of f

He(f) =( 2ftitj

)i,j=0,...,n

,

evaluated at the point a. The quadric Q(a) is singular if and only if the deter-minant of the matrix is equal to zero (the locus of singular points is equal tothe projectivization of the null-space of the matrix). The hypersurface

He(X) = V (det He(f))

describes the set of points a Pn such that the polar quadric Pad2(X) issingular. It is called the Hessian hypersurface of X . Its degree is equal to (d2)(n+ 1) unless it coincides with Pn.

Proposition 1.1.15 The following is equivalent:

(i) He(X) = Pn;(ii) there exists a nonzero polynomial g(z0, . . . , zn) such that

g(0f, . . . , nf) 0.

Proof This is a special case of a more general result about the Jacobian de-terminant (also known as the functional determinant) of n + 1 polynomialfunctions f0, . . . , fn defined by

J(f0, . . . , fn) = det((fitj

)).

Suppose J(f0, . . . , fn) 0. Then the map f : Cn+1 Cn+1 defined by thefunctions f0, . . . , fn is degenerate at each point (i.e. dfx is of rank < n + 1at each point x). Thus the closure of the image is a proper closed subset ofCn+1. Hence there is an irreducible polynomial that vanishes identically onthe image.

Conversely, assume that g(f0, . . . , fn) 0 for some polynomial g whichwe may assume to be irreducible. Then

g

ti=

nj=0

g

zj(f0, . . . , fn)

fjti

= 0, i = 0, . . . , n.

Since g is irreducible, its set of zeros is nonsingular on a Zariski open set U .

16 Polarity

Thus the vector( gz0

(f0(x), . . . , fn(x)), . . . ,g

zn(f0(x), . . . , fn(x)

)is a nontrivial solution of the system of linear equations with matrix (fitj (x)),where x U . Therefore, the determinant of this matrix must be equal to zero.This implies that J(f0, . . . , fn) = 0 on U , hence it is identically zero.

Remark 1.1.16 It was claimed by O. Hesse that the vanishing of the Hessianimplies that the partial derivatives are linearly dependent. Unfortunately, hisattempted proof was wrong. The first counterexample was given by P. Gordanand M. Noether in [253]. Consider the polynomial

f = t2t20 + t3t

21 + t4t0t1.

Note that the partial derivatives

f

t2= t20,

f

t3= t21,

f

t4= t0t1

are algebraically dependent. This implies that the Hessian is identically equalto zero. We have

f

t0= 2t0t2 + t4t1,

f

t1= 2t1t3 + t4t0.

Suppose that a linear combination of the partials is equal to zero. Then

c0t20 + c1t

21 + c2t0t1 + c3(2t0t2 + t4t1) + c4(2t1t3 + t4t0) = 0.

Collecting the terms in which t2, t3, t4 enter, we get

2c3t0 = 0, 2c4t1 = 0, c3t1 + c4t0 = 0.

This gives c3 = c4 = 0. Since the polynomials t20, t21, t0t1 are linearly inde-

pendent, we also get c0 = c1 = c2 = 0.The known cases when the assertion of Hesse is true are d = 2 (any n) and

n 3 (any d) (see [253], [370], [102]).

Recall that the set of singular quadrics in Pn is the discriminant hypersur-face D2(n) in Pn(n+3)/2 defined by the equation

det

t00 t01 . . . t0nt01 t11 . . . t1n...

......

...t0n t1n . . . tnn

= 0.By differentiating, we easily find that its singular points are defined by the


determinants of nn minors of the matrix. This shows that the singular locusof D2(n) parameterizes quadrics defined by quadratic forms of rank n 1(or corank 2). Abusing the terminology, we say that a quadric Q is of rankk if the corresponding quadratic form is of this rank. Note that

dim Sing(Q) = corank Q 1.

Assume that He(f) 6= 0. Consider the rational map p : |E| |S2(E)|defined by a 7 Pad2(X). Note that Pad2(f) = 0 implies Pad1(f) = 0and hence

ni=0 biif(a) = 0 for all b. This shows that a is a singular point

of X . Thus p is defined everywhere except maybe at singular points of X . Sothe map p is regular if X is nonsingular, and the preimage of the discriminanthypersurface is equal to the Hessian of X . The preimage of the singular locusSing(D2(n)) is the subset of points a He(f) such that Sing(Pad2(X)) is ofpositive dimension.

Here is another description of the Hessian hypersurface.

Proposition 1.1.17 The Hessian hypersurface He(X) is the locus of singularpoints of the first polars of X .

Proof Let a He(X) and let b Sing(Pad2(X)). Then

Db(Dad2(f)) = Dad2(Db(f)) = 0.

Since Db(f) is of degree d 1, this means that Ta(Pb(X)) = Pn, i.e., a is asingular point of Pb(X).

Conversely, if a Sing(Pb(X)), then Dad2(Db(f)) = Db(Dad2(f)) =0. This means that b is a singular point of the polar quadric with respect to a.Hence a He(X).

Let us find the affine equation of the Hessian hypersurface. Applying theEuler formula (1.13), we can write

t0f0i = (d 1)if t1f1i tnfni,

t00f = df t11f tnnf,

where fij denote the second partial derivative. Multiplying the first row ofthe Hessian determinant by t0 and adding to it the linear combination of theremaining rows taken with the coefficients ti, we get the following equality:

det(He(f)) =d 1t0

det

0f 1f . . . nf

f10 f11 . . . f1n...

......

...fn0 fn1 . . . fnn

.

18 Polarity

Repeating the same procedure but this time with the columns, we finally get

det(He(f)) =(d 1)2

t20det

dd1f 1f . . . nf

1f f11 . . . f1n...

......

...nf fn1 . . . fnn

. (1.20)Let (z1, . . . , zn) be the dehomogenization of f with respect to t0, i.e.,

f(t0, . . . , td) = td0(

t1t0, . . . ,

tnt0

).

We have

f

ti= td10 i(z1, . . . , zn),

2f

titj= td20 ij(z1, . . . , zn), i, j = 1, . . . , n,

where

i =

zi, ij =

2

zizj.

Plugging these expressions in (1.20), we obtain, that up to a nonzero constantfactor,

t(n+1)(d2)0 det(He()) = det

dd1(z) 1(z) . . . n(z)

1(z) 11(z) . . . 1n(z)...

......

...n(z) n1(z) . . . nn(z)

,(1.21)

where z = (z1, . . . , zn), zi = ti/t0, i = 1, . . . , n.

Remark 1.1.18 If f(x, y) is a real polynomial in three variables, the value of(1.21) at a point v Rn with [v] V (f) multiplied by 1f1(a)2+f2(a)2+f3(a)2 isequal to the Gauss curvature of X(R) at the point a (see [221]).

1.1.5 Parabolic points

Let us see where He(X) intersectsX . We assume that He(X) is a hypersurfaceof degree (n + 1)(d 2) > 0. A glance at the expression (1.21) reveals thefollowing fact.

Proposition 1.1.19 Each singular point of X belongs to He(X).


Let us see now when a nonsingular point a X lies in its Hessian hyper-surface He(X).

By Corollary 1.1.12, the inflection tangents in Ta(X) sweep the intersectionof Ta(X) with the polar quadric Pad2(X). If a He(X), then the polarquadric is singular at some point b.

If n = 2, a singular quadric is the union of two lines, so this means that oneof the lines is an inflection tangent. A point a of a plane curve X such thatthere exists an inflection tangent at a is called an inflection point of X .

If n > 2, the inflection tangent lines at a point a XHe(X) sweep a coneover a singular quadric in Pn2 (or the whole Pn2 if the point is singular).Such a point is called a parabolic point ofX . The closure of the set of parabolicpoints is the parabolic hypersurface in X (it could be the whole X).

Theorem 1.1.20 Let X be a hypersurface of degree d > 2 in Pn. If n = 2,then He(X) X consists of inflection points of X . In particular, each nonsin-gular curve of degree 3 has an inflection point, and the number of inflectionspoints is either infinite or less than or equal to 3d(d 2). If n > 2, then theset X He(X) consists of parabolic points. The parabolic hypersurface in Xis either the whole X or a subvariety of degree (n+ 1)d(d 2) in Pn.

Example 1.1.21 Let X be a surface of degree d in P3. If a is a parabolicpoint of X , then Ta(X) X is a singular curve whose singularity at a is ofmultiplicity higher than 3 or it has only one branch. In fact, otherwise X hasat least two distinct inflection tangent lines which cannot sweep a cone over asingular quadric in P1. The converse is also true. For example, a nonsingularquadric has no parabolic points, and all nonsingular points of a singular quadricare parabolic.

A generalization of a quadratic cone is a developable surface. It is a specialkind of a ruled surface which characterized by the condition that the tangentplane does not change along a ruling. We will discuss these surfaces later inChapter 10. The Hessian surface of a developable surface contains this surface.The residual surface of degree 2d 8 is called the pro-Hessian surface. Anexample of a developable surface is the quartic surface

(t0t3t1t2)24(t21t0t2)(t22t1t3) = 6t0t1t2t3+4t31t3+4t0t32+t20t233t21t22 = 0.

It is the surface swept out by the tangent lines of a rational normal curve ofdegree 3. It is also the discriminant surface of a binary cubic, i.e. the surfaceparameterizing binary cubics a0u3 + 3a1u2v+ 3a2uv2 +a3v3 with a multipleroot. The pro-Hessian of any quartic developable surface is the surface itself[84].

20 Polarity

Assume now that X is a curve. Let us see when it has infinitely many in-flection points. Certainly, this happens when X contains a line component;each of its points is an inflection point. It must be also an irreducible compo-nent of He(X). The set of inflection points is a closed subset of X . So, if Xhas infinitely many inflection points, it must have an irreducible componentconsisting of inflection points. Each such component is contained in He(X).Conversely, each common irreducible component of X and He(X) consists ofinflection points.

We will prove the converse in a little more general form taking care of notnecessarily reduced curves.

Proposition 1.1.22 A polynomial f(t0, t1, t2) divides its Hessian polynomialHe(f) if and only if each of its multiple factors is a linear polynomial.

Proof Since each point on a non-reduced component ofXred V (f) is a sin-gular point (i.e. all the first partials vanish), and each point on a line componentis an inflection point, we see that the condition is sufficient for X He(f).Suppose this happens and let R be a reduced irreducible component of thecurve X which is contained in the Hessian. Take a nonsingular point of R andconsider an affine equation of R with coordinates (x, y). We may assume thatOR,x is included in OR,x = C[[t]] such that x = t, y = tr, where (0) = 1.Thus the equation of R looks like

f(x, y) = y xr + g(x, y), (1.22)

where g(x, y) does not contain terms cy, c C. It is easy to see that (0, 0) isan inflection point if and only if r > 2 with the inflection tangent y = 0.

We use the affine equation of the Hessian (1.21), and obtain that the imageof

h(x, y) = det

dd1f f1 f2f1 f11 f12f2 f21 f22

in C[[t]] is equal to

det

0 rtr1 + g1 1 + g2rtr1 + g1 r(r 1)tr2 + g11 g121 + g2 g12 g22

.Since every monomial entering in g is divisible by y2, xy or xi, i > r, we

see that gy is divisible by t andgx is divisible by t

r1. Also g11 is divisible


by tr1. This shows that

h(x, y) = det

0 atr1 + 1 + atr1 + r(r 1)tr2 + g121 + g12 g22

,where denotes terms of higher degree in t. We compute the determinantand see that it is equal to r(r 1)tr2 + . This means that its image inC[[t]] is not equal to zero, unless the equation of the curve is equal to y = 0,i.e. the curve is a line.

In fact, we have proved more. We say that a nonsingular point of X is an in-flection point of order r2 and denote the order by ordflxX if one can choosean equation of the curve as in (1.22) with r 3. It follows from the previousproof that r 2 is equal to the multiplicity i(X,He)x of the intersection of thecurve and its Hessian at the point x. It is clear that ordflxX = i(`,X)x 2,where ` is the inflection tangent line of X at x. If X is nonsingular, we have

xXi(X,He)x =

xX

ordflxX = 3d(d 2). (1.23)

1.1.6 The Steinerian hypersurface

Recall that the Hessian hypersurface of a hypersurface X = V (f) is the locusof points a such that the polar quadric Pad2(X) is singular. The Steinerianhypersurface St(X) of X is the locus of singular points of the polar quadrics.Thus

St(X) =

aHe(X)

Sing(Pad2(X)). (1.24)

The proof of Proposition 1.1.17 shows that it can be equivalently defined as

St(X) = {a Pn : Pa(X) is singular}. (1.25)

We also have

He(X) =

aSt(X)

Sing(Pa(X)). (1.26)

A point b = [b0, . . . , bn] St(X) satisfies the equation

He(f)(a)

b0...bn

= 0, (1.27)

22 Polarity

where a He(X). This equation defines a subvariety

HS(X) Pn Pn (1.28)

given by n+ 1 equations of bidegree (d 2, 1). When the Steinerian map (seebelow) is defined, it is just its graph. The projection to the second factor is aclosed subscheme of Pn with support at St(X). This gives a scheme-theoreticaldefinition of the Steinerian hypersurface which we will accept from now on. Italso makes clear why St(X) is a hypersurface, not obvious from the definition.The expected dimension of the image of the second projection is n 1.

The following argument confirms our expectation. It is known (see, for ex-ample, [239]) that the locus of singular hypersurfaces of degree d in |E| is ahypersurface

Dd(n) |Sd(E)|

of degree (n + 1)(d 1)n defined by the discriminant of a general degree dhomogeneous polynomial in n + 1 variables (the discriminant hypersurface).Let L be the projective subspace of |Sd1(E)| that consists of first polars ofX . Assume that no polar Pa(X) is equal to Pn. Then

St(X) = L Dn(d 1).

So, unless L is contained in Dn(d 1), we get a hypersurface. Moreover, weobtain

deg(St(X)) = (n+ 1)(d 2)n. (1.29)

Assume that the quadric Pad2(X) is of corank 1. Then it has a uniquesingular point b with the coordinates [b0, . . . , bn] proportional to any columnor a row of the adjugate matrix adj(He(f)) evaluated at the point a. Thus,St(X) coincides with the image of the Hessian hypersurface under the rationalmap

st : He(X) 99K St(X), a 7 Sing(Pad2(X)),

given by polynomials of degree n(d 2). We call it the Steinerian map. Ofcourse, it is not defined when all polar quadrics are of corank > 1. Also, ifthe first polar hypersurface Pa(X) has an isolated singular point for a generalpoint a, we get a rational map

st1 : St(X) 99K He(X), a 7 Sing(Pa(X)).

These maps are obviously inverse to each other. It is a difficult question todetermine the sets of indeterminacy points for both maps.


Proposition 1.1.23 Let X be a reduced hypersurface. The Steinerian hyper-surface of X coincides with Pn if X has a singular point of multiplicity 3.The converse is true if we additionally assume that X has only isolated singu-lar points.

Proof Assume that X has a point of multiplicity 3. We may harmlesslyassume that the point is p = [1, 0, . . . , 0]. Write the equation of X in the form

f = tk0gdk(t1, . . . , tn)+ tk10 gdk+1(t1, . . . , tn)+ +gd(t1, . . . , tn) = 0,

(1.30)where the subscript indicates the degree of the polynomial. Since the multi-plicity of p is greater than or equal to 3, we must have d k 3. Then a firstpolar Pa(X) has the equation

a0

ki=0

(k i)tk1i0 gdk+i +ns=1

as

ki=0

tki0gdk+its

= 0. (1.31)

It is clear that the point p is a singular point of Pa(X) of multiplicity d k 1 2.

Conversely, assume that all polars are singular. By Bertinis Theorem (see[278], Theorem 17.16), the singular locus of a general polar is contained inthe base locus of the linear system of polars. The latter is equal to the singularlocus of X . By assumption, it consists of isolated points, hence we can finda singular point of X at which a general polar has a singular point. We mayassume that the singular point is p = [1, 0, . . . , 0] and (1.30) is the equation ofX . Then the first polar Pa(X) is given by Equation (1.31). The largest power oft0 in this expression is at most k. The degree of the equation is d 1. Thus thepoint p is a singular point of Pa(X) if and only if k d 3, or, equivalently,if p is at least triple point of X .

Example 1.1.24 The assumption on the singular locus is essential. First, it iseasy to check that X = V (f2), where V (f) is a nonsingular hypersurface hasno points of multiplicity 3 and its Steinerian coincides with Pn. An exampleof a reduced hypersurface X with the same property is a surface of degree 6 inP3 given by the equation

(

3i=0

t3i )2 + (

3i=0

t2i )3 = 0.

Its singular locus is the curve V (3i=0 t

3i ) V (

3i=0 t

2i ). Each of its points is

24 Polarity

a double point on X . Easy calculation shows that

Pa(X) = V((

3i=0

t3i )

3i=0

ait2i + (

3i=0

t2i )2

3i=0

aiti).

and

V (

3i=0

t3i ) V (3i=0

t2i ) V (3i=0

ait2i ) Sing(Pa(X)).

By Proposition 1.1.7, Sing(X) is contained in St(X). Since the same is truefor He(X), we obtain the following.

Proposition 1.1.25 The intersection He(X) St(X) contains the singularlocus of X .

One can assign one more variety to a hypersurface X = V (f). This is theCayleyan variety. It is defined as the image Cay(X) of the rational map

HS(X) 99K G1(Pn), (a, b) 7 ab,

where Gr(Pn) denotes the Grassmannian of r-dimensional subspaces in Pn.In the sequel we will also use the notation G(r + 1, E) = Gr(|E|) for thevariety of linear r + 1-dimensional subspaces of a linear space E. The mapis not defined at the intersection of the diagonal with HS(X). We know thatHS(a, a) = 0 means that Pad1(X) = 0, and the latter means that a is a singu-lar point of X . Thus the map is a regular map for a nonsingular hypersurfaceX .

Note that in the case n = 2, the Cayleyan variety is a plane curve in the dualplane, the Cayleyan curve of X .

Proposition 1.1.26 Let X be a general hypersurface of degree d 3. Then

deg Cay(X) =

{ni=1(d 2)i

(n+1i

)(n1i1)

if d > 3,12

ni=1

(n+1i

)(n1i1)

if d = 3,

where the degree is considered with respect to the Plucker embedding of theGrassmannian G1(Pn).

Proof Since St(X) 6= Pn, the correspondence HS(X) is a complete inter-section of n + 1 hypersurfaces in Pn Pn of bidegree (d 2, 1). Sincea Sing(Pa(X)) implies that a Sing(X), the intersection of HS(X) withthe diagonal is empty. Consider the regular map

r : HS(X) G1(Pn), (a, b) 7 ab. (1.32)

It is given by the linear system of divisors of type (1, 1) on Pn Pn restricted


to HS(X). The genericity assumption implies that this map is of degree 1 ontothe image if d > 3 and of degree 2 if d = 3 (in this case the map factorsthrough the involution of Pn Pn that switches the factors).

It is known that the set of lines intersecting a codimension 2 linear sub-space is a hyperplane section of the Grassmannian G1(Pn) in its Pluckerembedding. Write Pn = |E| and = |L|. Let = v1 . . . vn1 for somebasis (v1, . . . , vn1) of L. The locus of pairs of points (a, b) = ([w1], [w2]) inPnPn such that the line ab intersects is given by the equationw1w2 =0. This is a hypersurface of bidegree (1, 1) in PnPn. This shows that the map(1.32) is given by a linear system of divisors of type (1, 1). Its degree (or twiceof the degree) is equal to the intersection ((d 2)h1 + h2)n+1(h1 + h2)n1,where h1, h2 are the natural generators of H2(Pn Pn,Z). We have

((d 2)h1 + h2)n+1(h1 + h2)n1 =

(n+1i=0

(n+1i

)(d 2)ihi1hn+1i2

)(n1j=0

(n1j

)hn1j1 h

j2

)

=

ni=1

(d 2)i(n+1i

)(n1i1).

For example, if n = 2, d > 3, we obtain a classical result

deg Cay(X) = 3(d 2) + 3(d 2)2 = 3(d 2)(d 1),

and deg Cay(X) = 3 if d = 3.

Remark 1.1.27 The homogeneous forms defining the Hessian and Steinerianhypersurfaces of V (f) are examples of covariants of f . We already discussedthem in the case n = 1. The form defining the Cayleyan of a plane curve is anexample of a contravariant of f .

1.1.7 The Jacobian hypersurface

In the previous sections we discussed some natural varieties attached to the lin-ear system of first polars of a hypersurface. We can extend these constructionsto arbitrary n-dimensional linear systems of hypersurfaces in Pn = |E|. Weassume that the linear system has no fixed components, i.e. its general memberis an irreducible hypersurface of some degree d. Let L Sd(E) be a linearsubspace of dimension n + 1 and |L| be the corresponding linear system ofhypersurfaces of degree d. Note that, in the case of linear system of polars of a

26 Polarity

hypersurfaceX of degree d+1, the linear subspace L can be canonically iden-tified withE and the inclusion |E| |Sd(E)| corresponds to the polarizationmap a 7 Pa(X).

Let Dd(n) |Sd(E)| be the discriminant hypersurface. The intersection

D(|L|) = |L| Dd(n)

is called the discriminant hypersurface of |L|. We assume that it is not equalto Pn, i.e. not all members of |L| are singular. Let

D(|L|) = {(x,D) Pn |L| : x Sing(D)}

with two projections p : D D(|L|) and q : D |L|. We define the Jacobianhypersurface of |L| as

Jac(|L|) = q(D(|L|)).

It parameterizes singular points of singular members of |L|. Again, it maycoincide with the whole Pn. In the case of polar linear systems, the discrim-inant hypersurface is equal to the Steinerian hypersurface, and the Jacobianhypersurface is equal to the Hessian hypersurface.

The Steinerian hypersurface St(|L|) is defined as the locus of points x Pnsuch that there exists a Pn such that x D|L|Pan1(D). Since dimL =n+1, the intersection is empty, unless there existsD such that Pan1(D) = 0.Thus Pan(D) = 0 and a Sing(D), hence a Jac(|L|) and D D(|L|).Conversely, if a Jac(|L|), then D|L|Pan1(D) 6= and it is contained inSt(|L|). By duality (1.12),

x

D|L|

Pan1(D) a

D|L|

Px(D).

Thus the Jacobian hypersurface is equal to the locus of points which belong tothe intersection of the first polars of divisors in |L| with respect to some pointx St(X). Let

HS(|L|) = {(a, b) He(|L|) St(|L|) : a

D|L|

Pb(D)}

= {(a, b) He(|L|) St(|L|) : b

D|L|

Pad1(D)}.

It is clear that HS(|L|) PnPn is a complete intersection of n+ 1 divisorsof type (d 1, 1). In particular,

HS(|L|) = pr1(OPn((d 2)(n+ 1))). (1.33)

One expects that, for a general point x St(|L|), there exists a unique a


Jac(|L|) and a unique D D(|L|) as above. In this case, the correspondenceHS(|L|) defines a birational isomorphism between the Jacobian and Steinerianhypersurface. Also, it is clear that He(|L|) = St(|L|) if d = 2.

Assume that |L| has no base points. Then HS(|L|) does not intersect thediagonal of Pn Pn. This defines a map

HS(|L|) G1(Pn), (a, b) 7 ab.

Its image Cay(|L|) is called the Cayleyan variety of |L|.A line ` Cay(|L|) is called a Reye line of |L|. It follows from the defini-

tions that a Reye line is characterized by the property that it contains a pointsuch that there is a hyperplane in |L| of hypersurfaces tangent to ` at this point.For example, if d = 2 this is equivalent to the property that ` is contained is alinear subsystem of |L| of codimension 2 (instead of expected codimension 3).

The proof of Proposition 1.1.26 applies to our more general situation togive the degree of Cay(|L|) for a general n-dimensional linear system |L| ofhypersurfaces of degree d.

deg Cay(|L|) =

{ni=1(d 1)i

(n+1i

)(n1i1)

if d > 2,12

ni=1

(n+1i

)(n1i1)

if d = 2.(1.34)

Let f = (f0, . . . , fn) be a basis of L. Choose coordinates in Pn to iden-tify Sd(E) with the polynomial ring C[t0, . . . , tn]. A well-known fact fromthe complex analysis asserts that Jac(|L|) is given by the determinant of theJacobian matrix

J(f) =

0f0 1f0 . . . nf00f1 1f1 . . . nf1

......

......

0fn 1fn . . . nfn

.In particular, we expect that

deg Jac(|L|) = (n+ 1)(d 1).

If a Jac(|L|), then a nonzero vector in the null-space of J(f) defines a pointx such that Px(f0)(a) = . . . = Px(fn)(a) = 0. Equivalently,

Pan1(f0)(x) = . . . = Pan1(fn)(x) = 0.

This shows that St(|L|) is equal to the projectivization of the union of the null-spaces Null(Jac(f(a))), a Cn+1. Also, a nonzero vector in the null space ofthe transpose matrix tJ(f) defines a hypersurface in D(|L|) with singularity atthe point a.

28 Polarity

Let Jac(|L|)0 be the open subset of points where the corank of the jacobianmatrix is equal to 1. We assume that it is a dense subset of Jac(|L|). Then,taking the right and the left kernels of the Jacobian matrix, defines two maps

Jac(|L|)0 D(|L|), Jac(|L|)0 St(|L|).

Explicitly, the maps are defined by the nonzero rows (resp. columns) of theadjugate matrix adj(He(f)).

Let |L| : Pn 99K |L| be the rational map defined by the linear system |L|.Under some assumptions of generality which we do not want to spell out, onecan identify Jac(|L|) with the ramification divisor of the map and D(|L|) withthe dual hypersurface of the branch divisor.

Let us now define a new variety attached to a n-dimensional linear systemin Pn. Consider the inclusion map L Sd(E) and let

L Sd(E), f 7 f ,

be the restriction of the total polarization map (1.2) to L. Now we can consider|L| as a n-dimensional linear system |L| on |E|d of divisors of type (1, . . . , 1).Let

PB(|L|) =

D|L|

D |E|d

be the base scheme of |L|. We call it the polar base locus of |L|. It is equal tothe complete intersection of n+ 1 effective divisors of type (1, . . . , 1). By theadjunction formula,

PB(|L|) = OPB(|L|).

If smooth, PB(|L|) is a Calabi-Yau variety of dimension (d 1)n 1.For any f L, let N(f) be the set of points x = ([v(1)], . . . , [v(d)]) |E|d

such that

f(v(1), . . . , v(j1), v, v(j+1), . . . , v(d)) = 0

for every j = 1, . . . , d and v E. Since

f(v(1), . . . , v(j1), v, v(j+1), . . . , v(d)) = Dv(1)v(j1)v(j+1)v(d)(Dv(f)),

This can be also expressed in the form

jf(v(1), . . . , v(j1), v(j+1), . . . , v(d)) = 0, j = 0, . . . , n. (1.35)

Choose coordinates u0, . . . , un in L and coordinates t0, . . . , tn in E. Let f be


the image of a basis f of L in (E)d. Then PB(|L|) is a subvariety of (Pn)dgiven by a system of d multilinear equations

f0(t(1), . . . , t(d)) = . . . = fn(t

(1), . . . , t(d)) = 0,

where t(j) = (t(j)0 , . . . , t(j)n ), j = 1, . . . , d. For any = (0, . . . , n), set

f =ni=0 ifi.

Proposition 1.1.28 The following is equivalent:

(i) x PB(|L|) is a singular point,(ii) x N(f) for some 6= 0.

Proof The variety PB(|L|) is smooth at a point x if and only if the rank ofthe d(n+ 1) (n+ 1)-size matrix

(akij) =( fkt

(j)i

(x))i,k=0,...,n,j=1,...,d

is equal to n + 1. Let fu = u0f0 + + unfn, where u0, . . . , un are un-knowns. Then the nullspace of the matrix is equal to the space of solutionsu = (0, . . . , n) of the system of linear equations

fuu0

(x) = . . . =fuun

(x) =fu

t(j)i

(x) = 0. (1.36)

For a fixed , in terminology of [239], p. 445, the system has a solution x in|E|d if f =

ifi is a degenerate multilinear form. By Proposition 1.1 from

Chapter 14 of loc.cit., f is degenerate if and only ifN(f) is non-empty. Thisproves the assertion.

For any non-empty subset I of {1, . . . , d}, let I be the subset of pointsx |E|d with equal projections to i-th factors with i I . Let k be the unionof I with #I = k. The set d is denoted by (the small diagonal).

Observe that PB(|L|) = HS(|L|) if d = 2 and PB(|L|) d1 consists ofd copies isomorphic to HS(|L|) if d > 2.

Definition 1.1.29 A n-dimensional linear system |L| |Sd(E)| is calledregular if PB(|L|) is smooth at each point of d1.

Proposition 1.1.30 Assume |L| is regular. Then

(i) |L| has no base points,(ii) D(|L|) is smooth.

30 Polarity

Proof (i) Assume that x = ([v0], . . . , [v0]) PB(|L|) . Consider the lin-ear map L E defined by evaluating f at a point (v0, . . . , v0, v, v0, . . . , v0),where v E. This map factors through a linear map L E/[v0], and hencehas a nonzero f in its kernel. This implies that x N(f), and hence x is asingular point of PB(|L|).

(ii) In coordinates, the variety D(|L|) is a subvariety of type (1, d 1) ofPn Pn given by the equations

nk=0

ukfkt0

= . . . =

nk=0

ukfktn

= 0.

The tangent space at a point ([], [a]) is given by the system of n + 1 linearequations in 2n+ 2 variables (X0, . . . , Xn, Y0, . . . , Yn)

nk=0

fkti

(a)Xk +

nj=0

2ftitj

(a)Yj = 0, i = 0, . . . , n, (1.37)

where f =nk=0 kfk. Suppose ([], [a]) is a singular point. Then the

equations are linearly dependent. Thus there exists a nonzero vector v =(0, . . . , n) such that

ni=0

ifkti

(a) = Dv(fk)(a) = fk(a, . . . , a, v) = 0, k = 0, . . . , n

andni

i2ftitj

(a) = Dv(ftj

)(a) = Dad2v(ftj

) = 0, j = 0, . . . , n,

where f =kfk. The first equation implies that x = ([a], . . . , [a], [v])

belongs to PB(|L|). Since a Sing(f), we have Dad1(ftj ) = 0, j =0, . . . , n. By (1.35), this and the second equation now imply that x N(f).By Proposition 1.1.28, PB(|L|) is singular at x, contradicting the assumption.

Corollary 1.1.31 Suppose |L| is regular. Then the projection

q : D(|L|) D(|L|)

is a resolution of singularities.

Consider the projection p : D(|L|) Jac(|L|), (D,x) 7 x. Its fibres arelinear spaces of divisors in |L| singular at the point [a]. Conversely, supposeD(|L|) contains a linear subspace, in particular, a line. Then, by Bertinis The-orem all singular divisors parameterized by the line have a common singular


point. This implies that the morphism p has positive dimensional fibres. Thissimple observation gives the following.

Proposition 1.1.32 Suppose D(|L|) does not contain lines. Then D(|L|) issmooth if and only if Jac(|L|) is smooth. Moreover, HS(|L|) = St(|L|) =Jac(|L|).

Remark 1.1.33 We will prove later in Example 1.2.3 that the tangent space ofthe discriminant hypersurface Dd(n) at a point corresponding to a hypersurfaceX = V (f) with only one ordinary double point x is naturally isomorphic tothe linear space of homogeneous forms of degree d vanishing at the point xmodulo Cf . This implies that D(|L|) is nonsingular at a point correspondingto a hypersurface with one ordinary double point unless this double point is abase point of |L|. If |L| has no base points, the singular points of D(|L|) areof two sorts: either they correspond to divisors with worse singularities thanone ordinary double point, or the linear space |L| is tangent to Dd(n) at itsnonsingular point.

Consider the natural action of the symmetric group Sd on (Pn)d. It leavesPB(|L|) invarian. The quotient variety

Rey(|L|) = PB(|L|)/Sd

is called the Reye variety of |L|. If d > 2 and n > 1, the Reye variety issingular.

Example 1.1.34 Assume d = 2. Then PB(|L|) = HS(|L|) and Jac(|L|) =St(|L|). Moreover, Rey(|L|) = Cay(|L|). We have

deg Jac(|L|) = degD(|L|) = n+ 1, deg Cay(|L|) =ni=1

(n+1i

)(n1i1).

The linear system is regular if and only if PB(|L|) is smooth. This coincideswith the notion of regularity of a web of quadrics in P3 discussed in [132].

A Reye line ` is contained in a codimension 2 subspace (`) of |L|, andis characterized by this condition. The linear subsystem (`) of dimensionn 2 contains ` in its base locus. The residual component is a curve of degree2n1 1 which intersects ` at two points. The points are the two ramificationpoints of the pencil Q `,Q |L|. The two singular points of the base locusof (`) define two singular points of the intersection (`) D(|L|). Thus(`) is a codimension 2 subspace of |L| which is tangent to the determinantalhypersurface at two points.

If |L| is regular and n = 3, PB(|L|) is a K3 surface, and its quotient Rey(|L|)is an Enriques surface. The Cayley variety is a congruence (i.e. a surface) of

32 Polarity

lines in G1(P3) of order 7 and class 3 (this means that there are 7 Reye linesthrough a general point in P3 and there 3 Reye lines in a general plane). TheReye lines are bitangents of the quartic surface D(|L|). The quartic surface has10 nodes and is called Cayley quartic symmetroid. We refer for the details to[132]. The Reye congruence of lines is also discussed in [267].

1.2 The dual hypersurface

1.2.1 The polar map

Let X = V (f) for some f Sd(E). We assume that it is not a cone. Thepolarization map

E Sd1(E), v 7 Dv(f),

allows us to identify |E| with an n-dimensional linear system of hypersurfacesof degree d 1. This linear system defines a rational map

pX : |E| 99K P(E).

It follows from (1.12) that the map is given by assigning to a point a the linearpolar Pad1(X). We call the map p the polar map defined by the hypersurfaceX . In coordinates, the polar map is given by

[x0, . . . , xn] 7[ ft0

, . . . ,f

tn

].

Recall that a hyperplane Ha = V (aii) in the dual projective space (Pn)

is the point a = [a0, . . . , an] Pn. The preimage of the hyperplane Ha underpX is the polar Pa(X) = V (

aifti

).If X is nonsingular, the polar map is a regular map given by polynomials of

degree d 1. Since it is a composition of the Veronese map and a projection,it is a finite map of degree (d 1)n.

Proposition 1.2.1 Assume X is nonsingular. The ramification divisor of thepolar map is equal to He(X).

Proof Note that, for any finite map : X Y of nonsingular varieties, theramification divisor Ram() is defined locally by the determinant of the linearmap of locally free sheaves (1Y ) 1X . The image of Ram() in Y iscalled the branch divisor. Both of the divisors may be nonreduced. We havethe Hurwitz formula

KX = (KY ) + Ram(). (1.38)

1.2 The dual hypersurface 33

The map is etale outside Ram(), i.e., for any point x X the homomor-phism of local ring OY,(x) OX,x defines an isomorphism of their formalcompletions. In particular, the preimage 1(Z) of a nonsingular subvarietyZ Y is nonsingular outside the support of Ram(). Applying this to thepolar map we see that the singular points of Pa(X) = p1X (Ha) are containedin the ramification locus Ram(pX) of the polar map. On the other hand, weknow that the set of singular points of first polars is the Hessian He(X). Thisshows that He(X) Ram(pX). Applying the Hurwitz formula for the canon-ical sheaf

KPn = pX(K(Pn)) + Ram(pX).

we obtain that deg(Ram(pX)) = (n+ 1)(d 2) = deg(He(X)). This showsthat He(X) = Ram(pX).

What is the branch divisor? One can show that the preimage of a hyperplaneHa in P(E) corresponding to a point a |E| is singular if and only if itsintersection with the branch divisor is not transversal. This means that the dualhypersurface of the branch divisor is the Steinerian hypersurface. Equivalently,the branch divisor is the dual of the Steinerian hypersurface.

1.2.2 Dual varieties

Recall that the dual variety X of a subvariety X in Pn = |E| is the closurein the dual projective space (Pn) = |E| of the locus of hyperplanes in Pnwhich are tangent to X at some nonsingular point of X .

The dual variety of a hypersurface X = V (f) is the image of X under therational map given by the first polars. In fact, the point [0f(x), . . . , nf(x)]in (Pn) is the hyperplane V (

ni=0 if(x)ti) in Pn which is tangent to X at

the point x.The following result is called the Reflexivity theorem. One can find its proof

in many modern text-books (e.g. [239], [278], [562], [607]).

Theorem 1.2.2 (Reflexivity Theorem)

(X) = X.

It follows from any proof in loc. cit. that, for any nonsingular point y Xand any nonsingular point x X ,

Tx(X) Hy Ty(X) Hx.

Here we continue to identify a point a in |E| with a hyperplane Ha in P(E).The set of all hyperplanes in (Pn) containing the linear subspace Ty(X) is

34 Polarity

the dual linear space of Ty(X) in Pn. Thus the fiber of the duality map (orGauss map)

: Xns X, x 7 Tx(X), (1.39)

over a nonsingular point y X is an open subset of the projective subspacein Pn equal to the dual of the tangent space Ty(X). Here and later Xns de-notes the set of nonsingular points of a variety X . In particular, if X is ahypersurface, the dual space of Ty(X) must be a point, and hence the map is birational.

Let us apply this to the case when X is a nonsingular hypersurface. Thepolar map is a finite map, hence the dual of a nonsingular hypersurface is ahypersurface. The duality map is a birational morphism

pX |X : X X.

The degree of the dual hypersurface X (if it is a hypersurface) is calledthe class of X . For example, the class of any plane curve of degree > 1 iswell-defined.

Example 1.2.3 Let Dd(n) be the discriminant hypersurface in |Sd(E)|. Wewould like to describe explicitly the tangent hyperplane of Dd(n) at its nonsin-gular point. Let

Dd(n) = {(X,x) |OPn(d)| Pn : x Sing(X)}.

Let us see that Dd(n) is nonsingular and the projection to the first factor

: Dd(n) Dd(n) (1.40)

is a resolution of singularities. In particular, is an isomorphism over the openset Dd(n)ns of nonsingular points of Dd(n).

The fact that Dd(n) is nonsingular follows easily from considering the pro-jection to Pn. For any point x Pn the fiber of the projection is the projectivespace of hypersurfaces which have a singular point at x (this amounts to n+ 1linear conditions on the coefficients). Thus Dd(n) is a projective bundle overPn and hence is nonsingular.

Let us see where is an isomorphism. Let Ai, |i| = d, be the projectivecoordinates in

OPn(d) = |Sd(E)| corresponding to the coefficients of ahypersurface of degree d and let t0, . . . , tn be projective coordinates in Pn.Then Dd(n) is given by n+1 bihomogeneous equations of bidegree (1, d1):

|i|=d

isAities = 0, s = 0, . . . , n, (1.41)


Here es is the s-th unit vector in Zn+1.A point (X, [v0]) = (V (f), [v0]) |OPn(d)| Pn belongs to Dd(n) if

and only if, replacing Ai with the coefficient of f at ti and ti with the i-thcoefficient of v0, we get the identities.

We identify the tangent space of |Sd(E)||E| at a point (X, [v0]) with thespace Sd(E)/Cf E/Cv0. In coordinates, a vector in the tangent space is apair (g, [v]), where g =

|i|=d ait

i, v = (x0, . . . , xn) are considered modulopairs (f, v0). Differentiating equations (1.41), we see that the tangent spaceis defined by the (n+ 1)

(n+dd

)-matrix

. . . i0xie0 . . .

|i|=d i0i0Aix

ie0e0 . . .|i|=d i0inAix

ie0en

......

......

......

. . . inxien . . .

|i|=d ini0Aix

iene0 . . .|i|=d ininAix

ienen ,

where xies = 0 if i es is not a non-negative vector. It is easy to interpret

solutions of these equations as pairs (g, v) from the above such that

(g)(v0) + He(f)(v0) v = 0. (1.42)

Since [v0] is a singular point of V (f), (f)([v0]) = 0. Also He(f)(v0) v0 = 0, as follows from Theorem 1.1.20. This confirms that pairs (f, v0) arealways solutions. The tangent map d at the point (V (f), [v0]) is given by theprojection (g, v) 7 g, where (g, v) is a solution of (1.42). Its kernel consistsof the pairs (f, v) modulo pairs (f, v0). For such pairs the equations (1.42)give

He(f)(v0) v = 0. (1.43)

We may assume that v0 = (1, 0, . . . , 0). Since [v0] is a singular point of V (f),we can write f = td20 f2(t1, . . . , tn) + . . .. Computing the Hessian matrix atthe point v0 we see that it is equal to

0 . . . . . . 0

0 a11 . . . a1n...

......

...0 an1 . . . ann

, (1.44)where f2(t1, . . . , tn) =

0i,jn aijtitj . Thus a solution of (1.43), not pro-

portional to v0 exists if and only if det He(f2) = 0. By definition, this meansthat the singular point of X at x is not an ordinary double point. Thus we ob-tain that the projection map (1.40) is an isomorphism over the open subset ofDd(n) representing hypersurfaces with an isolated ordinary singularity.

We can also find the description of the tangent space of Dd(n) at its point

36 Polarity

X = V (f) representing a hypersurface with a unique ordinary singular pointx. It follows from calculation of the Hessian matrix in (1.44), that its corankat the ordinary singular point is equal to 1. Since the matrix is symmetric, avector in its nullspace is orthogonal to the column of the matrix. We know thatHe(f)(v0) v0 = 0. Thus the dot-product (g)(v0) v0 is equal to zero. ByEulers formula, we obtain g(v0) = 0. The converse is also true. This provesthat

T (Dd(n))X = {g Sd(E)/Cf : g(x) = 0}. (1.45)

Now we are ready to compute the dual variety of Dd(n). The conditiong(b) = 0, where Sing(X) = {b} is equivalent toDbd(f) = 0. Thus the tangenthyperplane, considered as a point in the dual space |Sd(E)| = |Sd(E)|corresponds to the envelope bd = (

ns=0 bsi)

d. The set of such envelopes isthe Veronese variety Vnd , the image of |E| under the Veronese map vd : |E| |Sd(E)|. Thus

Dd(n) = d(Pn), (1.46)

Of course, it is predictable. Recall that the Veronese variety is embeddednaturally in |OPn(d)|. Its hyperplane section can be identified with a hyper-surface of degree d in Pn. A tangent hyperplane is a hypersurface with a sin-gular point, i.e. a point in Dd(n). Thus the dual of Vnd is isomorphic to Dd(n),and hence, by duality, the dual of Dd(n) is isomorphic to Vnd .

Example 1.2.4 Let Q = V (q) be a nonsingular quadric in Pn. Let A = (aij)be a symmetric matrix defining q. The tangent hyperplane of Q at a point[x] Pn is the hyperplane

t0

nj=0

a0jxj + + tnnj=0

anjxj = 0.

Thus the vector of coordinates y = (y0, . . . , yn) of the tangent hyperplane isequal to the vector A x. Since A is invertible, we can write x = A1 y. Wehave

0 = x A x = (y A1) A (A1 y) = y A1 y = 0.

Here we treat x or y as a row-matrix or as a column-matrix in order the matrixmultiplication makes sense. Since A1 = det(A)1adj(A), we obtain that thedual variety of Q is also a quadric given by the adjugate matrix adj(A).

The description of the tangent space of the discriminant hypersurface fromExample 1.2.3 has the following nice application (see also Remark 1.1.33).


Proposition 1.2.5 Let X be a hypersurface of degree d in Pn. Suppose a isa nonsingular point of the Steinerian hypersurface St(X). Then Sing(Pa(X))consists of an ordinary singular point b and

Ta(St(X)) = Pbd1(X).

1.2.3 Plucker formulas

Let X = V (f) be a nonsingular irreducible hypersurface that is not a cone.Fix n 1 general points a1, . . . , an1 in Pn. Consider the intersection

X Pa1(X) . . . Pan1(X) = {b Pn : a1, . . . , an1 Tb(X)}.

The set of hyperplanes through a general set of n1 points is a line in the dualspace. This shows that

degX = #X Pa1(X) . . . Pan1(X) = d(d 1)n1. (1.47)

The computation does not apply to singular X since all polars Pa(X) passthrough singular points of X . In the case when X has only isolated singular-ities, the intersection of n 1 polars with X contains singular points whichcorrespond to hyperplanes which we excluded from the definition of the dualhypersurface. So we get the following formula

deg(X) = d(d1)n1

xSing(X)

i(X,Pa1(X), . . . , Pan1(X))x. (1.48)

To state an explicit formula we need some definition. Let = (1, . . . , k)be a set of polynomials in C[z1, . . . , zn]. We assume that the holomorphic mapCn Ck defined by these polynomials has an isolated critical point at theorigin. Let J() be the jacobian matrix. The ideal J () in the ring of formalpower series C[[z1, . . . , zn]] generated by the maximal minors of the Jacobianmatrix is called the Jacobian ideal of . The number

() = dimC[[z1, . . . , zn]]/J ()

is called the Milnor number of . Passing to affine coordinates, this definitioneasily extends to the definition of the Milnor number (X,x) of an isolatedsingularity of a complete intersection subvariety X in Pn.

We will need the following result of Le Dung Trang [360], Theorem 3.7.1.

Lemma 1.2.6 Let Z be a complete intersection in Cn defined by polynomials

38 Polarity

1, . . . , k with isolated singularity at the origin. Let Z1 = V (1, . . . , k1).Then

(1, . . . , k1) + (1, . . . , k1, k)

= dimC[[z1, . . . , zn]]/(1, . . . , k1,J (1, . . . , k)).

Now we can state and prove the Plucker-Teissier formula for a hypersurfacewith isolated singularities:

Theorem 1.2.7 Let X be a hypersurface in Pn of degree d. Suppose X hasonly isolated singularities. For any point x Sing(X), let

e(X,x) = (X,x) + (H X,x),

where H is a general hyperplane section of X containing x. Then

degX = d(d 1)n1

xSing(X)

e(X,x).

.

Proof We have to show that e(X,x) = i(X,Pa1(X), . . . , Pan1(X))x. Wemay assume that x = [1, 0, . . . , 0] and choose affine coordinates with zi =ti/t0. Let f(t0, . . . , tn) = td0g(z1, . . . , zn). Easy calculations employing theChain Rule, give the formula for the dehomogenized partial derivatives

xd0f

t0= dg +

gzi

zi,

xd0f

ti=

g

zi, i = 1, . . . , n.

Let H = V (h) be a general hyperplane spanned by n 1 general pointsa1, . . . , an1, and h : Cn C be the projection defined by the linear functionh =

izi. Let

F : Cn C2, z = (z1, . . . , zn) 7 (g(z), h(z)).

Consider the Jacobian determinant of the two functions (f, h)

J(g, h) =

(gz1

. . . gzn1 . . . n

).

The ideal (g, J(g, h)) defines the set of critical points of the restriction of themap F to X \ V (t0). We have

(g, J(g, h)) = (g, ig

zj j

g

zi)1i


The points (0, . . . , 0, j , 0, . . . , 0,i, 0, . . . , 0) span the hyperplane H . Wemay assume that these points are our points a1, . . . , an1. So, we see that(g, J(g, h)) coincides with the ideal in the completion of local ringOPn,x gen-erated by f and the polars Pai(f). By definition of the index of intersection,we have

i(X,Pa1(X), . . . , Pan1(X))x = (g, h).

It remains for us to apply Lemma 1.2.6, where Z = V (g) and Z1 = V (g) V (h).

Example 1.2.8 An isolated singular point x of a hypersurface X in Pn iscalled an Ak-singularity (or a singular point of type Ak) if the formal comple-tion ofOX,x is isomorphic to C[[z1, . . . , zn]]/(zk+11 +z22 + +z2n). If k = 1,it is an ordinary quadratic singularity (or a node), if k = 2, it is an ordinarycusp. We get

(X,x) = k, (X H,x) = 1.

This gives the Plucker formula for hypersurfaces with s singularities of typeAk1 , . . . , Aks

degX = d(d 1)n1 (k1 + 1) (ks + 1). (1.49)

In particularly, when X is a plane curve C with nodes and ordinary cusps,we get a familiar Plucker formula

degC = d(d 1) 2 3. (1.50)

Note that, in case of plane curves, (H X,x) is always equal to multxX1,where multxX is the multiplicity of X at x. This gives the Plucker formula forplane curves with arbitrary singularities

degC = d(d 1)

xSing(X)

((X,x) + multxX 1). (1.51)

Note that the dual curve C of a nonsingular curve C of degree d > 2 isalways singular. This follows from the formula for the genus of a nonsingularplane curve and the fact that C and C are birationally isomorphic. The po-lar map C C is equal to the normalization map. A singular point of Ccorresponds to a line which is either tangent to C at several points, or is an in-flection tangent. We skip a local computation which shows that a line which isan inflection tangent at one point with ordfl = 1 (an honest inflection tangent)gives an ordinary cusp of C and a line which is tangent at two points whichare not inflection points (honest bitangent) gives a node. Thus we obtain thatthe number of nodes of C is equal to the number of honest bitangents of

40 Polarity

C and the number of ordinary cusps of C is equal to the number of honestinflection tangents to C.

Assume that C is nonsingular and C has no other singular points exceptordinary nodes and cusps. We know that the number of inflection points isequal to 3d(d 2). Applying Plucker formula (1.50) to C, we get that

=1

2

(d(d1)(d(d1)1)d9d(d2)

)=

1

2d(d2)(d29). (1.52)

This is the (expected) number of bitangents of a nonsingular plane curve. Forexample, we expect that a nonsingular plane quartic has 28 bitangents.

We refer for discussions of Plucker formulas to many modern text-books(e.g. [220], [231], [267], [239]). A proof of Plucker-Teissiere formula can befound in [558]. A generalization of the Plucker-Teissier formula to completeintersections in projective space was given by S. Kleiman [334]

1.3 Polar s-hedra

1.3.1 Apolar schemes

We continue to use E to denote a complex vector space of dimension n + 1.Consider the polarization pairing (1.2)

Sd(E) Sk(E) Sdk(E), (f, ) 7 D(f).

Definition 1.3.1 Sk(E) is called apolar to f Sd(E) if D(f) = 0.We extend this definition to hypersurfaces in the obvious way.

Lemma 1.3.2 For any Sk(E), Sm(E) and f Sd(E),

D(D(f)) = D(f).

Proof By linearity and induction on the degree, it suffices to verify the asser-tion in the case when = i and = j . In this case it is obvious.

Corollary 1.3.3 Let f Sd(E). Let APk(f) be the subspace of Sk(E)spanned by forms of degree k apolar to f . Then

AP(f) =k=0

APk(f)

is a homogeneous ideal in the symmetric algebra S(E).

1.3 Polar s-hedra 41

Definition 1.3.4 The quotient ring

Af = S(E)/AP(f)

is called the apolar ring of f .

The ringAf inherits the grading of S(E). Since any polynomial Sr(E)with r > d is apolar to f

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Classical Algebraic Geometry: a modern view · Preface The main purpose of the present treatise is...

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