Field reduction in finite projective geometry
Geertrui Van de VoordeGhent University & Free University Brussels (VUB)
Fq11July 22–26 2013, Magdeburg
INTRODUCTION: FINITE PROJECTIVE GEOMETRY
FIELD REDUCTION AND LINEAR SETSThe equivalence of linear setsScattered linear sets and pseudoreguli
APPLICATIONSBlocking setsPseudo-arcsSemifields
CONCLUSION
OUTLINE
INTRODUCTION: FINITE PROJECTIVE GEOMETRY
FIELD REDUCTION AND LINEAR SETSThe equivalence of linear setsScattered linear sets and pseudoreguli
APPLICATIONSBlocking setsPseudo-arcsSemifields
CONCLUSION
FINITE PROJECTIVE GEOMETRY
NOTATION
I V: Vector spaceI PG(V ): Corresponding projective space
I Fq = GF (q), q = ph, p prime.I Fd
q : vector space in d dimensions over Fq.I PG(Fd
q ) = PG(d − 1,q)
I if d = 3: projective plane, which is Desarguesian
FINITE PROJECTIVE GEOMETRY
NOTATION
I V: Vector spaceI PG(V ): Corresponding projective spaceI Fq = GF (q), q = ph, p prime.I Fd
q : vector space in d dimensions over Fq.
I PG(Fdq ) = PG(d − 1,q)
I if d = 3: projective plane, which is Desarguesian
FINITE PROJECTIVE GEOMETRY
NOTATION
I V: Vector spaceI PG(V ): Corresponding projective spaceI Fq = GF (q), q = ph, p prime.I Fd
q : vector space in d dimensions over Fq.I PG(Fd
q ) = PG(d − 1,q)
I if d = 3: projective plane, which is Desarguesian
FINITE PROJECTIVE GEOMETRY
NOTATION
I V: Vector spaceI PG(V ): Corresponding projective spaceI Fq = GF (q), q = ph, p prime.I Fd
q : vector space in d dimensions over Fq.I PG(Fd
q ) = PG(d − 1,q)
I if d = 3: projective plane, which is Desarguesian
AXIOMATIC PROJECTIVE PLANES
Points, lines and three axioms
(a) ∀r 6= s ∃!L (b) ∀L 6= M ∃!r (c) ∃r , s, t , u
If Π is a projective plane, then interchanging points and lines,we obtain the dual plane ΠD.
AXIOMATIC PROJECTIVE PLANES
DEFINITIONThe order of a projective plane is the number of points on a lineminus 1.
EASY TO CHECK
I The order of PG(2,q) is q.I A projective plane of order s has s2 + s + 1 points and
s2 + s + 1 lines.
THE SMALLEST PROJECTIVE PLANE: PG(2,2)The projective plane of order 2, the Fano plane, has:
I q + 1 = 2 + 1 = 3 points on a line,I 3 lines through a point.
And it is the unique plane of order 2.
THE PROJECTIVE PLANE PG(2,3)
The projective plane PG(2,3) has:I q + 1 = 3 + 1 = 4 points on a line,I 4 lines through a point.
And it is the unique plane of order 3.
FINITE PROJECTIVE PLANES
The projective planes of orders 2,3,4,5,7 and 8 are unique,
but there are 4 non-isomorphic planes of order 9.
I Many easy to state problems are wide open, e.g.I is the order of a projective plane necessarily a prime
power?I is every projective plane of prime order Desarguesian (i.e.
a PG(2,p))?I Easy to construct general class of not necessarily
Desarguesian planes of order ph, p prime, h > 1:translation planes.
I Translation plane whose dual is also a translation plane:semifield plane.
FINITE PROJECTIVE PLANES
The projective planes of orders 2,3,4,5,7 and 8 are unique,but there are 4 non-isomorphic planes of order 9.
I Many easy to state problems are wide open, e.g.I is the order of a projective plane necessarily a prime
power?I is every projective plane of prime order Desarguesian (i.e.
a PG(2,p))?I Easy to construct general class of not necessarily
Desarguesian planes of order ph, p prime, h > 1:translation planes.
I Translation plane whose dual is also a translation plane:semifield plane.
FINITE PROJECTIVE PLANES
The projective planes of orders 2,3,4,5,7 and 8 are unique,but there are 4 non-isomorphic planes of order 9.
I Many easy to state problems are wide open, e.g.I is the order of a projective plane necessarily a prime
power?
I is every projective plane of prime order Desarguesian (i.e.a PG(2,p))?
I Easy to construct general class of not necessarilyDesarguesian planes of order ph, p prime, h > 1:translation planes.
I Translation plane whose dual is also a translation plane:semifield plane.
FINITE PROJECTIVE PLANES
The projective planes of orders 2,3,4,5,7 and 8 are unique,but there are 4 non-isomorphic planes of order 9.
I Many easy to state problems are wide open, e.g.I is the order of a projective plane necessarily a prime
power?I is every projective plane of prime order Desarguesian (i.e.
a PG(2,p))?
I Easy to construct general class of not necessarilyDesarguesian planes of order ph, p prime, h > 1:translation planes.
I Translation plane whose dual is also a translation plane:semifield plane.
FINITE PROJECTIVE PLANES
The projective planes of orders 2,3,4,5,7 and 8 are unique,but there are 4 non-isomorphic planes of order 9.
I Many easy to state problems are wide open, e.g.I is the order of a projective plane necessarily a prime
power?I is every projective plane of prime order Desarguesian (i.e.
a PG(2,p))?I Easy to construct general class of not necessarily
Desarguesian planes of order ph, p prime, h > 1:translation planes.
I Translation plane whose dual is also a translation plane:semifield plane.
FINITE PROJECTIVE PLANES
The projective planes of orders 2,3,4,5,7 and 8 are unique,but there are 4 non-isomorphic planes of order 9.
I Many easy to state problems are wide open, e.g.I is the order of a projective plane necessarily a prime
power?I is every projective plane of prime order Desarguesian (i.e.
a PG(2,p))?I Easy to construct general class of not necessarily
Desarguesian planes of order ph, p prime, h > 1:translation planes.
I Translation plane whose dual is also a translation plane:semifield plane.
THE ANDRÉ/BRUCK-BOSE-CONSTRUCTION
I Let S be a partition of thepoints of PG(3,q) in q2 + 1lines. (a line spread ofPG(3,q)).
I Embed H∞ = PG(3,q) inPG(4,q).
THE ANDRÉ/BRUCK-BOSE-CONSTRUCTION
I Define the followingincidence structure:
I Points:I type 1: the points of
PG(4, q) \ H∞I type 2: the lines of S.
I Lines:I type 1: planes
intersecting H∞exactly in a line of S.
I type 2: the space H∞.I Incidence: containment
This gives a projective plane of order q2, which is a translationplane.Translation planes of order qt : start with (t − 1)-spread inPG(2t − 1,q).
THE ANDRÉ/BRUCK-BOSE-CONSTRUCTION
I Define the followingincidence structure:
I Points:I type 1: the points of
PG(4, q) \ H∞I type 2: the lines of S.
I Lines:I type 1: planes
intersecting H∞exactly in a line of S.
I type 2: the space H∞.I Incidence: containment
This gives a projective plane of order q2, which is a translationplane.Translation planes of order qt : start with (t − 1)-spread inPG(2t − 1,q).
THE ANDRÉ/BRUCK-BOSE-CONSTRUCTION
I Define the followingincidence structure:
I Points:I type 1: the points of
PG(4, q) \ H∞I type 2: the lines of S.
I Lines:I type 1: planes
intersecting H∞exactly in a line of S.
I type 2: the space H∞.I Incidence: containment
This gives a projective plane of order q2, which is a translationplane.
Translation planes of order qt : start with (t − 1)-spread inPG(2t − 1,q).
THE ANDRÉ/BRUCK-BOSE-CONSTRUCTION
I Define the followingincidence structure:
I Points:I type 1: the points of
PG(4, q) \ H∞I type 2: the lines of S.
I Lines:I type 1: planes
intersecting H∞exactly in a line of S.
I type 2: the space H∞.I Incidence: containment
This gives a projective plane of order q2, which is a translationplane.Translation planes of order qt : start with (t − 1)-spread inPG(2t − 1,q).
OUTLINE
INTRODUCTION: FINITE PROJECTIVE GEOMETRY
FIELD REDUCTION AND LINEAR SETSThe equivalence of linear setsScattered linear sets and pseudoreguli
APPLICATIONSBlocking setsPseudo-arcsSemifields
CONCLUSION
FIELD REDUCTION
FIELD REDUCTIONA point of PG(r − 1,qt ) corresponds to a (t − 1)-space ofPG(rt − 1,q):
PG(0,qt )→ Fqt → Ftq → PG(t − 1,q)
This process is called field reduction.We also say we blow up a point.
FIELD REDUCTION
FIELD REDUCTIONA point of PG(r − 1,qt ) corresponds to a (t − 1)-space ofPG(rt − 1,q):
PG(0,qt )→ Fqt → Ftq → PG(t − 1,q)
This process is called field reduction.We also say we blow up a point.
FIELD REDUCTION
EXAMPLE
I PG(F3q3)→ PG(F9
q)
I PG(2,q3)→ PG(8,q)
I (α, β, γ) 7→ (a0,a1,a2,b0,b1,b2, c0, c1, c2)α = a0 + a1ω + a2ω
2
β = b0 + b1ω + b2ω2
γ = c0 + c1ω + c2ω2
and ω is a primitive element of Fq3 over Fq.I The point (1,0,0) of PG(2,q3) corresponds to a plane of
PG(8,q) since(λ,0,0)→ (a,b, c,0,0,0,0,0,0), λ ∈ F∗q3 , a,b, c ∈ Fq.
FIELD REDUCTION
EXAMPLE
I PG(F3q3)→ PG(F9
q)
I PG(2,q3)→ PG(8,q)
I (α, β, γ) 7→ (a0,a1,a2,b0,b1,b2, c0, c1, c2)α = a0 + a1ω + a2ω
2
β = b0 + b1ω + b2ω2
γ = c0 + c1ω + c2ω2
and ω is a primitive element of Fq3 over Fq.
I The point (1,0,0) of PG(2,q3) corresponds to a plane ofPG(8,q) since(λ,0,0)→ (a,b, c,0,0,0,0,0,0), λ ∈ F∗q3 , a,b, c ∈ Fq.
FIELD REDUCTION
EXAMPLE
I PG(F3q3)→ PG(F9
q)
I PG(2,q3)→ PG(8,q)
I (α, β, γ) 7→ (a0,a1,a2,b0,b1,b2, c0, c1, c2)α = a0 + a1ω + a2ω
2
β = b0 + b1ω + b2ω2
γ = c0 + c1ω + c2ω2
and ω is a primitive element of Fq3 over Fq.I The point (1,0,0) of PG(2,q3) corresponds to a plane of
PG(8,q) since(λ,0,0)→ (a,b, c,0,0,0,0,0,0), λ ∈ F∗q3 , a,b, c ∈ Fq.
FIELD REDUCTION
The set of points of PG(n,qt ) corresponds to a (t − 1)-spread ofPG((n + 1)t − 1,q).A spread constructed in this way is called a Desarguesianspread.
FIELD REDUCTION
The set of points of PG(n,qt ) corresponds to a (t − 1)-spread ofPG((n + 1)t − 1,q).A spread constructed in this way is called a Desarguesianspread.
FIELD REDUCTION
Points of PG(1,qt )→ Desarguesian (t − 1)-spread ofPG(2t − 1,q).
If we use a Desarguesianspread S at infinity, the
translation plane obtained is theDesarguesian plane PG(2,qt ).
THREE EQUIVALENT VIEWS ON LINEAR SETS
I Definition via vector spacesI Definition via Desarguesian spreadsI Definition via projection
LINEAR SETS: A NATURAL OBJECT TO CONSIDER
Vectors of (Fq)− subspace U ⊆ Fnq
Vectors of (Fq)− subspace U ⊆ Fnq ⊆ Fn
qt
↓ PG
Subset of points LU ⊆ PG(n − 1,qt )
This subset LU is called an Fq-linear set.
LINEAR SETS: A NATURAL OBJECT TO CONSIDER
Vectors of (Fq)− subspace U ⊆ Fnq
Vectors of (Fq)− subspace U ⊆ Fnq ⊆ Fn
qt
↓ PG
Subset of points LU ⊆ PG(n − 1,qt )
This subset LU is called an Fq-linear set.
LINEAR SETS: A NATURAL OBJECT TO CONSIDER
Vectors of (Fq)− subspace U ⊆ Fnq
Vectors of (Fq)− subspace U ⊆ Fnq ⊆ Fn
qt
↓ PG
Subset of points LU ⊆ PG(n − 1,qt )
This subset LU is called an Fq-linear set.
LINEAR SETS: A NATURAL OBJECT TO CONSIDER
Vectors of (Fq)− subspace U ⊆ Fnq
Vectors of (Fq)− subspace U ⊆ Fnq ⊆ Fn
qt
↓ PG
Subset of points LU ⊆ PG(n − 1,qt )
This subset LU is called an Fq-linear set.
LINEAR SETS: A NATURAL OBJECT TO CONSIDER
Vectors of (Fq)− subspace U ⊆ Fnq
Vectors of (Fq)− subspace U ⊆ Fnq ⊆ Fn
qt
↓ PG
Subset of points LU ⊆ PG(n − 1,qt )
This subset LU is called an Fq-linear set.
VECTOR SPACE DEFINITION
MORE FORMALLYLet W = Fn
qt . S is an Fq-linear set in PG(W ) iff there exists anFq-vectorsubspace U ⊂W such that S = B(U) with
B(U) = 〈u〉Fqt : u ∈ U \ 0.
If U has dimension k , then we say that S has rank k.
EXAMPLE: LINEAR SETS OF RANK 3W = F3
q3 , U = F3q, ω : primitive element of Fq3
U = 〈(1,0,0), (0,1,0), (0,0,1)〉Fq ⇒
B(U) is a subplane PG(2,q) of PG(2,q3)
U = 〈(1,1,0), (ω, ωq,0), (ω2, ω2q,0)〉Fq ⇒
B(U) is a set of q2 + q + 1 points on a line PG(1,q3)
U = 〈(1,0,0), (ω,0,0), (0,1,0)〉Fq ⇒
B(U) is a set of q2 + 1 points on a line PG(1,q3)
U = 〈(1,0,0), (ω,0,0), (ω2,0,0)〉Fq ⇒
B(U) is the point (1,0,0) in PG(2,q3)
EXAMPLE: LINEAR SETS OF RANK 3W = F3
q3 , U = F3q, ω : primitive element of Fq3
U = 〈(1,0,0), (0,1,0), (0,0,1)〉Fq ⇒
B(U) is a subplane PG(2,q) of PG(2,q3)
U = 〈(1,1,0), (ω, ωq,0), (ω2, ω2q,0)〉Fq ⇒
B(U) is a set of q2 + q + 1 points on a line PG(1,q3)
U = 〈(1,0,0), (ω,0,0), (0,1,0)〉Fq ⇒
B(U) is a set of q2 + 1 points on a line PG(1,q3)
U = 〈(1,0,0), (ω,0,0), (ω2,0,0)〉Fq ⇒
B(U) is the point (1,0,0) in PG(2,q3)
EXAMPLE: LINEAR SETS OF RANK 3W = F3
q3 , U = F3q, ω : primitive element of Fq3
U = 〈(1,0,0), (0,1,0), (0,0,1)〉Fq ⇒
B(U) is a subplane PG(2,q) of PG(2,q3)
U = 〈(1,1,0), (ω, ωq,0), (ω2, ω2q,0)〉Fq ⇒
B(U) is a set of q2 + q + 1 points on a line PG(1,q3)
U = 〈(1,0,0), (ω,0,0), (0,1,0)〉Fq ⇒
B(U) is a set of q2 + 1 points on a line PG(1,q3)
U = 〈(1,0,0), (ω,0,0), (ω2,0,0)〉Fq ⇒
B(U) is the point (1,0,0) in PG(2,q3)
EXAMPLE: LINEAR SETS OF RANK 3W = F3
q3 , U = F3q, ω : primitive element of Fq3
U = 〈(1,0,0), (0,1,0), (0,0,1)〉Fq ⇒
B(U) is a subplane PG(2,q) of PG(2,q3)
U = 〈(1,1,0), (ω, ωq,0), (ω2, ω2q,0)〉Fq ⇒
B(U) is a set of q2 + q + 1 points on a line PG(1,q3)
U = 〈(1,0,0), (ω,0,0), (0,1,0)〉Fq ⇒
B(U) is a set of q2 + 1 points on a line PG(1,q3)
U = 〈(1,0,0), (ω,0,0), (ω2,0,0)〉Fq ⇒
B(U) is the point (1,0,0) in PG(2,q3)
DEFINITION VIA DESARGUESIAN SPREADS
I D : Desarguesian (t − 1)-spread of PG((n + 1)t − 1,q)
I π : a k − 1-dimensional subspace of PG((n + 1)t − 1,q)
I B(π): set of elements of D intersecting π
Then B(π) is an Fq-linear set of rank k .
DEFINITION VIA DESARGUESIAN SPREADS
I D : Desarguesian (t − 1)-spread of PG((n + 1)t − 1,q)
I π : a k − 1-dimensional subspace of PG((n + 1)t − 1,q)
I B(π): set of elements of D intersecting π
Then B(π) is an Fq-linear set of rank k .
LINEAR SETS OF RANK 3
Fq-linear set S of rank 3: set of spread elements intersecting afixed plane π.
I a spread element or
LINEAR SETS OF RANK 3
I a set of q2 + 1 spread elements, one intersecting π in aline, the others intersect π in a point.
LINEAR SETS OF RANK 3
I a set of q2 + q + 1 spread elements, each intersecting π ina point.
B(π): either one point, q2 + 1 points, or q2 + q + 1 points.
DEFINITION VIA PROJECTION
THEOREM [G. LUNARDON, O. POLVERINO (2004)]An Fq-linear set of rank k in PG(n,qt ) is a subgeometryPG(k − 1,q) or the projection of a subgeometry PG(k − 1,q)from a suitable subspace.
DEFINITION VIA PROJECTION
Rank 3: projection of a subplane
A scattered linear set of rank 3: q2 + q + 1 points.
LINEAR SETS
Directions for research:I Equivalence of linear setsI The size of linear setsI Intersection of linear setsI Classification of particular linear setsI . . .
I often motivated by the applications
THE EQUIVALENCE PROBLEM
Subgeometries of the same dimension and order: alwaysPGL-equivalent
Linear sets of same rank=Projections of subgeometries ofsame order: not always equivalent.
THE EQUIVALENCE PROBLEM: SETTING
Σi : PG(m,q), subgeometry of Σ∗ = PG(m,qt ).Ω∗i : (m − n − 1)-space in Σ∗.Ωi : n-space in Σ∗, skew to Ω∗i .
THE EQUIVALENCE PROBLEM: PROJECTING
.
.!i
"!i
x
"i
!!
pi(x)
Si : linear set: projection of Σi from Ω∗i into Ωi .The pre-image of pi(x) can be a point, a line, a plane, a solid...
THE EQUIVALENCE PROBLEM: THE THEOREM
S1 = Σ1/Ω∗1, S2 = Σ2/Ω∗2, linear sets of rank m, and not ofsmaller rank.
THEOREM [M. LAVRAUW - G.VDV (2010)]S1 is PΓL-equivalent to S2 ifand only if there is an elementφ of PΓL(n,qt ) such that
φ(Σ1) = Σ2, φ(Ω∗1) = Ω∗2
I For linear sets of rank t + 1 in PG(2,qt ), this was proven byBonoli and Polverino. This is the case of linear blockingsets.
THE EQUIVALENCE PROBLEM: THE THEOREM
S1 = Σ1/Ω∗1, S2 = Σ2/Ω∗2, linear sets of rank m, and not ofsmaller rank.
THEOREM [M. LAVRAUW - G.VDV (2010)]S1 is PΓL-equivalent to S2 ifand only if there is an elementφ of PΓL(n,qt ) such that
φ(Σ1) = Σ2, φ(Ω∗1) = Ω∗2
I For linear sets of rank t + 1 in PG(2,qt ), this was proven byBonoli and Polverino.
This is the case of linear blockingsets.
THE EQUIVALENCE PROBLEM: THE THEOREM
S1 = Σ1/Ω∗1, S2 = Σ2/Ω∗2, linear sets of rank m, and not ofsmaller rank.
THEOREM [M. LAVRAUW - G.VDV (2010)]S1 is PΓL-equivalent to S2 ifand only if there is an elementφ of PΓL(n,qt ) such that
φ(Σ1) = Σ2, φ(Ω∗1) = Ω∗2
I For linear sets of rank t + 1 in PG(2,qt ), this was proven byBonoli and Polverino. This is the case of linear blockingsets.
SCATTERED LINEAR SETS
EASY TO SEE
I Maximum size of Fq-linear set of rank k :qk−1 + qk−2 + . . .+ q + 1
I If this bound is reached: scattered linear set.
SCATTERED LINEAR SETS: A BOUND ON THE RANK
What is the possible rank of a scattered linear set L?
WHAT ONE WOULD GUESS:The higher the rank of the linear set, the harder it is for L to bescattered.
SCATTERED LINEAR SETS: A BOUND ON THE RANK
What is the possible rank of a scattered linear set L?
WHAT ONE WOULD GUESS:The higher the rank of the linear set, the harder it is for L to bescattered.
SCATTERED LINEAR SETS: A BOUND ON THE RANK
THEOREM [A. BLOKHUIS AND M. LAVRAUW (2000)]If L is a scattered linear set of rank d in PG(r − 1,qt ), thend ≤ rt/2.
Scattered space reaching this bound↔Two-weight code↔two-intersection set↔ strongly regular graph
EXAMPLEIn PG(3,q3): a maximum scattered linear set has rank 6.
SCATTERED LINEAR SETS: A BOUND ON THE RANK
THEOREM [A. BLOKHUIS AND M. LAVRAUW (2000)]If L is a scattered linear set of rank d in PG(r − 1,qt ), thend ≤ rt/2.
Scattered space reaching this bound↔Two-weight code↔two-intersection set↔ strongly regular graph
EXAMPLEIn PG(3,q3): a maximum scattered linear set has rank 6.
SCATTERED LINEAR SETS: A BOUND ON THE RANK
THEOREM [A. BLOKHUIS AND M. LAVRAUW (2000)]If L is a scattered linear set of rank d in PG(r − 1,qt ), thend ≤ rt/2.
Scattered space reaching this bound↔Two-weight code↔two-intersection set↔ strongly regular graph
EXAMPLEIn PG(3,q3): a maximum scattered linear set has rank 6.
PSEUDOREGULI IN PG(3,q3)
THEOREM [G. MARINO, O. POLVERINO, R. TROMBETTI(2007)]To every scattered linear set of rank 6 in PG(3,q3), there is anFq-pseudoregulus associated.
A pseudoregulus R is a set of q3 + 1 lines meeting thescattered linear set L in q2 + q + 1 points, such that thereexactly two transversal lines meeting the q3 + 1 lines of R.
THEOREM [M. LAVRAUW - G. VDV]To every scattered Fq-linear set of rank 3n in PG(2n − 1,q3),there is a pseudoregulus associated.
I Useful for the study of particular semifields.
PSEUDOREGULI IN PG(3,q3)
THEOREM [G. MARINO, O. POLVERINO, R. TROMBETTI(2007)]To every scattered linear set of rank 6 in PG(3,q3), there is anFq-pseudoregulus associated.A pseudoregulus R is a set of q3 + 1 lines meeting thescattered linear set L in q2 + q + 1 points, such that thereexactly two transversal lines meeting the q3 + 1 lines of R.
THEOREM [M. LAVRAUW - G. VDV]To every scattered Fq-linear set of rank 3n in PG(2n − 1,q3),there is a pseudoregulus associated.
I Useful for the study of particular semifields.
PSEUDOREGULI IN PG(3,q3)
THEOREM [G. MARINO, O. POLVERINO, R. TROMBETTI(2007)]To every scattered linear set of rank 6 in PG(3,q3), there is anFq-pseudoregulus associated.A pseudoregulus R is a set of q3 + 1 lines meeting thescattered linear set L in q2 + q + 1 points, such that thereexactly two transversal lines meeting the q3 + 1 lines of R.
THEOREM [M. LAVRAUW - G. VDV]To every scattered Fq-linear set of rank 3n in PG(2n − 1,q3),there is a pseudoregulus associated.
I Useful for the study of particular semifields.
PSEUDOREGULI IN PG(3,q3)
THEOREM [G. MARINO, O. POLVERINO, R. TROMBETTI(2007)]To every scattered linear set of rank 6 in PG(3,q3), there is anFq-pseudoregulus associated.A pseudoregulus R is a set of q3 + 1 lines meeting thescattered linear set L in q2 + q + 1 points, such that thereexactly two transversal lines meeting the q3 + 1 lines of R.
THEOREM [M. LAVRAUW - G. VDV]To every scattered Fq-linear set of rank 3n in PG(2n − 1,q3),there is a pseudoregulus associated.
I Useful for the study of particular semifields.
PSEUDOREGULI IN PG(3,q3)
QUESTION
Can we characterise a pseudoregulus in a geometric way?
THEOREM [M. LAVRAUW-G. VDV]Let P be a set of q3 + 1 mutually disjoint lines in PG(3,q3),q > 2 and let P be its point set. If the Fq-subline through anythree collinear points of P is contained in P, then P is a regulusor a pseudoregulus.
PSEUDOREGULI IN PG(3,q3)
QUESTION
Can we characterise a pseudoregulus in a geometric way?
THEOREM [M. LAVRAUW-G. VDV]Let P be a set of q3 + 1 mutually disjoint lines in PG(3,q3),q > 2 and let P be its point set.
If the Fq-subline through anythree collinear points of P is contained in P, then P is a regulusor a pseudoregulus.
PSEUDOREGULI IN PG(3,q3)
QUESTION
Can we characterise a pseudoregulus in a geometric way?
THEOREM [M. LAVRAUW-G. VDV]Let P be a set of q3 + 1 mutually disjoint lines in PG(3,q3),q > 2 and let P be its point set. If the Fq-subline through anythree collinear points of P is contained in P, then P is a regulusor a pseudoregulus.
OUTLINE
INTRODUCTION: FINITE PROJECTIVE GEOMETRY
FIELD REDUCTION AND LINEAR SETSThe equivalence of linear setsScattered linear sets and pseudoreguli
APPLICATIONSBlocking setsPseudo-arcsSemifields
CONCLUSION
CREDITS
Ball, Blokhuis, Eisfeld, Harrach, Lavrauw, Metsch, Polito,Polverino, Storme, Szonyi, Sziklai, Weiner, ...
BLOCKING SETS: DEFINITION
DEFINITIONA set of points B in PG(2,q) such that every line contains atleast 1 point of B is a blocking set.
MINIMAL BLOCKING SETSA blocking set B in PG(n,q) is called minimal if no propersubset of B is a blocking set.
BLOCKING SETS: DEFINITION
DEFINITIONA set of points B in PG(2,q) such that every line contains atleast 1 point of B is a blocking set.
MINIMAL BLOCKING SETSA blocking set B in PG(n,q) is called minimal if no propersubset of B is a blocking set.
EXAMPLES
A line: q + 1 points
A projective triangle in PG(2,q), q odd: 3(q + 1)/2 well-chosenpointsA Baer subplane PG(2,
√q), q square: q +
√q + 1 points.
SMALL BLOCKING SETSA blocking set B in PG(2,q) is called small if |B| < 3(q + 1)/2.
EXAMPLES
A line: q + 1 pointsA projective triangle in PG(2,q), q odd: 3(q + 1)/2 well-chosenpoints
A Baer subplane PG(2,√
q), q square: q +√
q + 1 points.
SMALL BLOCKING SETSA blocking set B in PG(2,q) is called small if |B| < 3(q + 1)/2.
EXAMPLES
A line: q + 1 pointsA projective triangle in PG(2,q), q odd: 3(q + 1)/2 well-chosenpointsA Baer subplane PG(2,
√q), q square: q +
√q + 1 points.
SMALL BLOCKING SETSA blocking set B in PG(2,q) is called small if |B| < 3(q + 1)/2.
EXAMPLES
A line: q + 1 pointsA projective triangle in PG(2,q), q odd: 3(q + 1)/2 well-chosenpointsA Baer subplane PG(2,
√q), q square: q +
√q + 1 points.
SMALL BLOCKING SETSA blocking set B in PG(2,q) is called small if |B| < 3(q + 1)/2.
BLOCKING SETS: RESULTS
THEOREM [A. BLOKHUIS (1994)]A small minimal blocking set in PG(2,p), p prime, is a line.
THEOREM [T. SZONYI (1997)]A small minimal blocking set in PG(2,p2), p prime, is a line or aBaer subplane.
BLOCKING SETS: RESULTS
THEOREM [A. BLOKHUIS (1994)]A small minimal blocking set in PG(2,p), p prime, is a line.
THEOREM [T. SZONYI (1997)]A small minimal blocking set in PG(2,p2), p prime, is a line or aBaer subplane.
BLOCKING SETS: RESULTS
THEOREM [O. POLVERINO(1998)]A small minimal blocking set in PG(2,p3), p prime, is a line or isprojectively equivalent to(x , xp,1)|x ∈ Fp3 ∪ (x , xp,0)|x ∈ Fp3 or(x , x + xp + xp2
,1)|x ∈ Fp3 ∪ (x , x + xp + xp2,0)|x ∈ Fp3.
REMARKS
I Either p3 + p2 + p + 1 points or p3 + p2 + 1 points.I of Rédei-type: there is a line with |B| − p3 points of the
blocking set B.
BLOCKING SETS: RESULTS
THEOREM [O. POLVERINO(1998)]A small minimal blocking set in PG(2,p3), p prime, is a line or isprojectively equivalent to(x , xp,1)|x ∈ Fp3 ∪ (x , xp,0)|x ∈ Fp3 or(x , x + xp + xp2
,1)|x ∈ Fp3 ∪ (x , x + xp + xp2,0)|x ∈ Fp3.
REMARKS
I Either p3 + p2 + p + 1 points or p3 + p2 + 1 points.
I of Rédei-type: there is a line with |B| − p3 points of theblocking set B.
BLOCKING SETS: RESULTS
THEOREM [O. POLVERINO(1998)]A small minimal blocking set in PG(2,p3), p prime, is a line or isprojectively equivalent to(x , xp,1)|x ∈ Fp3 ∪ (x , xp,0)|x ∈ Fp3 or(x , x + xp + xp2
,1)|x ∈ Fp3 ∪ (x , x + xp + xp2,0)|x ∈ Fp3.
REMARKS
I Either p3 + p2 + p + 1 points or p3 + p2 + 1 points.I of Rédei-type: there is a line with |B| − p3 points of the
blocking set B.
BLOCKING SETS: RESULTS
THEOREM [P. POLITO, O. POLVERINO (1999)]There exists a small minimal blocking set in PG(2,ph), p prime,h > 3, that is not of Rédei-type.
The constructed blocking sets are linear sets.
BLOCKING SETS: RESULTS
THEOREM [P. POLITO, O. POLVERINO (1999)]There exists a small minimal blocking set in PG(2,ph), p prime,h > 3, that is not of Rédei-type.
The constructed blocking sets are linear sets.
EXAMPLE: THE CONSTRUCTION OF (Fq-LINEAR)BLOCKING SETS IN PG(2,q3)
I D : Desarguesian 2-spread of PG(8,q)
I π : a 3-dimensional subspace of PG(8,q)
Then B(π) is an Fq-linear set blocking set of PG(2,q3).
EXAMPLE: THE CONSTRUCTION OF (Fq-LINEAR)BLOCKING SETS IN PG(2,q3)
I D : Desarguesian 2-spread of PG(8,q)
I π : a 3-dimensional subspace of PG(8,q)
Then B(π) is an Fq-linear set blocking set of PG(2,q3).
EXAMPLE: THE CONSTRUCTION OF (Fq-LINEAR)BLOCKING SETS IN PG(2,q3)
I Line of PG(2,q3)→ a 5-dimensional subspace of PG(8,q)
I A 3-space and a 5-space in PG(8,q) always meet.
The linear blocking set B(π) is minimal and small.
EXAMPLE: THE CONSTRUCTION OF (Fq-LINEAR)BLOCKING SETS IN PG(2,q3)
I Line of PG(2,q3)→ a 5-dimensional subspace of PG(8,q)
I A 3-space and a 5-space in PG(8,q) always meet.
The linear blocking set B(π) is minimal and small.
THE LINEARITY CONJECTURE
CONJECTURE [P. SZIKLAI (‘2008’)]All small minimal blocking sets in PG(2,q), q = ph, p prime, areFp-linear sets.
REMARKThe conjecture is stated more generally for multiple blockingsets with respect to k -spaces in PG(n,q).
THE LINEARITY CONJECTURE
CONJECTURE [P. SZIKLAI (‘2008’)]All small minimal blocking sets in PG(2,q), q = ph, p prime, areFp-linear sets.
REMARKThe conjecture is stated more generally for multiple blockingsets with respect to k -spaces in PG(n,q).
A REDUCTION THEOREM
THEOREM [G. VDV]If the linearity conjecture holds for planar blocking sets, it holdsfor blocking sets with respect to k -spaces in PG(n,q), q = ph,p ≥ h + 11.
The linearity conjecture for blocking sets in PG(2,ph), h ≥ 4 iswide open!
SOME CIRCUMSTANTIAL EVIDENCE
I Theorem [Szonyi (1997)]: A line meets a small minimalblocking set in PG(2,ph) in 1 mod p points.
I Theorem [Sziklai (2008)]: if a line meets a small minimalblocking set in 1 mod pe points, then e|h and thisintersection is an Fpe -subline.
ESSENTIAL OPEN PROBLEM (ON LINEAR SETS)Let B be a set of points such that every line meets B in a linearset. Is B itself a linear set?
The linearity conjecture for blocking sets in PG(2,ph), h ≥ 4 iswide open!
SOME CIRCUMSTANTIAL EVIDENCE
I Theorem [Szonyi (1997)]: A line meets a small minimalblocking set in PG(2,ph) in 1 mod p points.
I Theorem [Sziklai (2008)]: if a line meets a small minimalblocking set in 1 mod pe points, then e|h and thisintersection is an Fpe -subline.
ESSENTIAL OPEN PROBLEM (ON LINEAR SETS)Let B be a set of points such that every line meets B in a linearset. Is B itself a linear set?
The linearity conjecture for blocking sets in PG(2,ph), h ≥ 4 iswide open!
SOME CIRCUMSTANTIAL EVIDENCE
I Theorem [Szonyi (1997)]: A line meets a small minimalblocking set in PG(2,ph) in 1 mod p points.
I Theorem [Sziklai (2008)]: if a line meets a small minimalblocking set in 1 mod pe points, then e|h and thisintersection is an Fpe -subline.
ESSENTIAL OPEN PROBLEM (ON LINEAR SETS)Let B be a set of points such that every line meets B in a linearset. Is B itself a linear set?
The linearity conjecture for blocking sets in PG(2,ph), h ≥ 4 iswide open!
SOME CIRCUMSTANTIAL EVIDENCE
I Theorem [Szonyi (1997)]: A line meets a small minimalblocking set in PG(2,ph) in 1 mod p points.
I Theorem [Sziklai (2008)]: if a line meets a small minimalblocking set in 1 mod pe points, then e|h and thisintersection is an Fpe -subline.
ESSENTIAL OPEN PROBLEM (ON LINEAR SETS)Let B be a set of points such that every line meets B in a linearset. Is B itself a linear set?
BASIC IDEA
QUESTION
I When is a set of (t − 1)-dimensional subspaces inPG(rt − 1,q) the image of a set of points of PG(r − 1,qt )under field reduction?
I Start with a set that has the combinatorial properties of an‘interesting’ point set of PG(r − 1,qt ) after field reduction.
I Arcs, ovals, elliptic quadric,...
BASIC IDEA
QUESTION
I When is a set of (t − 1)-dimensional subspaces inPG(rt − 1,q) the image of a set of points of PG(r − 1,qt )under field reduction?
I Start with a set that has the combinatorial properties of an‘interesting’ point set of PG(r − 1,qt ) after field reduction.
I Arcs, ovals, elliptic quadric,...
BASIC IDEA
QUESTION
I When is a set of (t − 1)-dimensional subspaces inPG(rt − 1,q) the image of a set of points of PG(r − 1,qt )under field reduction?
I Start with a set that has the combinatorial properties of an‘interesting’ point set of PG(r − 1,qt ) after field reduction.
I Arcs, ovals, elliptic quadric,...
ARCS, (HYPER)OVALS AND CONICS IN PROJECTIVE
PLANES
A k -arc in a projective plane is a set of k points, no three ofwhich are collinear.
EASY TO SEEAn arc in a projective plane of order n has at most n + 2 points.
THEOREMIf n is odd, an arc has at most n + 1 points.If n is even, every (n + 1)-arc can be extended to an (n + 2)-arc.
Oval: (n + 1)-arc in a projective plane of order n.Hyperoval: (n + 2)-arc in a projective plane of order n.
ARCS, (HYPER)OVALS AND CONICS IN PROJECTIVE
PLANES
A k -arc in a projective plane is a set of k points, no three ofwhich are collinear.
EASY TO SEEAn arc in a projective plane of order n has at most n + 2 points.
THEOREMIf n is odd, an arc has at most n + 1 points.If n is even, every (n + 1)-arc can be extended to an (n + 2)-arc.
Oval: (n + 1)-arc in a projective plane of order n.Hyperoval: (n + 2)-arc in a projective plane of order n.
ARCS, (HYPER)OVALS AND CONICS IN PROJECTIVE
PLANES
A k -arc in a projective plane is a set of k points, no three ofwhich are collinear.
EASY TO SEEAn arc in a projective plane of order n has at most n + 2 points.
THEOREMIf n is odd, an arc has at most n + 1 points.
If n is even, every (n + 1)-arc can be extended to an (n + 2)-arc.
Oval: (n + 1)-arc in a projective plane of order n.Hyperoval: (n + 2)-arc in a projective plane of order n.
ARCS, (HYPER)OVALS AND CONICS IN PROJECTIVE
PLANES
A k -arc in a projective plane is a set of k points, no three ofwhich are collinear.
EASY TO SEEAn arc in a projective plane of order n has at most n + 2 points.
THEOREMIf n is odd, an arc has at most n + 1 points.If n is even, every (n + 1)-arc can be extended to an (n + 2)-arc.
Oval: (n + 1)-arc in a projective plane of order n.Hyperoval: (n + 2)-arc in a projective plane of order n.
ARCS, (HYPER)OVALS AND CONICS IN PROJECTIVE
PLANES
A k -arc in a projective plane is a set of k points, no three ofwhich are collinear.
EASY TO SEEAn arc in a projective plane of order n has at most n + 2 points.
THEOREMIf n is odd, an arc has at most n + 1 points.If n is even, every (n + 1)-arc can be extended to an (n + 2)-arc.
Oval: (n + 1)-arc in a projective plane of order n.
Hyperoval: (n + 2)-arc in a projective plane of order n.
ARCS, (HYPER)OVALS AND CONICS IN PROJECTIVE
PLANES
A k -arc in a projective plane is a set of k points, no three ofwhich are collinear.
EASY TO SEEAn arc in a projective plane of order n has at most n + 2 points.
THEOREMIf n is odd, an arc has at most n + 1 points.If n is even, every (n + 1)-arc can be extended to an (n + 2)-arc.
Oval: (n + 1)-arc in a projective plane of order n.Hyperoval: (n + 2)-arc in a projective plane of order n.
OVALS: THE DESARGUESIAN CASE
THEOREM [B. SEGRE (1954)]Every oval in PG(2,q), q odd, is a conic.
If q is even: other examples.
OVALS: THE DESARGUESIAN CASE
THEOREM [B. SEGRE (1954)]Every oval in PG(2,q), q odd, is a conic.If q is even: other examples.
PSEUDO-ARCS
RECALLAn arc A in PG(2,q) is a set of points such that any three pointsof A span PG(2,q).
DEFINITIONA pseudo-arc A in PG(3n− 1,q) is a set of (n− 1)-spaces suchthat any three elements of A span PG(3n − 1,q).
In other words, a pseudo-arc has the combinatorial propertiesof a field reduced arc of PG(2,qn).
QUESTION
How many elements can a pseudo-arc have?
PSEUDO-ARCS
RECALLAn arc A in PG(2,q) is a set of points such that any three pointsof A span PG(2,q).
DEFINITIONA pseudo-arc A in PG(3n− 1,q) is a set of (n− 1)-spaces suchthat any three elements of A span PG(3n − 1,q).
In other words, a pseudo-arc has the combinatorial propertiesof a field reduced arc of PG(2,qn).
QUESTION
How many elements can a pseudo-arc have?
PSEUDO-ARCS
RECALLAn arc A in PG(2,q) is a set of points such that any three pointsof A span PG(2,q).
DEFINITIONA pseudo-arc A in PG(3n− 1,q) is a set of (n− 1)-spaces suchthat any three elements of A span PG(3n − 1,q).
In other words, a pseudo-arc has the combinatorial propertiesof a field reduced arc of PG(2,qn).
QUESTION
How many elements can a pseudo-arc have?
PSEUDO-ARCS
RECALLAn arc A in PG(2,q) is a set of points such that any three pointsof A span PG(2,q).
DEFINITIONA pseudo-arc A in PG(3n− 1,q) is a set of (n− 1)-spaces suchthat any three elements of A span PG(3n − 1,q).
In other words, a pseudo-arc has the combinatorial propertiesof a field reduced arc of PG(2,qn).
QUESTION
How many elements can a pseudo-arc have?
PSEUDO-ARCS
EASY TO SEEA pseudo-arc in PG(3n − 1,q) has at most qn + 2 elements.
Projecting a pseudo-arc fromone element E :a partial spread S(E) of(n− 1)-spaces in PG(2n− 1,q)
→ at most (q2n−1)/(q−1)(qn−1)/(q−1) + 1
elements.
PSEUDO-ARCS
EASY TO SEEA pseudo-arc in PG(3n − 1,q) has at most qn + 2 elements.
Projecting a pseudo-arc fromone element E :a partial spread S(E) of(n− 1)-spaces in PG(2n− 1,q)
→ at most (q2n−1)/(q−1)(qn−1)/(q−1) + 1
elements.
PSEUDO-ARCS
EASY TO SEEA pseudo-arc in PG(3n − 1,q) has at most qn + 2 elements.
Projecting a pseudo-arc fromone element E :a partial spread S(E) of(n− 1)-spaces in PG(2n− 1,q)
→ at most (q2n−1)/(q−1)(qn−1)/(q−1) + 1
elements.
PSEUDO-ARCS
RECALLIf n is odd, an arc in a projective plane of order n has at mostn + 1 points.
THEOREM [J.A. THAS]If q is odd, a pseudo-arc in PG(3n − 1,q) has at most qn + 1elements.
RECALLIf n is even, an arc of size n + 1 in a projective plane of order nis uniquely extendable to an arc of size n + 2.
THEOREM [J.A. THAS]If q is even, a pseudo-arc of size qn + 1 is uniquely extendableto a pseudo-arc of size qn + 2.
PSEUDO-ARCS
RECALLIf n is odd, an arc in a projective plane of order n has at mostn + 1 points.
THEOREM [J.A. THAS]If q is odd, a pseudo-arc in PG(3n − 1,q) has at most qn + 1elements.
RECALLIf n is even, an arc of size n + 1 in a projective plane of order nis uniquely extendable to an arc of size n + 2.
THEOREM [J.A. THAS]If q is even, a pseudo-arc of size qn + 1 is uniquely extendableto a pseudo-arc of size qn + 2.
PSEUDO-ARCS
RECALLIf n is odd, an arc in a projective plane of order n has at mostn + 1 points.
THEOREM [J.A. THAS]If q is odd, a pseudo-arc in PG(3n − 1,q) has at most qn + 1elements.
RECALLIf n is even, an arc of size n + 1 in a projective plane of order nis uniquely extendable to an arc of size n + 2.
THEOREM [J.A. THAS]If q is even, a pseudo-arc of size qn + 1 is uniquely extendableto a pseudo-arc of size qn + 2.
PSEUDO-ARCS
RECALLIf n is odd, an arc in a projective plane of order n has at mostn + 1 points.
THEOREM [J.A. THAS]If q is odd, a pseudo-arc in PG(3n − 1,q) has at most qn + 1elements.
RECALLIf n is even, an arc of size n + 1 in a projective plane of order nis uniquely extendable to an arc of size n + 2.
THEOREM [J.A. THAS]If q is even, a pseudo-arc of size qn + 1 is uniquely extendableto a pseudo-arc of size qn + 2.
PSEUDO-OVALS AND PSEUDO-CONICS
TERMINOLOGYPseudo-oval: pseudo-arc of size qn + 1.
Pseudo-conic: pseudo-oval arising from a field reduced conic.
RECALLIn PG(2,q), q odd: every oval is a conic.
QUESTION
In PG(3n − 1,q), q odd: is every pseudo-oval a pseudo-conic?i.e. does every pseudo-oval arise from field reduction?
THEOREM [R. CASSE - J.A. THAS - P. R. WILD (1985)]Let O be a pseudo-oval in odd characteristic. If for one elementE of O, the partial spread S(E) extends to a Desarguesianspread, then it does so for every element and O is apseudo-conic, i.e., O arises from field reduction.
PSEUDO-OVALS AND PSEUDO-CONICS
TERMINOLOGYPseudo-oval: pseudo-arc of size qn + 1.Pseudo-conic: pseudo-oval arising from a field reduced conic.
RECALLIn PG(2,q), q odd: every oval is a conic.
QUESTION
In PG(3n − 1,q), q odd: is every pseudo-oval a pseudo-conic?i.e. does every pseudo-oval arise from field reduction?
THEOREM [R. CASSE - J.A. THAS - P. R. WILD (1985)]Let O be a pseudo-oval in odd characteristic. If for one elementE of O, the partial spread S(E) extends to a Desarguesianspread, then it does so for every element and O is apseudo-conic, i.e., O arises from field reduction.
PSEUDO-OVALS AND PSEUDO-CONICS
TERMINOLOGYPseudo-oval: pseudo-arc of size qn + 1.Pseudo-conic: pseudo-oval arising from a field reduced conic.
RECALLIn PG(2,q), q odd: every oval is a conic.
QUESTION
In PG(3n − 1,q), q odd: is every pseudo-oval a pseudo-conic?i.e. does every pseudo-oval arise from field reduction?
THEOREM [R. CASSE - J.A. THAS - P. R. WILD (1985)]Let O be a pseudo-oval in odd characteristic. If for one elementE of O, the partial spread S(E) extends to a Desarguesianspread, then it does so for every element and O is apseudo-conic, i.e., O arises from field reduction.
PSEUDO-OVALS AND PSEUDO-CONICS
TERMINOLOGYPseudo-oval: pseudo-arc of size qn + 1.Pseudo-conic: pseudo-oval arising from a field reduced conic.
RECALLIn PG(2,q), q odd: every oval is a conic.
QUESTION
In PG(3n − 1,q), q odd: is every pseudo-oval a pseudo-conic?i.e. does every pseudo-oval arise from field reduction?
THEOREM [R. CASSE - J.A. THAS - P. R. WILD (1985)]Let O be a pseudo-oval in odd characteristic. If for one elementE of O, the partial spread S(E) extends to a Desarguesianspread, then it does so for every element and O is apseudo-conic, i.e., O arises from field reduction.
PSEUDO-OVALS AND PSEUDO-CONICS
TERMINOLOGYPseudo-oval: pseudo-arc of size qn + 1.Pseudo-conic: pseudo-oval arising from a field reduced conic.
RECALLIn PG(2,q), q odd: every oval is a conic.
QUESTION
In PG(3n − 1,q), q odd: is every pseudo-oval a pseudo-conic?i.e. does every pseudo-oval arise from field reduction?
THEOREM [R. CASSE - J.A. THAS - P. R. WILD (1985)]Let O be a pseudo-oval in odd characteristic.
If for one elementE of O, the partial spread S(E) extends to a Desarguesianspread, then it does so for every element and O is apseudo-conic, i.e., O arises from field reduction.
PSEUDO-OVALS AND PSEUDO-CONICS
TERMINOLOGYPseudo-oval: pseudo-arc of size qn + 1.Pseudo-conic: pseudo-oval arising from a field reduced conic.
RECALLIn PG(2,q), q odd: every oval is a conic.
QUESTION
In PG(3n − 1,q), q odd: is every pseudo-oval a pseudo-conic?i.e. does every pseudo-oval arise from field reduction?
THEOREM [R. CASSE - J.A. THAS - P. R. WILD (1985)]Let O be a pseudo-oval in odd characteristic. If for one elementE of O, the partial spread S(E) extends to a Desarguesianspread, then it does so for every element
and O is apseudo-conic, i.e., O arises from field reduction.
PSEUDO-OVALS AND PSEUDO-CONICS
TERMINOLOGYPseudo-oval: pseudo-arc of size qn + 1.Pseudo-conic: pseudo-oval arising from a field reduced conic.
RECALLIn PG(2,q), q odd: every oval is a conic.
QUESTION
In PG(3n − 1,q), q odd: is every pseudo-oval a pseudo-conic?i.e. does every pseudo-oval arise from field reduction?
THEOREM [R. CASSE - J.A. THAS - P. R. WILD (1985)]Let O be a pseudo-oval in odd characteristic. If for one elementE of O, the partial spread S(E) extends to a Desarguesianspread, then it does so for every element and O is apseudo-conic, i.e., O arises from field reduction.
PSEUDO-OVALS AND PSEUDO-CONICS
If an arc in PG(2,q), q odd, has more than m′2(q) points, then itis uniquely extendable to a conic.
THEOREM [T. PENTTILA-G. VDV]Let A be a pseudo-arc in PG(3n − 1,q), q odd, of size morethan m′2(qn). If in one element E of A, the partial spread S(E)extends to a Desarguesian spread, then it does so in everyelement and A is uniquely extendable to a pseudo-conic, i.e., Aarises from field reduction.
PSEUDO-OVALS AND PSEUDO-CONICS
If an arc in PG(2,q), q odd, has more than m′2(q) points, then itis uniquely extendable to a conic.
THEOREM [T. PENTTILA-G. VDV]Let A be a pseudo-arc in PG(3n − 1,q), q odd, of size morethan m′2(qn).
If in one element E of A, the partial spread S(E)extends to a Desarguesian spread, then it does so in everyelement and A is uniquely extendable to a pseudo-conic, i.e., Aarises from field reduction.
PSEUDO-OVALS AND PSEUDO-CONICS
If an arc in PG(2,q), q odd, has more than m′2(q) points, then itis uniquely extendable to a conic.
THEOREM [T. PENTTILA-G. VDV]Let A be a pseudo-arc in PG(3n − 1,q), q odd, of size morethan m′2(qn). If in one element E of A, the partial spread S(E)extends to a Desarguesian spread, then it does so in everyelement
and A is uniquely extendable to a pseudo-conic, i.e., Aarises from field reduction.
PSEUDO-OVALS AND PSEUDO-CONICS
If an arc in PG(2,q), q odd, has more than m′2(q) points, then itis uniquely extendable to a conic.
THEOREM [T. PENTTILA-G. VDV]Let A be a pseudo-arc in PG(3n − 1,q), q odd, of size morethan m′2(qn). If in one element E of A, the partial spread S(E)extends to a Desarguesian spread, then it does so in everyelement and A is uniquely extendable to a pseudo-conic, i.e., Aarises from field reduction.
PSEUDO-OVOIDS AND EGGS
Similarly, a pseudo-ovoid possesses the combinatorialproperties of a field reduced elliptic quadric.
I Generalisation: eggsI The theory of eggs is equivalent to the theory of translation
generalised quadranglesI Not every pseudo-ovoid arises from field reduction! These
other pseudo-ovoids are related to semifields.
PSEUDO-OVOIDS AND EGGS
Similarly, a pseudo-ovoid possesses the combinatorialproperties of a field reduced elliptic quadric.
I Generalisation: eggs
I The theory of eggs is equivalent to the theory of translationgeneralised quadrangles
I Not every pseudo-ovoid arises from field reduction! Theseother pseudo-ovoids are related to semifields.
PSEUDO-OVOIDS AND EGGS
Similarly, a pseudo-ovoid possesses the combinatorialproperties of a field reduced elliptic quadric.
I Generalisation: eggsI The theory of eggs is equivalent to the theory of translation
generalised quadrangles
I Not every pseudo-ovoid arises from field reduction! Theseother pseudo-ovoids are related to semifields.
PSEUDO-OVOIDS AND EGGS
Similarly, a pseudo-ovoid possesses the combinatorialproperties of a field reduced elliptic quadric.
I Generalisation: eggsI The theory of eggs is equivalent to the theory of translation
generalised quadranglesI Not every pseudo-ovoid arises from field reduction! These
other pseudo-ovoids are related to semifields.
OPEN PROBLEMS
I Do all pseudo-ovals arise from field reduction?I Do all pseudo-ovoids in even characteristic arise from field
reduction?I Are all eggs pseudo-ovals or pseudo-ovoids?
CREDITS (GEOMETRIC APPROACH)
S. Ball, A. Blokhuis, G. Ebert, V. Jha, N. Johnson, W. Kantor, M.Lavrauw, G. Lunardon, G. Marino, O. Polverino, R. Trombetti, ...
FINITE SEMIFIELDS
A finite semifield S is a finite division algebra, which is notnecessarily associative
(S1) (S,+) is a finite group(S2) Left and right distributive laws hold(S3) (S, ) has no zero-divisors(S4) (S, ) has a unit
FINITE SEMIFIELDS
A finite semifield S is a finite division algebra, which is notnecessarily associative , i.e., (S,+, ) satisfying the followingaxioms:
(S1) (S,+) is a finite group
(S2) Left and right distributive laws hold(S3) (S, ) has no zero-divisors(S4) (S, ) has a unit
FINITE SEMIFIELDS
A finite semifield S is a finite division algebra, which is notnecessarily associative , i.e., (S,+, ) satisfying the followingaxioms:
(S1) (S,+) is a finite group(S2) Left and right distributive laws hold
(S3) (S, ) has no zero-divisors(S4) (S, ) has a unit
FINITE SEMIFIELDS
A finite semifield S is a finite division algebra, which is notnecessarily associative , i.e., (S,+, ) satisfying the followingaxioms:
(S1) (S,+) is a finite group(S2) Left and right distributive laws hold(S3) (S, ) has no zero-divisors
(S4) (S, ) has a unit
FINITE SEMIFIELDS
A finite semifield S is a finite division algebra, which is notnecessarily associative , i.e., (S,+, ) satisfying the followingaxioms:
(S1) (S,+) is a finite group(S2) Left and right distributive laws hold(S3) (S, ) has no zero-divisors(S4) (S, ) has a unit
FINITE SEMIFIELDS
A finite semifield S is a finite division algebra, which is notnecessarily associative , i.e., (S,+, ) satisfying the followingaxioms:
(S1) (S,+) is a finite group(S2) Left and right distributive laws hold(S3) (S, ) has no zero-divisors(S4) (S, ) has a unit
(without (S4)→ presemifield)
DEFINITIONTwo presemifields (V ,+, ) and (V ,+, ?) are said to be isotopicif there exist invertible linear transformations A,B,C : V → Vsuch that
A(x y) = B(x) ? C(y)
for all x , y ∈ V
I Every presemifield is isotopic to a semifield.I Semifields are isotopic if and only if they coordinatise
isomorphic projective planes
NUCLEI AND CENTRE
The left, middle and right nucleus are defined as
Nl = a ∈ S | (ab)c = a(bc) ∀b, c ∈ SNm = b ∈ S | (ab)c = a(bc) ∀a, c ∈ SNr = c ∈ S | (ab)c = a(bc) ∀a,b ∈ S
I S: left vector space over left nucleus, also denoted byVl(S).
I Rx : y 7→ y x is an endomorphism of Vl(S).
NUCLEI AND CENTRE
The left, middle and right nucleus are defined as
Nl = a ∈ S | (ab)c = a(bc) ∀b, c ∈ SNm = b ∈ S | (ab)c = a(bc) ∀a, c ∈ SNr = c ∈ S | (ab)c = a(bc) ∀a,b ∈ S
I S: left vector space over left nucleus, also denoted byVl(S).
I Rx : y 7→ y x is an endomorphism of Vl(S).
LINEAR SETS FROM A SEMIFIELD SI The set Rx : x ∈ S ⊂ End(Vl(S)) is an Fq-vector space
of dimension n.
⇒ Fq-linear set L(S) in PG(End(Vl(S))) = PG(l2 − 1,qn/l)of rank n.
I Since S has no zero divisors, Rx is non-singular and henceL(S) is disjoint from the (l − 2)nd secant variety of theSegre variety Sl,l(qn/l).
I Denote this secant variety by Ω.
LINEAR SETS FROM A SEMIFIELD SI The set Rx : x ∈ S ⊂ End(Vl(S)) is an Fq-vector space
of dimension n.
⇒ Fq-linear set L(S) in PG(End(Vl(S))) = PG(l2 − 1,qn/l)of rank n.
I Since S has no zero divisors, Rx is non-singular and henceL(S) is disjoint from the (l − 2)nd secant variety of theSegre variety Sl,l(qn/l).
I Denote this secant variety by Ω.
LINEAR SETS FROM A SEMIFIELD SI The set Rx : x ∈ S ⊂ End(Vl(S)) is an Fq-vector space
of dimension n.
⇒ Fq-linear set L(S) in PG(End(Vl(S))) = PG(l2 − 1,qn/l)of rank n.
I Since S has no zero divisors, Rx is non-singular and henceL(S) is disjoint from the (l − 2)nd secant variety of theSegre variety Sl,l(qn/l).
I Denote this secant variety by Ω.
LINEAR SETS FROM A SEMIFIELD SI The set Rx : x ∈ S ⊂ End(Vl(S)) is an Fq-vector space
of dimension n.
⇒ Fq-linear set L(S) in PG(End(Vl(S))) = PG(l2 − 1,qn/l)of rank n.
I Since S has no zero divisors, Rx is non-singular and henceL(S) is disjoint from the (l − 2)nd secant variety of theSegre variety Sl,l(qn/l).
I Denote this secant variety by Ω.
LINEAR SETS FROM A SEMIFIELD SI Let G denote the stabiliser of the two families of maximal
subpaces on Sl,l(qn/l).
I Let X denote the set of linear sets of rank n disjoint from Ω.
THEOREM (M. LAVRAUW (2011))There is a one-to-one correspondence between the isotopismclasses of semifields of order qn, l-dimensional over their leftnucleus and the orbits of G on the set X.
LINEAR SETS FROM A SEMIFIELD SI Let G denote the stabiliser of the two families of maximal
subpaces on Sl,l(qn/l).I Let X denote the set of linear sets of rank n disjoint from Ω.
THEOREM (M. LAVRAUW (2011))There is a one-to-one correspondence between the isotopismclasses of semifields of order qn, l-dimensional over their leftnucleus and the orbits of G on the set X.
LINEAR SETS FROM A SEMIFIELD SI Let G denote the stabiliser of the two families of maximal
subpaces on Sl,l(qn/l).I Let X denote the set of linear sets of rank n disjoint from Ω.
THEOREM (M. LAVRAUW (2011))There is a one-to-one correspondence between the isotopismclasses of semifields of order qn, l-dimensional over their leftnucleus and the orbits of G on the set X.
A GEOMETRIC VIEW ON SEMIFIELDS
I Semifields are constructed in various different ways (e.g.via planar functions)
I Study the associated linear sets to investigate whetherthey are new are not
A GEOMETRIC VIEW ON SEMIFIELDS: EXAMPLE
I. CARDINALI - O. POLVERINO - R. TROMBETTI (2006)Classification of semifields of order q4 with left nucleus Fq2 andcenter Fq.
THEIR METHODClassification of Fq-linear sets in PG(3,q2)
I disjoint from a hyperbolic quadric Q+(3,q2)
I under the action of the stabiliser of the reguli of Q+(3,q2).
A GEOMETRIC VIEW ON SEMIFIELDS: EXAMPLE
I. CARDINALI - O. POLVERINO - R. TROMBETTI (2006)Classification of semifields of order q4 with left nucleus Fq2 andcenter Fq.
THEIR METHODClassification of Fq-linear sets in PG(3,q2)
I disjoint from a hyperbolic quadric Q+(3,q2)
I under the action of the stabiliser of the reguli of Q+(3,q2).
A GEOMETRIC VIEW ON SEMIFIELDS: EXAMPLE
G. LUNARDON - G. MARINO - O. POLVERINO - R.TROMBETTI (2013)Study of semifields of pseudoregulus type.
AN APPLICATIONCharacterisation of generalised twisted fields: if
I the linear set corresponding to a semifield is ofpseudoregulus type and
I the two transversal spaces are conjugate and skew from Ω
then the semifield is (isotopic to) a generalised twisted field.
A GEOMETRIC VIEW ON SEMIFIELDS: EXAMPLE
G. LUNARDON - G. MARINO - O. POLVERINO - R.TROMBETTI (2013)Study of semifields of pseudoregulus type.
AN APPLICATIONCharacterisation of generalised twisted fields: if
I the linear set corresponding to a semifield is ofpseudoregulus type and
I the two transversal spaces are conjugate and skew from Ω
then the semifield is (isotopic to) a generalised twisted field.
OUTLINE
INTRODUCTION: FINITE PROJECTIVE GEOMETRY
FIELD REDUCTION AND LINEAR SETSThe equivalence of linear setsScattered linear sets and pseudoreguli
APPLICATIONSBlocking setsPseudo-arcsSemifields
CONCLUSION
CONCLUSION
I Linear sets are useful for the construction andcharacterisation of all kinds of objects
I Many open problems are left!