Field reduction in finite projective geometryfq11/talks/Van de Voorde.pdf · INTRODUCTION: FINITE PROJECTIVE GEOMETRY FIELD REDUCTION AND LINEAR SETS The equivalence of linear sets

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




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


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)



NOTATION



q ) = PG(d − 1,q)



NOTATION



q ) = PG(d − 1,q)


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.


The projective planes of orders 2,3,4,5,7 and 8 are unique,but there are 4 non-isomorphic planes of order 9.









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.






a PG(2,p))?

I Easy to construct general class of not necessarilyDesarguesian planes of order ph, p prime, h > 1:translation planes.
















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).


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).
















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




CONCLUSION

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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)


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)


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.




qt

↓ PG






qt

↓ PG






qt

↓ PG






qt

↓ PG



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)



U = 〈(1,0,0), (0,1,0), (0,0,1)〉Fq ⇒


U = 〈(1,1,0), (ω, ωq,0), (ω2, ω2q,0)〉Fq ⇒


U = 〈(1,0,0), (ω,0,0), (0,1,0)〉Fq ⇒


U = 〈(1,0,0), (ω,0,0), (ω2,0,0)〉Fq ⇒




U = 〈(1,0,0), (0,1,0), (0,0,1)〉Fq ⇒


U = 〈(1,1,0), (ω, ωq,0), (ω2, ω2q,0)〉Fq ⇒


U = 〈(1,0,0), (ω,0,0), (0,1,0)〉Fq ⇒


U = 〈(1,0,0), (ω,0,0), (ω2,0,0)〉Fq ⇒




U = 〈(1,0,0), (0,1,0), (0,0,1)〉Fq ⇒


U = 〈(1,1,0), (ω, ωq,0), (ω2, ω2q,0)〉Fq ⇒


U = 〈(1,0,0), (ω,0,0), (0,1,0)〉Fq ⇒


U = 〈(1,0,0), (ω,0,0), (ω2,0,0)〉Fq ⇒


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


I a set of q2 + 1 spread elements, one intersecting π in aline, the others intersect π in a point.


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.


Rank 3: projection of a subplane

A club: q2 + 1 points.


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.




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.




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.


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.


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.


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.













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.


QUESTION


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.


QUESTION


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




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.


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 + 1 points.


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 + 1 points.


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.


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.


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.



,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.



,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.

A CONJECTURE (A. BLOKHUIS)

Are all small minimal blocking sets of Rédei-type?


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.


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).


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).


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.


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?
















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




BASIC IDEA

QUESTION




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.


PLANES






PLANES



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.



PLANES






PLANES




Oval: (n + 1)-arc in a projective plane of order n.

Hyperoval: (n + 2)-arc in a projective plane of order n.


PLANES





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




QUESTION


PSEUDO-ARCS




QUESTION


PSEUDO-ARCS




QUESTION


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



→ at most (q2n−1)/(q−1)(qn−1)/(q−1) + 1

elements.

PSEUDO-ARCS



→ 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





PSEUDO-ARCS





PSEUDO-ARCS





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.


TERMINOLOGYPseudo-oval: pseudo-arc of size qn + 1.Pseudo-conic: pseudo-oval arising from a field reduced conic.


QUESTION






QUESTION






QUESTION






QUESTION


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.




QUESTION


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.




QUESTION




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.



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.



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.



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.



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.




generalised quadrangles

I Not every pseudo-ovoid arises from field reduction! Theseother pseudo-ovoids are related to semifields.




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


(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


(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



FINITE SEMIFIELDS



(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.


of dimension n.





of dimension n.





of dimension n.




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.


subpaces on Sl,l(qn/l).I Let X denote the set of linear sets of rank n disjoint from Ω.



subpaces on Sl,l(qn/l).I Let X denote the set of linear sets of rank n disjoint from Ω.


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).


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).


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.


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




CONCLUSION

CONCLUSION

I Linear sets are useful for the construction andcharacterisation of all kinds of objects

I Many open problems are left!

Thank you for your attention!

Field reduction in finite projective geometryfq11/talks/Van de Voorde.pdf · INTRODUCTION: FINITE PROJECTIVE GEOMETRY FIELD REDUCTION AND LINEAR SETS The equivalence of linear sets

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