Approaching Some Problems in Finite Geometry Through Algebraic - PowerPoint PPT Presentation

Approaching Some Problems in Finite Geometry Through Algebraic Geometry Eric Moorhouse http://www.uwyo.edu/moorhouse/

Algebraic Combinatorics • finite geometry (classical and nonclassical) • association schemes • algebraic graph theory • combinatorial designs • enumerative combinatorics (à la Rota, Stanley, etc.) • much more… Use of Gröbner Bases: Conceptual vs. Computational

Outline 1. Motivation / Background from Finite Geometry 2. p -ranks 3. Computing p -ranks via the Hilbert Function 4. Open Problems

1. Motivation / Background from Finite Geometry Classical projective n-space P n F q : n +1 incidence system formed by subspaces of F q = 1-spaces points = 2-spaces lines planes = 3-spaces etc. Non-classical projective planes (2-spaces) exist but spaces of dimension ≥ 3 are classical

1. Motivation / Background from Finite Geometry An ovoid in projective 3-space P 3 F q : a set O consisting of q 2 +1 points, no three collinear. Let C be a linear [ n ,4] code over F q . If C ┴ has minimum weight ≥ 4 then n ≤ q 2 +1. When equality occurs then a generator matrix G for C has as its columns an ovoid.

1. Motivation / Background from Finite Geometry An ovoid in projective 3-space P 3 F q : a set O consisting of q 2 +1 points, no three collinear. For q odd, an ovoid is an elliptic quadric [Barlotti (1955); Panella (1955)]. When q is even the known ovoids are the elliptic quadrics, and (when q =2 2 e +1 ) the Suzuki-Tits ovoids .

1. Motivation / Background from Finite Geometry A spread in projective (2 n − 1)-space P 2 n − 1 F q : a set S consisting of q n +1 projective ( n − 1)-subspaces, partitioning the points of (2 n − 1)-space. These exist for all n and q , and give rise to translation planes (the most prolific source of non-classical projective planes).

1. Motivation / Background from Finite Geometry Classical polar spaces of orthogonal, unitary, symplectic type : projective subspaces of P n F q totally singular/isotropic with respect to the appropriate form, which induces a polarity Orthogonal polar space: nondegenerate quadric Unitary polar space: Hermitian variety Projective and polar spaces constitute the Lie incidence geometries of types A n , B n , C n , D n

1. Motivation / Background from Finite Geometry Ovoid of a polar space P : a point set O meeting every maximal subspace of P exactly once Spread of a polar space P : a partition S of the point set into maximal subspaces Many existence questions for ovoids and spreads remain open. These may be regarded as dual packing problems:

bipartite graph:

Ovoid

Spread

Hyperbolic (i.e. ruled) quadrics in P 3 F

Hyperbolic (i.e. ruled) quadrics in P 3 F have spreads S 1 S 2

Hyperbolic (i.e. ruled) quadrics in P 3 F have ovoids All real quadrics have ovoids. Some have spreads.

Projective 3-space P 3 F points lines planes

Projective 3-space P 3 F P 5 F quadric type I planes points Plücker reflection duality points lines type II planes planes

Projective 3-space P 3 F P 5 F quadric type I planes points Plücker points lines type II planes planes spread: ovoid: q 2 + 1 lines, q 2 + 1 points, pairwise disjoint no two collinear

Projective 3-space P 3 F P 5 F quadric type I planes points points lines type II planes planes spread

Projective 3-space P 3 F P 5 F quadric type I planes points points lines type II planes planes Plücker q 2 + 1 points (or spread planes), no two collinear

Projective 3-space P 3 F P 5 F quadric type I planes points points lines type II planes planes Plücker q 2 + 1 points (or spread planes), no two collinear

Projective 3-space P 3 F P 5 F quadric type I planes points points lines type II planes planes Plücker q 2 + 1 points (or spread planes), no two collinear

P 7 F quadric type I solids duality (reflection) points lines type II solids

P 7 F quadric spread type I solids triality ovoid points lines type II solids spread

Spreads Spreads of P 7 F of P 7 F quadrics quadrics Ovoids Ovoids Ovoids Ovoids Spreads Spreads Projective Projective E 8 lattice E 8 lattice of P 7 F of P 5 F of P 7 F of P 5 F in P 3 F in P 3 F planes planes quadrics quadrics quadrics quadrics

Ovoids in Ovoids in Ovoids in P 3 F q , quadrics of P 7 F q , quadrics of P 6 F q , q =2 r q =2 r q =3 r Known examples: Known examples: Known examples: • Elliptic quadrics • Examples • Examples admitting PSL (2, q 2 ) admitting PSL (3, q ) admitting PSU (3, q ) • ( r odd) Suzuki-Tits ovoids • ( r odd) Examples • ( r odd) Ree-Tits ovoids admitting 2 B 2 ( q ) admitting PSU (3, q ) admitting 2 G 2 ( q ) • ( q =8) sporadic example Code spanned by Code spanned by planes Code spanned by tangent hyperplanes has dimension q 2 +1. tangent hyperplanes to quadric to quadric Basis: p ┴ , p ∈ O has dimension q 3 +1. has dimension q 3 +1. | O | = q 2 +1 Basis: p ┴ , p ∈ O Basis: p ┴ , p ∈ O | O | = q 3 +1 | O | = q 3 +1

Ovoids in quadrics of P n F q , q = p r • always exist for n =7 and r =1 (use E 8 root lattice) [J.H. Conway et. al. (1988); M. (1993)] • do not exist for p └ n / 2 ┘ > ( p + n − 1 ) − ( p + n − 3 ) n n [Blokhuis and M. (1995)] e.g. ovoids do not exist • for n =9, p =2,3; • for n =11, p =2,3,5,7; etc. Code spanned by tangent hyperplanes to quadric has dimension [( p + n − 1 ) − ( p + n − 3 )] r + 1 n n Subcode spanned by tangent hyperplanes to putative ovoid has dimension | O | = p └ n / 2 ┘ r + 1

Ovoids in quadrics of P n F q , q = p r • always exist for n =7 and r =1 (use E 8 root lattice) [J.H. Conway et. al. (1988); M. (1993)] • do not exist for p └ n / 2 ┘ > ( p + n − 1 ) − ( p + n − 3 ) n n [Blokhuis and M. (1995)] e.g. ovoids do not exist • for n =9, p =2,3; • for n =11, p =2,3,5,7; etc. • do not exist for n =8,10,12,14,16,… [Gunawardena and M. (1997)] Similar results for ovoids on Hermitian varieties [M. (1996)]

2. p -ranks F = F q , q = p r N = ( q n +1 − 1) / ( q − 1) = number of points of P n F The code over F = F q spanned by (characteristic vectors of) hyperplanes of P n F has dimension ( p + n − 1 ) r + 1 n [Goethals and Delsarte (1968); MacWilliams and Mann (1968); Smith (1969)] Stronger information: Smith Normal Form of point-hyperplane adjacency matrix [Black and List (1990)]

2. p -ranks F = F q , q = p r N = ( q n +1 − 1) / ( q − 1) = number of points of P n F More generally, let C = C ( n , k , p , r ) be the code over F of length N spanned by projective subspaces of codimension k . Then dim C = 1 + ( coeff. of t r in tr([ I – tA ] − 1 ) ) where A is the k × k matrix with ( i , j )-entry equal to the coefficient of t pj − i in (1+ t + t 2 +…+ t p − 1 ) n +1 . Original formula for dim C due to Hamada (1968). This improved form is implicit in Bardoe and Sin (2000). Smith Normal Form: Chandler, Sin and Xiang (2006).

2. p -ranks F = F q , q = p r Q : nondegenerate quadric in P 4 F N = ( q 4 − 1) / ( q − 1) = number of points of Q C = C ( n , p , r ) = the code over F = F q of length N spanned by (characteristic vectors of) lines which lie on Q 1 + ( 1 + √ 17 ) 2 r + ( 1 − √ 17 ) 2 r , p =2 2 2 [Sastry and Sin (1996)]; 1 + p ( p +1) 2 , q = p [de Caen and M. (1998)]; 2 dim C = p ( p +1) 2 p ( p 2 − 1) √ 17, q = p r 1 + α r + β r ; α , β = ± 4 12 [Chandler, Sin and Xiang (2006)].

3. Computing p -ranks via the Hilbert Function Consider the [ N , k +1] code over F = F q spanned by (characteristic vectors of) hyperplanes of P n F . q = p r N = number of points = ( q n +1 − 1) / ( q − 1) k = ( p + n − 1 ) r n The subcode C spanned by complements of hyperplanes has dimension k . V : subset of points of P n F C V : the code of length | V | consisting of puncturing : restricting C to the points of V dim( C V ) = ?

3. Computing p -ranks via the Hilbert Function F = F q R = F [ X 0 , X 1 ,…, X n ] = ⊕ R d , R d = d -homogeneous part of R d ≥ 0 Ideal I ⊆ R q q F -rational points V = V ( I + J ), J = ( X i X j − X i X j : 0 ≤ i < j ≤ n ) I = I ( V ) ⊆ R, I d = I ∩ R d Hilbert Function h I ( d ) = dim ( R d / I d ) = no. of standard monomials of degree d , i.e. no. of monomials of degree d not in LM( I ) Case q = p : dim( C V ) = h I ( p − 1)

3. Computing p -ranks via the Hilbert Function F = F q R = F [ X 0 , X 1 ,…, X n ] = ⊕ R d , R d = d -homogeneous part of R d ≥ 0 Ideal I ⊆ R q q F -rational points V = V ( I + J ), J = ( X i X j − X i X j : 0 ≤ i < j ≤ n ) I = I ( V ) ⊆ R, I d = I ∩ R d Modified Hilbert Function : * h I ( d ) = no. of monomials of Case q = p r : Recall p 2 p the form m 0 m 1 m 2 … Lucas’ Theorem . Write such that d = d 0 + pd 1 + p 2 d 2 +… c = c 0 + pc 1 + p 2 c 2 + … ; deg( m i ) = d i and m i standard d = d 0 + pd 1 + p 2 d 2 + … . Then * dim( C V ) = h I ( p − 1) r ( d ) ≡ Π ( d i ) mod p [M. (1997)] c c i i