STATEMENT OF PHD THESIS PROBLEM
JON XU
Let Fq2 be the nite eld with q 2 elements, where q = pk is a prime power.
The
bar involution
: Fq2 → Fq2 is the eld automorphism dened by
k
a = ap .
Let V = Fnq2 considered as a Fq2 -vector space.
Let h·, ·i : V × V → Fq2 be the Hermitian form dened by
hv, wi = v1 w1 + v2 w2 + . . . + vn wn .
A totally
The
isotropic subspace is a vector subspace W ⊆ V
Hermitian variety is
such that
if v, w ∈ W then hv, wi = 0.
H(n − 1, q 2 ) = (P, B, I)
where
P = {p ⊆ V | p is a 1-dimensional totally isotropic subspace} ,
B = {b ⊆ V | b is a maximal dimension totally isotropic proper subspace} ,
I = {(p, b) ⊆ P × B | p ⊆ b} .
The sets P , B, and I are called points,
blocks, and incidences respectively.
An ovoid in H(n − 1, q 2 ) is a subset of points O ⊆ P such that if b ∈ B then there exists
exactly one point p ∈ O such that (p, b) ∈ I .
Known results:
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•
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Ovoids exist in H(3, q 2 ) for all q .
There is a classication theorem for ovoids in H(3, 4).
No ovoids exist in H(2n, q 2 ) for n ≥ 2.
If no ovoids exist in H(5, q 2 ) then no ovoids exist in H(2n + 1, q 2 ) for n ≥ 3.
No ovoids exist in H(5, 4) [De Beule, Metsch, 2006].
Thesis question:
Does there exist an ovoid in the Hermitian space H(5, q 2 ) for q > 2?
:
Starting point
An example of an ovoid in H(3, q 2 ) is a nondegenerate hyperplane section of a Schubert variety.
Does this fail for H(5, q 2 )?
Date
: June 26, 2015.
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JON XU
References