ODTÜ-BİLKENT Algebraic
Geometry Seminar
(See all past talks ordered according
to speaker or date)
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**** 2022 Fall Talks ****
This
semester we plan to have most of our seminars online
tentatively we now list all talks as
online
check for last minute changes
Andrew
Sutherland-[MIT] - Sato-Tate groups of abelian varieties
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Abstract: Let A be an
abelian variety of dimension g defined over a
number field K. As defined by Serre, the
Sato-Tate group ST(A) is a compact subgroup of the
unitary symplectic group USp(2g) equipped with a
map that sends each Frobenius element of the
absolute Galois group of K at primes p of good
reduction for A to a conjugacy class of ST(A)
whose characteristic polynomial is determined by
the zeta function of the reduction of A at
p. Under a set of axioms proposed by Serre
that are known to hold for g <= 3, up to
conjugacy in Usp(2g) there is a finite list of
possible Sato-Tate groups that can arise for
abelian varieties of dimension g over number
fields. Under the Sato-Tate conjecture
(which is known for g=1 when K has degree 1 or 2),
the asymptotic distribution of normalized
Frobenius elements is controlled by the Haar
measure of the Sato-Tate group. |
Emre Coşkun-[ODTÜ] - McKay correspondence I
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Abstract: John
McKay observed, in 1980, that there is a
one-to-one correspondence between the nontrivial
finite subgroups of SU(2) (up to conjugation) and
connected Euclidean graphs (other than the Jordan
graph) up to isomorphism. In these talk, we shall
first examine the finite subgroups of SU(2) and
then establish this one-to-one correspondence,
using the representation theory of finite groups.
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Emre Coşkun-[ODTÜ] - McKay correspondence II
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Abstract: Let $G
\subset SU(2)$ be a finite subgroup containing
$-I$, and let $Q$ be the corresponding Euclidean
graph. Given an orientation on $Q$, one can define
the (bounded) derived category of the
representations of the resulting quiver. Let
$\bar{G} = G / {\pm I}$. Then one can also define
the category $Coh_{\bar{G}}(\mathbb{P}^1)$ of
$\bar{G}$-equivariant coherent sheaves on the
projective line; this abelian category also has a
(bounded) derived category. In the second of these
talks dedicated to the McKay correspondence, we
establish an equivalence between the two derived
categories mentioned above. |
Emre Can
Sertöz-[Hannover] - Computing limit mixed Hodge structures
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Abstract: Consider a
smooth family of varieties over a punctured disk
that is extended to a flat family over the whole
disk, e.g., consider a 1-parameter family of
hypersurfaces with a central singular fiber. The
Hodge structures (i.e. periods) of smooth fibers
exhibit a divergent behavior as you approach the
singular fiber. However, Schmid's nilpotent orbit
theorem states that this divergence can be
"regularized" to construct a limit mixed Hodge
structure. This limit mixed Hodge structure
contains detailed information about the geometry
and arithmetic of the singular fiber. I will
explain how one can compute such limit mixed Hodge
structures in practice and give a demonstration of
my code. |
Müfit Sezer-[Bilkent]
- Vector invariants of a permutation
group over characteristic zero
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Abstract: We consider a finite
permutation group acting naturally on a vector space V over
a field k. A
well known theorem of Göbel asserts that the
corresponding ring of invariants k[V]G is
generated by invariants of degree at most dim V choose 2.
We point out that if the characteristic of k is
zero then the top degree of the vector coinvariants k[mV]G is
also bounded above by n choose 2 implying
that Göbel's bound almost holds for vector invariants
as well in characteristic zero. |
Davide Cesare Veniani-[Stuttgart]
- Non-degeneracy of Enriques surfaces
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Abstract: Enriques'
original construction of Enriques surfaces
involves a 10-dimensional family of sextic
surfaces in the projective space which are
non-normal along the edges of a tetrahedron. The
question whether all Enriques surfaces arise
through Enriques' construction has remained open
for more than a century. |
Fatma Karaoğlu-[Gebze
Teknik] - Smooth cubic surfaces
with 15 lines
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Abstract: It is
well-known that a smooth cubic surface has 27
lines over an algebraically closed field. If the
field is not closed, however, fewer lines are
possible. The next possible case is that of smooth
cubic surfaces with 15 lines. This work is a
contribution to the problem of classifying smooth
cubic surfaces with 15 lines over fields of
positive characteristic. We present an algorithm
to classify such surfaces over small finite
fields. Our classification algorithm is based on a
new normal form of the equation of a cubic surface
with 15 lines and less than 10 Eckardt points. The
case of cubic surfaces with more than 10 Eckardt
points is dealt with separately. Classification
results for fields of order at most 13 are
presented and a verification using an enumerative
formula of Das is performed. Our work is based on
a generalization of the old result due to Cayley
and Salmon that there are 27 lines if the field is
algebraically closed. |
Meral
Tosun-[Galatasaray] - Jets schemes and toric embedded resolution of
rational triple points
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Abstract: One of the
aims of J.Nash in an article on the arcs spaces
(1968) was to understand resolutions of
singularities via the arcs living on the singular
variety. He conjectured that there is a
one-to-one relation between a family of the
irreducible components of the jet schemes of an
hypersurface centered at the singular point and
the essential divisors on every resolution.
J.Fernandez de Bobadilla and M.Pe Pereira (2011)
have shown his conjecture, but the proof is not
constructive to get the resolution from the arc
space. We will construct an embedded toric
resolution of singularities of type rtp from the
irreducible components of the jet schemes. |
Özhan Genç-[Jagiellonian]
- Finite Length Koszul Modules and
Vector Bundles
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Abstract: Let V be a complex vector space of dimension n≥ 2 and K be a subset of ⋀2V of dimension m. Denote the Koszul module by W(V,K) and its corresponding resonance variety by ℛ(V,K). Pap |