ODTÜ-BİLKENT Algebraic Geometry Seminar

(See all past talks
ordered according to speaker or date)

Refresh this page to see recent changes, if any


**** 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

  1.   ODTÜ+Zoom, 14 October 2022, Friday, 15:40

    Andrew Sutherland-[MIT] - Sato-Tate groups of abelian varieties

    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.

    In this talk I will present a complete classification of the Sato-Tate groups that can and do arise for g <= 3.

    This is joint work with Francesc Fite and Kiran Kedlaya.

     
      
  2. ODTÜ+Zoom, 21 October 2022, Friday, 15:40

    Emre Coşkun-[ODTÜ] - McKay correspondence I
       

    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.

        

  3. ODTÜ+Zoom, 4 November 2022, Friday, 15:40

    Emre Coşkun-[ODTÜ] - McKay correspondence II
        

    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.

      
      
  4. Zoom, 11 November 2022, Friday, 15:40

    Emre Can Sertöz-[Hannover] - Computing limit mixed Hodge structures
        

    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.

         

  5. Zoom, 18 November 2022, Friday, 15:40

    Müfit Sezer-[Bilkent] - Vector invariants of a permutation group over characteristic zero

    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.
    This work is joint with F. Reimers. 

      
      

  6. Zoom, 25 November 2022, Friday, 15:40

    Davide Cesare Veniani-[Stuttgart] - Non-degeneracy of Enriques surfaces

    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.

    In two joint works with G. Martin (Bonn) and G. Mezzedimi (Hannover), we have now settled this question in all characteristics by studying particular configurations of genus one fibrations, and two invariants called maximal and minimal non-degeneracy. The proof involves so-called `triangle graphs' and the distinction between special and non-special 3-sequences of half-fibers.

    In this talk, I will present the problem and explain its solution, illustrating further possible developments and applications.

         

  7. Zoom, 2 December 2022, Friday, 15:40

    Fatma Karaoğlu-[Gebze Teknik] - Smooth cubic surfaces with 15 lines
        

    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.

     
      
  8. Zoom, 9 December 2022 Friday, 15:40

    Meral Tosun-[Galatasaray] - Jets schemes and toric embedded resolution of rational triple points
        

    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.

    This is a joint work with B.Karadeniz, H. Mourtada and C.Plenat.

       
     
  9. Zoom, 16 December 2022, Friday, 15:40

    Özhan Genç-[Jagiellonian] - Finite Length Koszul Modules and Vector Bundles
        

    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