ODTÜ-BİLKENT Algebraic Geometry Seminar

(See all past talks ordered according to speaker)

Refresh this page to see recent changes, if any

**** 2023 Fall Talks ****

This semester we plan to have our seminars online

531.   Zoom, 13 October 2023, Friday, 15:40

     Alexander Degtyarev-[Bilkent] - Singular real plane sextic curves without real points
     

     Abstract:  (joint with Ilia Itenberg) It is a common understanding that any reasonable geometric question about $K3$-surfaces can be restated and solved in purely arithmetical terms, by means of an appropriately defined homological type. For example, this works well in the study of singular complex sextic curves in $\mathbb{P}^2$ or quartic surfaces in $\mathbb{P}^3$ (see [1,2]), as well as in that of smooth real ones (see [4,6]). However, when the two are combined (both singular and real curves or surfaces), the approach fails as the `"obvious'' concept of homological type does not fully reflect the geometry (cf., e.g., [3] or [5]). We show that the situation can be repaired if the curves in question have empty real part or, more generally, have no real singular points; then, one can indeed confine oneself to the homological types consisting of the exceptional divisors, polarization, and real structure. Still, the resulting arithmetical problem is not quite straightforward, but we manage to solve it and obtain a satisfactory classification in the case of empty real part; it matches all known results obtained by an alternative purely geometric approach. In the general case of smooth real part, we also have a formal classification; however, establishing a correspondence between arithmetic and geometric invariants (most notably, the distribution of ovals among the components of a reducible curve) still needs a certain amount of work. This project was conceived and partially completed during our joint stay at the Max-Planck-Institut für Mathematik, Bonn. The speaker is partially supported by TÜBİTAK project 123F111. REFERENCES [1]. Ayşegül Akyol and Alex Degtyarev, Geography of irreducible plane sextics, Proc. Lond. Math. Soc. (3) 111 (2015), no. 6, 13071337. MR 3447795 [2]. Çisem Güneş Aktaş, Classication of simple quartics up to equisingular deformation, Hiroshima Math. J. 47 (2017), no. 1, 87112. MR 3634263 [3]. I. V. Itenberg, Curves of degree 6 with one nondegenerate double point and groups of monodromy of nonsingular curves, Real algebraic geometry (Rennes, 1991), Lecture Notes in Math., vol. 1524, Springer, Berlin, 1992, pp. 267288. MR 1226259 [4]. V. M. Kharlamov, On the classication of nonsingular surfaces of degree 4 in $\mathbb{R}\mathbb{P}^3$ with respect to rigid isotopies, Funktsional. Anal. i Prilozhen. 18 (1984), no. 1, 4956. MR 739089 [5]. Sébastien Moriceau, Surfaces de degré 4 avec un point double non dégénéré dans l'espace projectif réel de dimension 3, Ph.D. thesis, 2004. [6]. V. V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 1, 111177, 238, English translation: Math USSR-Izv. 14 (1979), no. 1, 103167 (1980). MR 525944 (80j:10031)

      
       
     
532. Zoom, 20 October 2023, Friday, 15:40

     Turgay Akyar-[ODTÜ] - Special linear series on real trigonal curves
     

     Abstract:  For a given trigonal curve $C$,  geometric features of the Brill-Noether variety $W_d^r(C)$ parametrizing complete linear series of degree $d$ and dimension at least $r$ are well known. If the curve $C$ is real, then $W_d^r(C)$ is also defined over $\mathbb{R}$. In this talk we will see the basic properties of real linear series and discuss the topology of the real locus $W_d^r(C)(\mathbb{R})$ for some specific cases.

         
     
     
533. Zoom, 27 October 2023, Friday, 15:40
       
     İzzet Coşkun-[UIC] - Dense orbits of the PGL(n)-action on products of flag varieties
     
     

     Abstract:   It is a classical and very useful fact that any n+2 linearly general points in P^n are projectively equivalent. In this talk, I will consider generalizations of this statement to higher dimensional linear spaces. The group PGL(n) acts on products of Grassmannians or more generally flag varieties. I will discuss cases when this action has a dense orbit. This talk is based on joint work with Demir Eken, Abuzer Gündüz, Majid Hadian, Chris Yun and Dmitry Zakharov.

       
       
     
534. Zoom, 3 November 2023, Friday, 15:40
       
     Çisem Güneş Aktaş-[Abdullah Gül] - Geometry of equisingular strata of quartic surfaces with simple singularities   
     
     

     Abstract: The geometry of the equisingular strata of curves, surfaces, etc. is one of the central problems of K3-surfaces.  Thanks to the global Torelli theorem and surjectivity of the period map, the equisingular deformation classification of singular projective models of K3-surfaces with any given polarization becomes a mere computation. The most popular models studied intensively in the literature are plane sextic curves and spatial quartic surfaces. Using the arithmetical reduction, Akyol and Degtyarev [1] completed the problem of equisingular deformation classification of simple plane sextics. Simple quartic surfaces which play the same role in the realm of spatial surfaces as sextics do for curves, are a relatively new subject, promising interesting discoveries. In this talk, we discuss the problem of classifying quartic surfaces with simple singularities up to equisingular deformations by reducing the problem to an arithmetical problem about lattices. This research [3]  originates from our previous  study [2] where the classification was given only for nonspecial quartics,  in the spirit of Akyol ve Degtyarev [1]. Our principal result is extending the classification to the whole space of simple quartics and, thus, completing the equisingular deformation classification of simple quartic surfaces.            [1]  Akyol, A. ve Degtyarev, A., 2015. Geography of irreducible plane sex- tics. Proc. Lond. Math. Soc. (3), 111(6), 13071337. ISSN 0024-6115. doi:10.1112/plms/pdv053.            [2]  Güneş Aktaş, Ç, 2017. Classification of simple quartics up to equisin- gular deformation. Hiroshima Math. J., 47(1), 87112. ISSN 0018-2079. doi:10.32917/hmj/1492048849.            [3]  Güneş Aktaş, Ç, to appear in Deformation classification of quartic surfaces with simple singularities. Rev. Mat. Iberoam. doi:10.4171/RMI/1431

          
     
     
535. Zoom, 10 November 2023, Friday, 15:40
       
     Nurömür Hülya Argüz-[Georgia] - Quivers and curves in higher dimensions
     
     

     Abstract:  Quiver Donaldson-Thomas invariants are integers determined by the geometry of moduli spaces of quiver representations. I will describe a correspondence between quiver Donaldson-Thomas invariants and Gromov-Witten counts of rational curves in toric and cluster varieties. This is joint work with Pierrick Bousseau (arXiv:2302.02068 and arXiv:arXiv:2308.07270).

       
       
     
536. Zoom, 17 November 2023, Friday, 15:40
       
     Deniz Genlik-[OSU] - Holomorphic anomaly equations for $\mathbb{C}^n/\mathbb{Z}_n$
     
     

     Abstract:  In this talk, we present certain results regarding the higher genus Gromov-Witten theory of $\mathbb{C}^n/\mathbb{Z}_n$ obtained by studying its cohomological field theory structure in detail. Holomorphic anomaly equations are certain recursive partial differential equations predicted by physicists for the Gromov-Witten potential of a Calabi-Yau threefold. We prove holomorphic anomaly equations for $\mathbb{C}^n/\mathbb{Z}_n$ for any $n\geq 3$. In other words, we present a phenomenon of holomorphic anomaly equations in arbitrary dimension, a result beyond the consideration of physicists. The proof of this fact relies on showing that the Gromov-Witten potential of $\mathbb{C}^n/\mathbb{Z}_n$ lies in a certain polynomial ring. This talk is based on the joint work arXiv:2301.08389 with Hsian-Hua Tseng.

          
     
     
537. Zoom, 24 November 2023, Friday, 15:40
       
     Ali Ulaş Özgür Kişisel-[ODTÜ] - Random Algebraic Geometry and Random Amoebas
     
     

     Abstract: Random algebraic geometry studies variable properties of typical algebraic varieties as opposed to invariant properties or extremal properties. For instance, a complex algebraic projective plane curve is always topologically connected, which is an invariant property;  a real algebraic projective plane curve of degree $d$ has, by a classical theorem of Harnack, at most $g+1=(d-1)(d-2)/2+1$ connected components where $g$ denotes genus, which is an extremal property; whereas a random real algebraic projective degree $d$ plane curve in a suitable precise sense (to be explained in the talk) has an expected number of connected components of order $d$. In this talk, I will first present the setup and some of the main known results of the field of random algebraic geometry. I will then proceed to discuss some of our results on the expected properties of amoebas of random complex algebraic varieties, based on a joint work with Turgay Bayraktar, and another joint work with Jean-Yves Welschinger.

      
       
538. Zoom, 1 December 2023, Friday, 15:40
       
     Nil Şahin-[Bilkent] - Monotonicity of the Hilbert Functions of some monomial curves
     
     

     Abstract: Let $S$ be a 4-generated pseudo-symmetric semigroup generated by the positive integers $\{n_1, n_2, n_3, n_4\}$ where $\gcd(n_1, n_2, n_3, n_4) = 1$. $k$ being a field, let $k[S]$ be the corresponding semigroup ring and $I_S$ be the defining ideal of $S$. $f_*$ being the homogeneous summand of $f$, tangent cone of $S$ is $k[S]/{I_S}_*$ where ${I_S}_* =< f_*|f \in I_S >$. We will show that the  "Hilbert function of the local ring (which is isomorphic to the tangent cone) for a 4 generated pseudo-symmetric numerical semigroup $<n_1,n_2,n_3,n_4>$ is always non-decreasing when $n_1<n_2<n_3<n_4$" by an explicit Hilbert function computation.

        
      
     
539. Zoom, 8 December 2023, Friday, 15:40
       
     Kazım İlhan İkeda-[Boğaziçi] - Kapranov's higher-dimensional Langlands reciprocity principle for GL(n)
     
     
     

     Abstract:  Abelian class field theory, which describes (including the arithmetic of) all abelian extensions of local and global fields using algebraic and analytic objects related to the ground field via Artin reciprocity laws has undergone two generalizations. The first one, which is still largely conjectural, is the non-abelian class field theory of Langlands, is an extreme generalization of the abelian class field theory, describes the whole absolute Galois groups of local and global fields using automorphic objects related to the ground field via the celebrated Langlands reciprocity principles, (and more generally via functoriality principles). The second generalization is the higher-dimensional class field theory of Kato and Parshin, which describes (including the arithmetic of) all abelian extensions of higher-dimensional local fields and higher-dimensional global fields (function fields of schemes of finite type over ℤ) using this time K-groups of objects related to the ground field via Kato-Parshin reciprocity laws. So it is a very natural question to ask the possibility to construct higher-dimensional Langlands reciprocity principle. In this direction, as an answer to this question, Kapranov proposed a conjectural framework for higher-dimensional Langlands reciprocity principle for GL(n). In this talk, we plan to sketch this conjectural framework of Kapranov (where we plan to focus on the local case only).

     
     
     
540. Zoom, 15 December 2023, Friday, 15:40
       
     Alexander Degtyarev-[Bilkent] - Lines on singular quartic surfaces via Vinberg
     
     

     Abstract: Large configurations of lines (or, more generally, rational curves of low degree) on algebraic surfaces  appear in various contexts, but only in the case of cubic surfaces the picture is complete. Our principal goal is the classification of large configurations of lines on quasi-polarized K3-surfaces in the presence of singularities. To the best of our knowledge, no attempt has been made to attack this problem from the lattice-theoretical, based on the global Torelli theorem, point of view; some partial results were obtained  by various authors using ``classical'' algebraic geometry, but very little is known. The difficulty is that, given a polarized N\'eron--Severi lattice, computing the classes of smooth rational curves depends on the choice of a Weyl chamber of a certain root lattice, which is not unique. We show that this ambiguity disappears and the algorithm becomes deterministic provided that sufficiently many classes of lines are fixed. Based on this fact, Vinberg's algorithm, and a combinatorial version of elliptic pencils, we develop an algorithm that, in principle, would list all extended Fano graphs. After testing it on octic K3-surfaces, we turn to the most classical case of simple quartics where, prior to our work, only an upper bound of 64 lines (Veniani, same as in the smooth case) and an example of 52 lines (the speaker) were known. We show that, in the presence of singularities, the sharp upper bound is indeed 52, substantiating the long standing conjecture (by the speaker) that the upper bound is reduced by the presence of smooth rational curves of lower degree. We also extend the classification (I. Itenberg, A.S. Sertöz, and the speaker) of large configurations of lines on smooth quartics down to 49 lines. Remarkably, most of these configurations were known before. This project was conceived and partially completed during our joint stay at the Max-Planck-Institut f\ür Mathematik, Bonn. The speaker is partially supported by TÜBİTAK project 123F111.

ODTÜ talks are either at Hüseyin Demir Seminar room or at Gündüz İkeda seminar room at the Mathematics building of ODTÜ.
Bilkent talks are at room 141 of Faculty of Science A-building at Bilkent.
Zoom talks are online.