Algebraic Geometry Seminars

ODTÜ-BİLKENT Algebraic Geometry Seminar
(See all past talks ordered according to speaker and date)

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.

ODTÜ-BİLKENT Algebraic Geometry Seminar
(See all past talks ordered according to speaker and date)

ODTÜ-BİLKENT Algebraic Geometry Seminar
(See all past talks ordered according to speaker and date)

Refresh this page to see recent changes, if any

**** 2021 Spring Talks ****

(The New Yorker, Dec 7, 2020 Cover)

This semester we plan to have all our seminars online

Zoom, 5 February 2021, Friday, 15:40

Caner Koca-[City University of New York] - Kähler Geometry and Einstein-Maxwell Metrics

Abstract: A classical problem in Kähler Geometry is to determine a canonical representative in each Kähler class of a complex manifold. In this talk, I will introduce this problem in several well-known settings (Calabi-Yau, Kähler-Einstein, constant-scalar-curvature-Kähler, extremal Kähler). In light of recent examples and developments, I will elucidate a possible role of Einstein-Maxwell metrics in this problem.

Zoom, 12 February 2021, Friday, 15:40

Yıldıray Ozan-[ODTÜ] - Liftable homeomorphisms of finite abelian p-group regular branched covers over the 2-sphere and the projective plane

Abstract: This talk mainly is based our work joint with F. Atalan and E. Medetoğulları.

In 2017 Ghaswala and Winarski classified finite cyclic regular branched coverings of the 2-sphere, where every homeomorphism of the base (preserving the branch locus) lifts to a homeomorphism of the covering surface, answering a question of Birman and Hilden. In this talk, we will present generalizations of this result in two directions. First, we will replace finite cyclic groups with finite abelian p-groups. Second, we will replace the base surface with the real projective plane.

The main tool is the algebraic characterization of such coverings in terms of the automorphism groups of these finite abelian p-groups. Due to computational insufficiencies we have complete results only for groups of rank 1 and 2.

In particular, we prove that for a regular branched $A$ -covering $π : Σ \to S^{2}$ , where $A = Z_{p^{r}} \times Z_{p^{t}}, 1 \leq r \leq t$ , all homeomorphisms $f : S^{2} → S^{2}$ lift to those of $Σ$ , if and only if $t = r$ or $t = r + 1$ and $p = 3$ .

If time permits we will also present some applications to automorphisms of Riemann surfaces.

Zoom, 19 February 2021, Friday, 15:40

Meral Tosun-[Galatasaray] - A new root system and free divisors

Abstract: In this talk, we will construct a root system for the minimal resolution graph of some surface singularities and we will show that the new roots give linear free divisors.

Zoom, 26 February 2021, Friday, 15:40

Tony Scholl-[Cambridge] - Plectic structures on locally symmetric varieties

Abstract: In this talk I will discuss a class of locally symmetric complex varieties whose cohomology seems to behave as if they are products, even though they are not. This has geometric and number-theoretic consequences which I will describe.
This is joint work with Jan Nekovář (Paris).

Zoom, 5 March 2021, Friday, 15:40

Alexander Degtyarev-[Bilkent] - 800 conics in a smooth quartic surface

Abstract: Generalizing Bauer, define $N_{2 n} (d)$ as the maximal number of smooth rational curves of degree $d$ that can lie in a smooth degree- $2 n$ K3-surface in $P n + 1$ . (All varieties are over $C$ .) The bounds $N_{2 n} (1)$ have a long history and currently are well known, whereas for $d = 2$ the only known value is $N_{6} (2) = 285$ (my recent result reported in this seminar). In the most classical case $2 n = 4$ (spatial quartics), the best known examples have 352 or 432 conics (Barth and Bauer), whereas the best known upper bound is 5016 (Bauer with a reference to Strømme).

For $d = 1$ , the extremal configurations (for various values of $n$ ) tend to exhibit similar behavior. Hence, contemplating the findings concerning sextic surfaces, one may speculate that -- it is easier to count *all* conics, both irreducible and reducible, but -- nevertheless, in extremal configurations all conics are irreducible. On the other hand, famous Schur's quartic (the one on which the maximum $N_{4} (1)$ is attained) has 720 conics (mostly reducible), suggesting that 432 should be far from the maximum $N_{4} (2)$ . Therefore, in this talk I suggest a very simple (although also implicit) construction of a smooth quartic with 800 irreducible conics.

The quartic found is Kummer in the sense of Barth and Bauer: it contains 16 disjoint conics. I conjecture that $N_{4} (2) = 800$ and, moreover, 800 is the sharp upper bound on the total number of conics (irreducible or reducible) in a smooth spatial quartic.

Zoom, 12 March 2021, Friday, 15:40

Anar Dosi-[ODTU-Northern Cyprus] - Algebraic spectral theory and index of a variety

Abstract: The present talk is devoted to an algebraic treatment of the joint spectral theory within the framework of Noetherian modules over an algebra finite extension of an algebraically closed field. We discuss the spectral mapping theorem and analyse the index of tuples in purely algebraic case. The index function over tuples from the coordinate ring of a variety is naturally extended up to a numerical Tor-polynomial which behaves as the Hilbert polynomial and provides a link between the index and dimension of a variety.

Zoom, 19 March 2021, Friday, 15:40

Remziye Arzu Zabun-[Gaziantep] - Topology of Real Schläfli Six-Line Configurations on Cubic Surfaces and in ${R P}^{3}$

Abstract: A famous configuration of 27 lines on a non-singular cubic surface in $C P 3$ contains remarkable subconfigurations, and in particular the ones formed by six pairwise disjoint lines. We will discuss such six-line configurations in the case of real cubic surfaces from topological viewpoint, as configurations of six disjoint lines in the real projective 3-space, and show that the condition that they lie on a cubic surface implies a very special property which distinguishes them in the Mazurovskii list of 11 deformation types of configurations formed by six disjoint lines in $R P 3$ .
This is joint work with Sergey Finashin.

Zoom, 26 March 2021, Friday, 16:00

Türkü Özlüm Çelik-[Simon Fraser University] - Integrable Systems in Symbolic, Numerical and Combinatorial Algebraic Geometry

Abstract: The Kadomtsev-Petviashvili (KP) equation is a universal integrable system that describes nonlinear waves. It is known that algebro-geometric approaches to the KP equation provide solutions coming from a complex algebraic curve, in terms of the Riemann theta function associated with the curve. Reviewing this relation, I will introduce an algebraic object and discuss its algebraic and geometric features: the so-called Dubrovin threefold of an algebraic curve, which parametrizes the solutions. Mentioning the relation of this threefold with the classical algebraic geometry problem, namely the Schottky problem, I will report a procedure that is via the threefold and based on numerical algebraic geometric tools, which can be used to deal with the Schottky problem from the lens of computations. I will finally focus on the geometric behaviour of the threefold when the underlying curve degenerates.

Zoom, 2 April 2021, Friday, 15:40

Özhan Genç-[Jagiellonian] - Instanton Bundles on $P^{1} \times F_{1}$

Abstract: A $μ$ -stable vector bundle $E$ of rank 2 with $c_{1} (E) = 0$ on $P_{C}^{3}$ is called mathematical instanton bundle if $H^{1} (P^{3}, E (- 2)) = 0$ . In this talk, we will study the definiton of mathematical instanton bundles on Fano 3-folds and the construction of them on $P^{1} \times F_{1}$ where $F 1$ is the del Pezzo surface of degree 8. This talk is based on the joint work with Vincenzo Antonelli and Gianfranco Casnati.

Zoom, 9 April 2021, Friday, 15:40

Berrin Şentürk-[TEDU] - Free Group Action on Product of 3 Spheres

Abstract: A long-standing Rank Conjecture states that if an elementary abelian $p$ -group acts freely on a product of spheres, then the rank of the group is at most the number of spheres in the product. We will discuss the algebraic version of the Rank Conjecture given by Carlsson for a differential graded module $M$ over a polynomial ring. We will state a stronger conjecture concerning varieties of square-zero upper triangular matrices corresponding to the differentials of certain modules. By the work on free flags in $M$ introduced by Avramov, Buchweitz, and Iyengar, we will obtain some restriction on the rank of submodules of these matrices. By this argument we will show that $(Z / 2 Z)^{4}$ cannot act freely on product of $3$ spheres of any dimensions.

Big Blue Button, 16 April 2021, Friday, 15:40

Yankı Lekili-[Imperial College London] - A panorama of Mirror Symmetry

Abstract: Mirror symmetry is one of the most striking developments in modern mathematics whose scope extends to very different fields of pure mathematics. It predicts a broad correspondence between two subfields of geometry - symplectic geometry and algebraic geometry. Homological mirror symmetry uses the language of triangulated categories to give a mathematically precise meaning to this correspondence. Since its announcement, by Kontsevich in ICM (1994), it has attracted huge attention and over the years several important cases of it have been established. Despite significant progress, many central problems in the field remain open. After reviewing the general features, I will survey some of my recent results on mirror symmetry (with thanks to collaborators T. Perutz, A. Polishchuk, K. Ueda, D. Treumann).

Zoom, 30 April 2021, Friday, 15:40

Çisem Güneş Aktaş-[Abdullah Gül] - Real representatives of equisingular strata of projective models of K3-surfaces

Abstract: It is a wide open problem what kind of singularities a projective surface or a curve of a given degree can have. In general, this problem seems hopeless. However, in the case of K3-surfaces, the equisingular deformation classification of surfaces with any given polarization becomes a mere computation.

In this talk, we will discuss projective models of K3-surfaces of different polarizations together with the deformation classification problems. Although it is quite common that a real variety may have no real points, very few examples of equisingular deformation classes with this property are known. We will study an algorithm detecting real representatives in equisingular strata of projective models of K3-surfaces. Then, we will apply this algorithm to spatial quartics and find two new examples of real strata without real representatives where the only previously known example of this kind is in the space of plane sextics.

ODTÜ-BİLKENT Algebraic Geometry Seminar
(See all past talks ordered according to speaker and date)

Refresh this page to see recent changes, if any

**** 2021 Fall Talks ****

Funny Things All Online Learners Can Relate To |
Reader's Digest

This semester we plan to have most of our seminars online
tentatively we now list all talks as online
check for last minute changes

Zoom, 1 October 2021, Friday, 16:00

İzzet Coşkun-[UIC] - The cohomology of a general stable sheaf on a K3 surface

Abstract: In this talk, I will describe joint work with Howard Nuer and Kota Yoshioka on computing the cohomology of the general stable sheaf in a moduli space of semistable sheaves on a K3 surface of Picard rank 1. We find sharp bounds on the Mukai vector that guarantees that the cohomology can be determined from the Euler characteristic and classify all the Mukai vectors up to rank 20 where the generic sheaf does not exhibit the expected behavior. We make critical use of Bridgeland stability conditions in our computation.

Zoom, 8 October 2021, Friday, 15:40

Mesut Şahin-[Hacettepe] - Linear Codes on Subgroups of Weighted Projective Tori

Abstract: Toric varieties are interesting geometric objects lying on the crossroad of algebra, geometry and combinatorics containing a dense torus which is an algebraic group acting on the toric variety. Many champion codes obtained from toric varieties appeared in the literature.

The simplest examples of toric varieties include classical and weighted projective spaces. Parameters of linear codes obtained by evaluating rational functions on a projective torus are computed in 2011 by Sarmiento, Vaz Pinto and Villarreal. This idea is transferred to weighted projective tori and some parameters are computed in 2015 by Dias and Neves.

The purpose of this talk is to introduce some linear codes on toric varieties. We focus on codes obtained from certain subgroups of the weighted projective torus over a finite field, and to share some formulas for their parameters in some cases. We restrict to two dimensional case to obtain more explicit formula for the minimum distance of the code on the weighted projective torus T(1,1,a) over F_q.

This is a joint work with Oğuz Yayla of METU.

Zoom, 15 October 2021, Friday, 15:40

Oğuzhan Yürük-[TU-Berlin] - Nonnegativity of the polynomials supported on circuits

Abstract: A real multivariate polynomial is called nonnegative if its evaluation at any given point in R^n is nonnegative. Checking the nonnegativity of a real polynomial is a not only a mathematically challenging task, but also is an effective tool both for mathematics and for sciences. Often one uses nonnegativity certificates in order to tackle this problem, i.e., easily verifiable conditions that imply the nonnegativity for a large class of polynomials. The typical nonnegativity certificates usually make use of the fact that a polynomial is nonnegative if it is a sum of squares of polynomials (SOS polynomial), however not every nonnegative polynomial is of this form. In the first part this talk, we focus on a relatively new nonnegativity certificate based on the arithmetic and geometric means (AM-GM) inequality, and we elaborate on the fact that this class of polynomials neither contains nor is contained in the class of SOS polynomials. Unlike the SOS certificates, one is only interested in the exponents that show up in the support while working with AM-GM certificates. In particular, this gives us a framework to write sufficient symbolic conditions for the nonnegativity of a given sparse polynomial in terms of its coefficients. We utilize the aforementioned AM-GM framework in the second part of the talk, and present an application to a particular problem from the chemical reaction networks theory.

Zoom, 22 October 2021, Friday, 15:40

Alp Bassa-[Boğaziçi] - Curves over finite fields and error correcting codes

Abstract: Historically, questions about rational points on curves over finite fields occupy a prominent place in number theory. The introduction of the zeta functions for these curves by Artin led to an increased interest in this field, which culminated in the proof of the corresponding Riemann hypothesis by Hasse and Weil in the first half of the 20th century. After a long period, interest in this field was again reawakened in the 80's, when Goppa showed how this machinery from algebraic geometry can be used in the constructions of long codes allowing reliable communication over channels in the presence of errors. Using algebraic curves it became possible to beat the best constructions known to coding theorists and in the following decades many other applications in coding theory and cryptography followed. In this talk I will talk about recent results on the number of rational points on curves of large genus and their applications in the theory of error correcting codes.

Zoom, 5 November 2021, Friday, 15:40

Sergey Finashin-[ODTÜ] - Two kinds of real lines on real del Pezzo surfaces and invariance of their signed count

Abstract: In his classical treatise on real cubic surfaces, Segre discovered two kinds of real lines which he called elliptic and hyperbolic.

His enumeration indicated that the number of hyperbolic is greater by 3 than the number of elliptic ones independently of a real structure on the cubic surface.

However this property did not receive a conceptual explanation until recently: in a joint work with V.Kharlamov we interpreted a signed count of lines as a signed count of zeroes of some vector field in a Grassmannian (and so, it is Euler’s number of the corresponding vector bundle).

In the current work that I will present, we develop an alternative approach to counting lines on real del Pezzo surfaces $X$ of degrees 1 and 2 (a projective plane blown up at 8 or 7 generic points, respectively). The two types of real lines are distinguished by certain canonical Pin-structure on the real locus $X_{R}$ (this looks different from the approach of Segre, but is equivalent to it in the case of cubic surfaces).

A signed count of real lines is interpreted as some lattice root enumeration, which lets us prove our invariance properties for del Pezzo of degree 1 and 2, like in the case of cubic surfaces.

Zoom, 12 November 2021, Friday, 15:40

Berkan Üze-[Boğaziçi] - : A Glimpse of Noncommutative Motives

Abstract: The theory of motives was conceived as a universal cohomology theory for algebraic varieties. Today it is a vast subject systematically developed in many directions spanning algebraic geometry, arithmetic geometry, homotopy theory and higher category theory. Following ideas of Kontsevich, Tabuada and Robalo independently developed a theory of “noncommutative” motives for DG-categories (such as enhanced derived categories of schemes) which encompasses the classical theory of motives and helps assemble so-called additive invariants such as Algebraic K-Theory, Hochschild Homology and Topological Cyclic Homology into a motivic formalism in the very precise sense of the word. We will review the fundamental concepts at work, which will inevitably involve a foray into the formalism of enhanced and higher categories. We will then discuss Kontsevich’s notion of a noncommutative space and introduce noncommutative motives as “universal additive invariants” of noncommutative spaces. We will conclude by offering a brief sketch of Robalo’s construction of the noncommutative stable homotopy category, which is directly in the spirit of Voevodsky’s original construction.

Zoom, 19 November 2021, Friday, 15:40

Sadık Terzi-[ODTÜ] - Some Special Torsors and Its Relation to BMY-Inequality

Abstract:

Zoom, 26 November 2021, Friday, 15:40

Alexander Degtyarev-[Bilkent] - Conics on polarized K3-surfaces

Abstract: Generalizing Barth and Bauer, denote by $N_{2 n} (d)$ the maximal number of smooth degree $d$ rational curves that can lie on a smooth $2 n$ -polarized $K 3$ -surface $X \subset P n$ . Originally, the question was raised in conjunction with smooth spatial quartics, which are $K 3$ -surfaces.

The numbers $N_{2 n} (1)$ are well understood, whereas the only known value for $d = 2$ is $N_{6} (2) = 285$ . I will discuss my recent discoveries that support the following conjecture on the conic counts in the remaining interesting degrees.

Conjecture. One has $N_{2} (2) = 8910$ , $N_{4} (2) = 800$ , and $N_{8} (2) = 176$ .

The approach used does not distinguish (till the very last moment) between reducible and irreducible conics. However, extensive experimental evidence suggests that all conics are irreducible whenever their number is large enough.

Conjecture. There exists a bound $N_{2 n}^{*} (2) < N_{2 n} (2)$ such that, whenever a smooth $2 n$ -polarized $K 3$ -surface $X$ has more than $N_{2 n}^{*} (2)$ conics, it has no lines and, in particular, all conics on $X$ are irreducible.

We know that $249 \leq N_{6}^{*} (2) \leq 260$ is indeed well defined, and it seems feasible that $N_{2}^{*} (2) \geq 8100$ and $N_{4}^{*} (2) \geq 720$ are also defined; furthermore, conjecturally, the lower bounds above are the exact values.

Zoom, 3 December 2021, Friday, 15:40

Emre Coşkun-[ODTÜ] - An Introduction to Hall Algebras of Quivers

Abstract: In this talk, we shall define and study some basic properties of Hall algebras, and prove a theorem of Ringel on the structure of the Hall algebras of Dynkin quivers.

Zoom, 10 December 2021, Friday, 15:40

Susumu Tanabé-[Galatasaray] - Asymptotic critical values of a polynomial map

Abstract: The bifurcation locus of a polynomial map f is the smallest subset B(f) such that f realises a local trivialisation in the neighbourhood of each point of the complement to B(f).

It is known that the bifurcation locus B(f) is the union of the set of critical values f(Sing f) and the set of bifurcation values at infinity which may be non-empty and disjoint from the critical value set f(Sing f). It is a difficult task to find the bifurcation locus in the cases for a polynomial depending on more than three variables. Nevertheless, one can obtain approximations by supersets of B(f) from exploiting asymptotical regularity conditions. Jelonek and Kurdyka established an algorithm for finding a superset of B(f): the set of asymptotic critical values.

In this talk, we survey the history of the research of the bifurcation locus and discuss recent results on the asymptotic critical values.

Zoom, 17 December 2021, Friday, 15:40

Ichiro Shimada-[Hiroshima] - Computation of automorphism groups of Enriques surfaces

Abstract: By Torelli's theorem for K3 surfaces, the automorphism group of a complex Enriques surface is determined by the Hodge structure of the covering K3 surface. However, in many cases, explicit computations are very heavy and practically infeasible.
We give a method by which one can compute a finitely generated set of automorphism groups of various Enriques surfaces
and their nef cones.
This is a joint work with Simon Brandhorst.

Zoom, 24 December 2021, Friday, 15:40

Kadri İlker Berktav-[ODTÜ] - Higher structures in Einstein gravity

Abstract: This is a talk on a recent investigation about higher structures in the theory of General Relativity. The talk essentially features higher categorical constructions and their consequences in various Einstein's gravity theories. In this talk, for the sake of completeness, we shall begin with a summary of key ideas from moduli theory and derived algebraic geometry. We, indeed, overview the basics of derived algebraic geometry and its essential role in encoding the formal geometric aspects of moduli problems in physics. With this spirit, we will then investigate higher spaces and structures in various scenarios and present some of our works in this research direction.

Year			Year
1	2000 Fall Talks (1-15)	2001 Spring Talks (16-28)	2	2001 Fall Talks (29-42)	2002 Spring Talks (43-54)
3	2002 Fall Talks (55-66)	2003 Spring Talks (67-79)	4	2003 Fall Talks (80-90)	2004 Spring Talks (91-99)
5	2004 Fall Talks (100-111)	2005 Spring Talks (112-121)	6	2005 Fall Talks (122-133)	2006 Spring Talks (134-145)
7	2006 Fall Talks (146-157)	2007 Spring Talks (158-168)	8	2007 Fall Talks (169-178)	2008 Spring Talks (179-189)
9	2008 Fall Talks (190-204)	2009 Spring Talks (205-217)	10	2009 Fall Talks (218-226)	2010 Spring Talks (227-238)
11	2010 Fall Talks (239-248)	2011 Spring Talks (249-260)	12	2011 Fall Talks (261-272)	2012 Spring Talks (273-283)
13	2012 Fall Talks (284-296)	2013 Spring Talks (297-308)	14	2013 Fall Talks (309-319)	2014 Spring Talks (320-334)
15	2014 Fall Talks (335-348)	2015 Spring Talks (349-360)	16	2015 Fall Talks (361-371)	2016 Spring Talks (372-379)
17	2016 Fall Talks (380-389)	2017 Spring Talks (390-401)	18	2017 Fall Talks (402-413)	2018 Spring Talks (414-425)
19	2018 Fall Talks (426-434)	2019 Spring Talks (435-445)	20	2019 Fall Talks (446-456)	2020 Spring Talks (457-465)
21	2020 Fall Talks (467-476)	2021 Spring Talks (477-488)	22	2021 Fall Talks (478-500)

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 Spring 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

Zoom, 18 February 2022, Friday, 15:40

Deniz Kutluay-[Indiana] - Winding homology of knotoids

Abstract: Knotoids were introduced by Turaev as open-ended knot-type diagrams that generalize knots. Turaev defined a two-variable polynomial invariant of knotoids generalizing the Jones knot polynomial to knotoids. We will give a construction of a triply-graded homological invariant of knotoids categorifying the Turaev polynomial, called the winding homology. Forgetting one of the three gradings gives a generalization of the Khovanov knot homology to knotoids. We will briefly review the basics of the theory of knotoids and also explain the notion of categorification which plays an important role in contemporary knot theory -- no prior knowledge will be assumed.

Zoom, 25 February 2022, Friday, 15:40

Turgay Bayraktar-[Sabancı] - Universality results for zeros of random holomorphic sections

Abstract: In this talk, I will present some recent results on the asymptotic expansion of the Bergman kernel associated with sequences of singular Hermitian holomorphic line bundles $(L_{p}, h_{p})$ over compact Kähler manifolds. As an application, I will also present several universality results regarding the equidistribution of zeros of random holomorphic sections in this geometric setup.
The talk is based on the joint work with Dan Coman and George Marinescu.

Zoom, 4 March 2022, Friday, 15:40

Ilia Itenberg-[imj-prg] - Real enumerative invariants and their refinement

Abstract: The talk is devoted to several real and tropical enumerative problems. We suggest new invariants of the projective plane (and, more generally, of toric surfaces) that arise as results of an appropriate enumeration of real elliptic curves.
These invariants admit a refinement (according to the quantum index) similar to the one introduced by Grigory Mikhalkin in the rational case. We discuss tropical counterparts of the elliptic invariants under consideration and establish a tropical algorithm allowing one to compute them.
This is a joint work with Eugenii Shustin.

Zoom, 11 March 2022, Friday, 15:40

Alexander Degtyarev-[Bilkent] - Towards 800 conics on a smooth quartic surfaces

Abstract: This will be a technical talk where I will discuss a few computational aspects of my work in progress towards the following conjecture.

Conjecture: A smooth quartic surface in $P^{3}$ may contain at most $800$ conics.

I will suggest and compare several arithmetical reductions of the problem. Then, for the chosen one, I will discuss a few preliminary combinatorial bounds and some techniques used to handle the few cases where those bounds are not sufficient.

At the moment, I am quite confident that the conjecture holds. However, I am trying to find all smooth quartics containing 720 or more conics, in the hope to find the real quartic maximizing the number of real lines and to settle yet another conjecture (recall that we count all conics, both irreducible and reducible).

Conjecture: If a smooth quartic $X \subset P^{3}$ contains more than 720 conics, then $X$ has no lines; in particular, all conics are irreducible.

Currently, similar bounds are known only for sextic $K 3$ -surfaces in $P^{4}$ .

As a by-product, I have found a few examples of large configurations of conics that are not Barth--Bauer, i.e., do not contain
a $16$ -tuple of pairwise disjoint conics or, equivalently, are not Kummer surfaces with all 16 Kummer divisors conics.

Zoom, 18 March 2022, Friday, 15:40

Matthias Schütt-[Hannover] - Finite symplectic automorphism groups of supersingular K3 surfaces

Abstract: Automorphism groups form a classical object of study in algebraic geometry. In recent years, a special focus has been put on automorphisms of K3 surface, the most famous example being Mukai’s classification of finite symplectic automorphism groups on complex K3 surfaces. Building on work of Dolgachev-Keum, I will discuss a joint project with Hisanori Ohashi (Tokyo) extending Mukai’s results to fields positive characteristic. Notably, we will retain the close connection to the Mathieu group $M_{23}$ while realizing many larger groups compared to the complex setting.

Zoom, 25 March 2022, Friday, 15:40

Emre Can Sertöz-[Hannover] - Heights, periods, and arithmetic on curves

Abstract: The size of an explicit representation of a given rational point on an algebraic curve is captured by its canonical height. However, the canonical height is defined through the dynamics on the Jacobian and is not particularly accessible to computation. In 1984, Faltings related the canonical height to the transcendental "self-intersection" number of the point, which was recently used by van Bommel-- Holmes--Müller (2020) to give a general algorithm to compute heights. The corresponding notion for heights in higher dimensions is inaccessible to computation. We present a new method for computing heights that promises to generalize well to higher dimensions. This is joint work with Spencer Bloch and Robin de Jong.

Zoom, 1 April 2022, Friday, 15:40

Halil İbrahim Karakaş-[Başkent] - Arf Partitions of Integers

Abstract: The colection of partitions of positive integers, the collection of Young diagrams and the collection of numerical sets are in one to one correspondance with each other. Therefore any concept in one of these collections has its counterpart in the other collections. For example the concept of Arf numerical semigroup in the collection of numerical sets, gives rise to the concept of Arf partition of a positive integer in the collection of partitions. Several characterizations of Arf partitions have been given in recent works. In this talk we wil characterize Arf partitions of maximal length of positive integers.
This is a joint work with Nesrin Tutaş and Nihal Gümüşbaş from Akdeniz University.

Zoom, 8 April 2022 Friday, 15:40

Yıldıray Ozan-[ODTÜ] - Picard Groups of the Moduli Spaces of Riemann Surfaces with Certain Finite Abelian Symmetry Groups

Abstract: In 2021, H. Chen determined all finite abelian regular branched covers of the 2-sphere with the property that all homeomorphisms of the base preserving the branch set lift to the cover, extending the previous works of Ghaswala-Winarski and Atalan-Medettoğulları-Ozan. In this talk, we will present a consequence of this classification to the computation of Picard groups of moduli spaces of complex projective curves with certain symmetries. Indeed, we will use the work by K. Kordek already used by him for similar computations. During the talk we will try to explain the necessary concepts and tools following Kordek's work.

Zoom, 15 April 2022, Friday, 15:40

Ali Ulaş Özgür Kişisel-[ODTÜ] - An upper bound on the expected areas of amoebas of plane algebraic curves

Abstract:The amoeba of a complex plane algebraic curve has an area bounded above by $π^{2} d^{2} / 2$ . This is a deterministic upper bound due to Passare and Rullgard. In this talk I will argue that if the plane curve is chosen randomly with respect to the Kostlan distribution, then the expected area cannot be more than $O (d)$ . The results in the talk will be based on our joint work in progress with Turgay Bayraktar.

Zoom, 22 April 2022, Friday, 15:40

Muhammed Uludağ-[Galatasaray] - Heyula

Abstract: This talk is about the construction of a space H and its boundary on which the group PGL(2,Q) acts. The ultimate aim is to recover the action of PSL(2,Z) on the hyperbolic plane as a kind of boundary action.

Zoom, 29 April 2022, Friday, 15:40

Melih Üçer-[Yıldırım Beyazıt] - Burau Monodromy Groups of Trigonal Curves

Abstract: For a trigonal curve on a Hirzebruch surface, there are several notions of monodromy ranging from a very coarse one in S_3 to a very fine one in a certain subgroup of Aut(F_3), and one group in this range is PSL(2,Z). Except for the special case of isotrivial curves, the monodromy group (the subgroup generated by all monodromy actions) in PSL(2,Z) is a subgroup of genus-zero and conversely any genus-zero subgroup is the monodromy group of a trigonal curve (This is a result of Degtyarev).

A slightly finer notion in the same range is the monodromy in the Burau group Bu_3. The aforementioned result of Degtyarev imposes obvious restrictions on the monodromy group in this case but without a converse result. Here we show that there are additional non-obvious restrictions as well and, with these restrictions, we show the converse as well.

Year			Year
1	2000 Fall Talks (1-15)	2001 Spring Talks (16-28)	2	2001 Fall Talks (29-42)	2002 Spring Talks (43-54)
3	2002 Fall Talks (55-66)	2003 Spring Talks (67-79)	4	2003 Fall Talks (80-90)	2004 Spring Talks (91-99)
5	2004 Fall Talks (100-111)	2005 Spring Talks (112-121)	6	2005 Fall Talks (122-133)	2006 Spring Talks (134-145)
7	2006 Fall Talks (146-157)	2007 Spring Talks (158-168)	8	2007 Fall Talks (169-178)	2008 Spring Talks (179-189)
9	2008 Fall Talks (190-204)	2009 Spring Talks (205-217)	10	2009 Fall Talks (218-226)	2010 Spring Talks (227-238)
11	2010 Fall Talks (239-248)	2011 Spring Talks (249-260)	12	2011 Fall Talks (261-272)	2012 Spring Talks (273-283)
13	2012 Fall Talks (284-296)	2013 Spring Talks (297-308)	14	2013 Fall Talks (309-319)	2014 Spring Talks (320-334)
15	2014 Fall Talks (335-348)	2015 Spring Talks (349-360)	16	2015 Fall Talks (361-371)	2016 Spring Talks (372-379)
17	2016 Fall Talks (380-389)	2017 Spring Talks (390-401)	18	2017 Fall Talks (402-413)	2018 Spring Talks (414-425)
19	2018 Fall Talks (426-434)	2019 Spring Talks (435-445)	20	2019 Fall Talks (446-456)	2020 Spring Talks (457-465)
21	2020 Fall Talks (467-476)	2021 Spring Talks (477-488)	22	2021 Fall Talks (478-500)	2022 Spring Talks (501-511)

ODTÜ-BİLKENT Algebraic Geometry Seminar

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

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**** 2022 Fall Talks ****

Types of Orchids: 47 Different Orchid
Varieties With Names and Pictures

This semester we plan to have most of our seminars online
tentatively we now list all talks as online
check for last minute changes

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.

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.

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

Emre Coşkun-[ODTÜ] - McKay correspondence II

Abstract: Let $G \subset S U (2)$ 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 $C o h_{\bar{G}} (P^{1})$ 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.

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.

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.

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.

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.

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.

Zoom, 16 December 2022, Friday, 15:40

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

Abstract: Let Vbe a complex vector space of dimension n≥ 2 and Kbe a subset of ⋀²V of dimension m. Denote the Koszul module by W(V,K) and its corresponding resonance variety by ℛ(V,K) . Papadima and Suciu showed that there exists a uniform bound q(n,m) such that the graded component of the Koszul module W_q(V,K)=0 for all q≥ q(n,m) and for all (V,K) satisfying ℛ(V,K)={0} . In this talk, we will determine this bound q(n,m) precisely, and find an upper bound for the Hilbert series of these Koszul modules. Then we will consider a class of Koszul modules associated to vector bundles.

Year			Year
1	2000 Fall Talks (1-15)	2001 Spring Talks (16-28)	2	2001 Fall Talks (29-42)	2002 Spring Talks (43-54)
3	2002 Fall Talks (55-66)	2003 Spring Talks (67-79)	4	2003 Fall Talks (80-90)	2004 Spring Talks (91-99)
5	2004 Fall Talks (100-111)	2005 Spring Talks (112-121)	6	2005 Fall Talks (122-133)	2006 Spring Talks (134-145)
7	2006 Fall Talks (146-157)	2007 Spring Talks (158-168)	8	2007 Fall Talks (169-178)	2008 Spring Talks (179-189)
9	2008 Fall Talks (190-204)	2009 Spring Talks (205-217)	10	2009 Fall Talks (218-226)	2010 Spring Talks (227-238)
11	2010 Fall Talks (239-248)	2011 Spring Talks (249-260)	12	2011 Fall Talks (261-272)	2012 Spring Talks (273-283)
13	2012 Fall Talks (284-296)	2013 Spring Talks (297-308)	14	2013 Fall Talks (309-319)	2014 Spring Talks (320-334)
15	2014 Fall Talks (335-348)	2015 Spring Talks (349-360)	16	2015 Fall Talks (361-371)	2016 Spring Talks (372-379)
17	2016 Fall Talks (380-389)	2017 Spring Talks (390-401)	18	2017 Fall Talks (402-413)	2018 Spring Talks (414-425)
19	2018 Fall Talks (426-434)	2019 Spring Talks (435-445)	20	2019 Fall Talks (446-456)	2020 Spring Talks (457-465)
21	2020 Fall Talks (467-476)	2021 Spring Talks (477-488)	22	2021 Fall Talks (478-500)	2022 Spring Talks (501-511)
23	2022 Fall Talks (512-520)