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

**** 2016 Fall Talks ****

 Learning seminar on K3 surfaces

1. Bilkent, 7 October 2016, Friday, 15:40

Alexander Degtyarev-[Bilkent] - Lines in K3 surfaces

 Abstract: The unifying theme of this series of talks is the classical problem of counting lines in the projective models of $K3$-surfaces of small degree. Starting with such classical results as Schur's quartic and Segre's bound (proved by Rams and Schütt) of $64$ lines in a nonsingular quartic, I will discuss briefly our recent contribution (with I. Itenberg and A. S. Sertöz), i.e., the complete classification of nonsingular quartics with many lines. There are limitless opportunities in extending and generalizing these results. First, one can switch from $\Bbb C$ to an algebraically closed field of characteristic $p>0$. Here, of course, most interesting are the so-called (Shioda) supersingular surfaces. I will discuss the properties of (quasi-)elliptic pencils on such surfaces, culminating in the classification of large configurations of lines for $p=2,3$. Alternatively, one may consider non-closed fields such as $\Bbb R$ or $\Bbb Q$. For the former, the sharp bound is $56$ real lines in a real quartic; for the latter, the current bound is $52$, and the best known example has $46$ lines. Most quartics found (in an implicit way) in our work are new'', attracting the attention of experts in the field (Rams, Schütt, Shimada, Shioda, Veniani). For example, one of them turned out an alternative nonsingular quartic model of the famous Fermat surface $\Phi_4:=\{z_0^4+z_1^4+z_2^4+z_3^4=0\},$ raising the natural question if there are other such models. An extensive search (Shimada, Shioda) returned no results, and we show that, although there are over a thousand singular models, only two models are smooth! Taking this line of research slightly further, one can classify all smooth quartic models of singular $K3$-surfaces of small discriminant, arriving at a remarkable alternative characterisation of Schur's quartic---the champion carrying $64$ lines: it is also the (only) smooth quartic of the smallest possible discriminant, which is $48$. Going even further, we can study other projective models of small degree; counting lines in these models, we arrive at the following conjectures: a smooth sextic curve in $\Bbb P^2$ has at most $72$ tritangents; a smooth sextic surface in $\Bbb P^4$ has at most $42$ lines; a smooth octic surface in $\Bbb P^5$ has at most $36$ lines. These conjectures are still wide open; I only have but a few examples. A few other sporadic problems may be mentioned in the talks: growth of the number of smooth models, hyperelliptic models, Mukai groups, explicit equations, lines in singular quartics (including the current champion with $52$ lines), etc. I hope to conclude with a brief account of the tools used in the proofs (the global Torelli theorem and surjectivity of the period map, both over $\Bbb C$ and over $p>0$, elliptic and quasi-elliptic pencils, arithmetic of integral lattices and Nikulin's theory, Niemeier lattices, etc.), raising the audience's interest in a semester long learning seminar.

2. ODTÜ, 14 October 2016, Friday, 15:40

Alexander Degtyarev-[Bilkent- Lines in K3 surfaces-II

 Abstract:  This is the continuation of last week's talk.

3. Bilkent, 21 October 2016, Friday, 15:40

Ali Sinan Sertöz-[Bilkent] - Introduction to complex K3 surfaces

 Abstract: We will start reviewing and explaining as the case might be some introductory concepts in K3 surface theory. The level will be introductory so it is a good opportunity so jump on the "wagon".

**** No talks are scheduled for 28 October 2016 Friday *****

4. ODTÜ, 4 November   2016, Friday, 15:40

Ali Sinan Sertöz-[Bilkent]  - K3 surfaces and lattices

 Abstract:We will introduce some basic concepts of lattice theory that are used to understand K3 surfaces with a view towards Torelli type theorems.

5. Bilkent, 11 November 2016, Friday, 15:40

Ali Sinan Sertöz-[Bilkent]- K3 lattice of a K3 surface

 Abstract: We will continue our series on K3 surfaces by examining the cohomology of K3 surfaces and finding out how this cohomology structure characterizes the surface.

6. ODTÜ, 18 November 2016, Friday, 15:40

Çisem Güneþ-[Bilkent] - Classification of simple quartics up to equisingular deformation-I

 Abstract: In this talk we discuss the problem of classifying complex non-special simple quartics up to equisingular deformation by reducing the problem to an arithmetical problem about lattices. On this arithmetical side, after applying Nikulin's existence theorem, our computation based on the Miranda-Morrison's theory computing the genus groups. We give a complete description of equisingular strata of non-special simple quartics. First we recall fundamentals of Nikulin's theory of discriminant forms and lattice extensions and give a brief introduction to Miranda-Morrison's theory and recast some of their results in a form more suitable for our computations. Then we recall the notion of abstract homological types and arithmetical reduction of classification problem. Finally we give ideas of the proof of our principal result.

7. Bilkent, 25 November 2016, Friday, 15:40

Çisem Güneþ-[Bilkent] - Classification of simple quartics up to equisingular deformation-II

 Abstract: This is the continuation of last week's talk.

8. ODTU, 2 December 2016, Friday, 15:40

Oðuzhan Yörük-[Bilkent] - Which K3 surfaces of Picard rank 19 cover an Enriques surface?

 Abstract: The parities of the entries of the  transcendental lattice of a K3 surface $X$ determine, in most cases, if $X$ covers an Enriques surface or not. We will summarize what is known about this problem and talk about the missing case when $\rho=19$.

9. Bilkent, 9 December 2016, Friday, 15:40

Alexander Degtyarev-[Bilkent] - Projective models of $K3$-surfaces

 Abstract: Now, that we know everything about abstract $K3$-surfaces, I will try to take a closer look at Saint-Donat's seminal paper$^\ast$ and share my findings. This paper is the foundation for all arithmetical reductions of geometric problems about projective $K3$-surfaces: it gives the conditions for an algebraic class to be (very) ample, i.e., to define a map from the $K3$-surface to a projective space, serving as the hyperplane section. If time permits, we will also discuss various properties of the maps obtained in this way: whether they are embeddings, the degree of the image, the generators of the defining ideal, etc. ($\ast$) Saint-Donat, B.  Projective models of K−3 surfaces,  Amer. J. Math.  96  (1974), 602--639.

10. ODTU, 16 December 2016, Friday, 15:40

Ali Ulaþ Özgür Kiþisel - [ODTÜ] - Arithmetic of K3 surfaces

 Abstract:  I'll try to outline some of the results in the survey paper of M. Schütt with the same title.

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

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