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

**2015 Spring Talks**

We will mainly continue our learning seminar on |

__Bilkent, 13 February 2015, Friday, 15:40__

**Davide Cesare Veniani**-**[***Leibniz University of Hanover***]**-**Lines on K3 quartic surfaces**

**Abstract:**Counting lines on surfaces of fixed degree in projective space is a topic in algebraic geometry with a long history. The fact that on every smooth cubic there are exactly 27 lines, combined in a highly symmetrical way, was already known by 19th century geometers. In 1943 Beniamino Segre stated correctly that the maximum number of lines on a smooth quartic surface over an algebraically closed field of characteristic zero is 64, but his proof was wrong. It has been corrected in 2013 by Slawomir Rams and Matthias Schütt using techniques unknown to Segre, such as the theory of elliptic fibrations. The talk will focus on the generalization of these techniques to quartics admitting isolated ADE singularities.__ODTÜ, 20 February 2015, Friday, 15:40__

**Ferruh Özbudak**-**[***ODTÜ***]**-**Perfect nonlinear and quadratic maps on finite fields and some connections to finite semifields, algebraic curves and cryptography**

**Abstract:**Let $K$ be a finite field with $q$ elements, where $q$ is odd. Let $E3$ and $E2$ be extensions of $K$ of index $3$ and $2$. We show that all*perfect nonlinear*$K$-quadratic maps from $E3$ to $E2$ are extended affine equivalent (and also CCZ-equivalent). These notions are naturally connected to finite semifields (and to finite projective planes) and to certain important functions in cryptography. The proof is based on Bezout's Theorem of algebraic curves. We also give a related non-extendability result.__Bilkent, 27 February 2015, Friday, 15:40__**Ali Sinan Sertöz**-**[***Bilkent***]**-**Belyi's Theorem-IV****Abstract:**In the previous talks, the proof of Belyi's theorem was completed modulo a finiteness criterion. In this talk we will prove that criterion. Namely, we will prove that a compact Riemann surface $S$ is defined over the algebraic numbers $\bar{\mathbb{Q}}$ if and only if the orbit of $S$ under the action of the Galois group $Gal(\mathbb{C}/\mathbb{Q})$ is finite.__ODTÜ, 6 March 2015, Friday, 15:40__**Davide Cesare Veniani**-**[***Leibniz University of Hanover***]**-

**An introduction to elliptic fibrations - part I: Singular Fibres****Abstract:**The theory of elliptic fibrations is an important tool in the study of algebraic and complex surfaces. The talk will focus on Kodaira's classification of possible singular fibres. I will construct some examples of rational and K3 elliptic surfaces to illustrate the theory, coming from pencils of plane cubics and lines on quartic surfaces.

The talk will be aimed at students who took a first course in algebraic geometry.__Bilkent, 13 March 2015, Friday, 15:40__**Davide Cesare Veniani**-**[***Leibniz University of Hanover***] -**

An introduction to elliptic fibrations - part II: Mordell-Weil group and torsion sections**Abstract:**Given an elliptic surface, the set of sections of its fibration forms a group called the Mordell-Weil group. After recalling the main concepts from part I, I will expose the main properties of this group, with a special focus on torsion sections. I will give two constructions on quartic surfaces which appear naturally in the study of the enumerative geometry of lines, where torsion sections play a prominent role.

The talk will be aimed at students who took a first course in algebraic geometry.__ODTÜ, 20 March 2015, Friday, 15:40__**Ali Sinan Sertöz**-**[***Bilkent***]**-**Belyi's Theorem-V****Abstract:**This is the last talk in our series of talks on Belyi's theorem. In this talk I will outline the proof of the fact that a compact Riemann surface $S$ is defined over the algebraic number field if and only if the orbit of $S$ under the Galois group $Gal(\mathbb{C}/\mathbb{Q})$ contains only finitely many isomorphism classes of Riemann surfaces. Once this is established, we will show that having a Belyi map for $S$ leads to the finiteness of the isomorphism classes in $\{ S^\sigma\}_{\sigma\in Gal(\mathbb{C}/\mathbb{Q})}$. This will conclude our study of the first three chapters of Girondo and Gonzalez-Diez's book.__Bilkent, 27 March 2015, Friday, 15:40__**Ali Sinan Sertöz**-**[***Bilkent***]**-**Exit Belyi, enter dessins d'enfants****Abstract:**This week I will first clarify some of the conceptual details of the proof of Belyi's theorem that were left on faith last week. After that we will start talking about dessins d'enfants.__ODTÜ, 3 April 2015, Friday, 15:40__**Ali Sinan Sertöz**-**[***Bilkent***]**-**From dessins d'enfants to Belyi pairs****Abstract:**We will describe the process of obtaining a Belyi pair starting from a dessin d'enfant.__Bilkent, 10 April 2015, Friday, 15:40__**Ali Sinan Sertöz**-**[***Bilkent***]**-**Calculating the Belyi function associated to a dessin****Abstract:**I will go over the calculation of the Belyi pair corresponding to a particular dessin given in the book, see Example 4.21. Time permitting, I will briefly talk about constructing a dessin from a Belyi pair.__ODTÜ, 17 April 2015, Friday, 15:40__**Ali Sinan Sertöz**-**[***Bilkent***]**-**From Belyi pairs to dessins****Abstract:**We will talk about obtaining a dessin from a Belyi function.__Bilkent, 24 April 2015, Friday, 15:40__**Alexander Klyachko**-**[***Bilkent***]**-**Exceptional Belyi coverings****Abstract:****(**This is a joint project with Cemile Kürkođlu.)Exceptional covering is a connected Belyi coverings uniquely determined by its ramification scheme. Well known examples are cyclic, dihedral, and Chebyshev coverings. We add to this list a new infinite series of rational exceptional coverings together with the respective Belyi functions.

We shortly discuss the minimal field of definition of a rational exceptional covering and show that it is either $\mathbb{Q}$ or its quadratic extension.

Existing theories give no upper bound on degree of the field of definition of an exceptional covering of genus 1. It is an open question whether the number of such coverings is finite or infinite.

Maple search for an exceptional covering of $g>1$ found none of degree 18 or less. Absence of exceptional hyperbolic coverings is a mystery we couldn’t explain.

__ODTÜ, 8 May 2015, Friday, 15:40__**Alexander Degtyarev**-**[***Bilkent***]**-**Dessins d'enfants and topology of algebraic curves****Abstract:**I will give a brief introduction into the very fruitful interplay between Grothendieck's dessins d'enfants, subgroups of the modular group, and topology and geometry of trigonal curves/elliptic surfaces/Lefschetz fibrations.

**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.**

2000-2001 Talks
(1-28) |
2001 Fall Talks
(29-42) |
2002 Spring Talks
(43-54) |
2002 Fall Talks
(55-66) |

2003 Spring Talks
(67-79) |
2003 Fall Talks
(80-90) |
2004 Spring Talks
(91-99) |
2004 Fall Talks (100-111) |

2005 Spring Talks
(112-121) |
2005 Fall Talks
(122-133) |
2006 Spring Talks
(134-145) |
2006 Fall Talks (146-157) |

2007 Spring Talks
(158-168) |
2007 Fall Talks
(169-178) |
2008 Spring Talks (179-189) |
2008 Fall Talks (190-204) |

2009 Spring Talks
(205-217) |
2009 Fall Talks
(218-226) |
2010
Spring Talks (227-238) |
2010
Fall Talks (239-248) |

2011
Spring Talks (249-260) |
2011
Fall Talks (261-272) |
2012
Spring Talks (273-283) |
2012
Fall Talks (284-296) |

2013
Spring Talks (297-308) |
2013
Fall Talks (309-319) |
2014
Spring Talks (320-334) |
2014
Fall Talks (335-348) |