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

2012 Fall Talks

284. Bilkent, 21 September  2012, Friday, 15:40
     
     Alexander Degtyarev-[Bilkent] - On lines on smooth quartics
            

     Abstract:  (a never ending joint project with I. Itenberg and S. Sertoz) It is a common understanding that, thanks to the global Torelli theorem and the surjectivity of the period map, any reasonable question concerning the topology and geometry of $K3$-surfaces can be reduced to a certain arithmetical problem. We tried to apply this ideology to the study of the possible configurations of straight lines on a nonsingular quartic surface in $ \mathbb{P}^3$. According to C. Segre, a nonsingular quartic in $\mathbb{P}^3$ may contain at most 64 lines, and one explicit example of a surface with exactly 64 lines is known. The original proof, using classical algebraic geometry in the Italian school style, is very complicated. We managed to reprove Segre's result using the contemporary arithmetical approach. In addition, we prove that, up to projective equivalence, a nonsingular quartic with 64 lines is unique. Furthermore, we show that a real nonsingular quartic may contain at most 56 real lines and, conjecturally, such a quartic is also unique (although the latter statement is not quite definite yet).  Alas, the proof is transparent but heavily computer aided, the principal achievement being a stage at which my laptop can handle it in finite time (although a human still cannot).

      

      

285. ODTU, 28 September   2012, Friday, 15:40
     
     Ali Sinan Sertöz-[Bilkent] - The Chow ring of $\mathbb{G}(1,3)$
            

     Abstract:  I will start with Chapter 2 of Eisenbud-Harris notes and after a brief introduction I will describe the Chow ring of $\mathbb{G}(1,3)$, with a view toward counting the number of lines which meet four general lines in $\mathbb{P}^3$.

     
     
     

286. Bilkent, 5 October  2012, Friday, 15:40
     
     Ali Sinan Sertöz-[Bilkent]  - The Chow ring of $\mathbb{G}(1,3)$, part II
            

     Abstract:  I will continue to explore the geometry of Grassmannians, after which I will start discussing the Chow ring of $\mathbb{G}(1,3)$. I hope to have time to talk about the number of lines meeting four general lines in space.

     
     

287. ODTU, 12 October  2012, Friday, 15:40
     
     Ali Sinan Sertöz-[Bilkent]  - The Chow ring of $\mathbb{G}(1,3)$, part III
            

     Abstract:  I will  start by describing the Chow ring of  $\mathbb{G}(1,3)$ and then attack the "Keynote Questions" quoted at the beginning of the chapter.

     
     

288. Bilkent, 19 October   2012, Friday, 15:40
     
     Ali Sinan Sertöz-[Bilkent]  - The Chow ring of $\mathbb{G}(1,3)$, part IV-last!
            

     Abstract:  I will complete the multiplication table of the Chow ring of  $\mathbb{G}(1,3)$ and then attack the "Keynote Questions" quoted at the beginning of the chapter. Rain or shine, I will finish my talk series this week!

     
     

289. ODTU, 2 November  2012, Friday, 15:40
     
     Tolga Karayayla-[ODTU] - Schubert calculus, Chow ring of Grassmannians, part I
           

     Abstract: We generalize the discussion on the intersection theory on G(1,3) given in the previous seminars to the Grassmanian variety G(k,n). We are going to define the Schubert cells and cycles and discuss their properties known as Schubert Calculus. We will determine the Chow Ring A(G(k,n)) and mention some applications to intersection theory problems.

     
     

290. Bilkent, 9 November  2012, Friday, 15:40
     
     Tolga Karayayla-[ODTU] - Schubert calculus, Chow ring of Grassmannians, part II
            

     Abstract:  We generalize the discussion on the intersection theory on G(1,3) given in the previous seminars to the Grassmanian variety G(k,n). We are going to define the Schubert cells and cycles and discuss their properties known as Schubert Calculus. We will determine the Chow Ring A(G(k,n)) and mention some applications to intersection theory problems.

     
     

291. ODTU, 16 November  2012, Friday, 15:40
     
     Müfit Sezer-[Bilkent] - Invariants of the Klein four group in chracteristic two
            

     Abstract:  We consider an indecomposable representation of the Klein four group over a field of characteristic two and compute a generating set for the corresponding invariant ring up to a localization. We also obtain a homogeneous system of parameters consisting of twisted norms and show that the ideal generated by positive degree invariants is a complete intersection. (joint with J. Shank)

     
     

292. Bılkent, 23 November  2012, Friday, 15:40
     
     Emre Şen-[Bilkent] - Chow Groups of Rational Equivalence Classes of Cycles
         

     Abstract:  First we start with defining rational equivalence between two cycles. Then we define the chow group as a group of rational equivalence classes. Then we will present essential theorems and propositions which are developed at the fourth chapter (D. Eisenbud and J. Harris, All That Intersection Theory in Algebraic Geometry) to solve the keynote question  b:  "Let $L,Q\subset \mathbb{P}^3$ be a line and a nonsingular conic in $ \mathbb{P}^3$. Is $\left( \mathbb{P}^3\setminus L\right)\cong\left( \mathbb{P}^3\setminus Q\right)$ as schemes?" (ref. page 139)

      

293. Bilkent, 30 November  2012, Friday, 15:40
     
     Özgür Kişisel-[ODTU] - Tropical Intersections
       

     Abstract:  After an introductory discussion of tropical varieties, I intend to talk about tropical intersections and in particular the tropical Grassmannian.

     
     7 December 2012, Friday
     This week's seminar is cancelled due to the traffic of Docent juries taking place this week.
            
     

294. ODTU, 14 December  2012, Friday, 15:40
     
     Tolga Karayayla-[ODTU] - Schubert calculus, Chow ring of Grassmannians, part III
            

     Abstract:  We generalize the discussion on the intersection theory on G(1,3) given in the previous seminars to the Grassmanian variety G(k,n). We are going to define the Schubert cells and cycles and discuss their properties known as Schubert Calculus. We will determine the Chow Ring A(G(k,n)) and mention some applications to intersection theory problems.

      

295. ODTU, 21 December  2012, Friday, 15:40
     
     Nil Şahin-[ODTU] - Singularity Theory and Arf Rings
            

     Abstract:  Arf Closure of a local ring corresponding to a curve branch, which carries a lot of information about the branch, is an important object of study, and both Arf rings and Arf semigroups are being studied by many mathematicians, but there is not an implementable fast algorithm for constructing the Arf closure. The main aim of this work is to give an easily implementable fast algorithm for constructing the Arf closure of a given local ring. The speed of the algorithm is a result of the fact that the algorithm avoids computing the semigroup of the local ring. Moreover, in doing this, we give a bound for the conductor of the semigroup of the Arf Closure without computing the Arf Closure by using the theory of plane branches. We also give an exposition of plane algebroid curves and present the SINGULAR library written by us to compute the invariants of plane algebroid curves.

      

296. Bilkent, 28 December  2012, Friday, 15:40
     
     Mustafa Kalafat-[Tunceli] - Einstein-Hermitian 4-Manifolds of Positive Bisectional Curvature
            

     Abstract:  We show that a compact complex surface together with an Einstein-Hermitian metric of positive holomorphic bisectional curvature is biholomorphically isometric to the complex projective plane with its Fubini-Study metric up to rescaling.  (Joint work with C.Koca.)

      

      

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.