Humboldt Universität zu Berlin
Mathem.-Naturwissenschaftliche Fakultät II
Institut für Mathematik
Sommersemester 2009
| 14.04.2009 | Bernd Siebert (Univ. Hamburg) |
| A toolkit for Calabi-Yau and related varieties | |
| Abstract: Toric geometry constructs rational varieties out of polytopes. While this provides a rich source of generalizations of projective space and an important link between combinatorics and algebraic geometry, toric geometry is limited to a very special class of varieties. The reason is that, in a sense, toric geometry is a linear theory. A construction developed in my joint work with Mark Gross on mirror symmetry provides a non-linear generalization of toric geometry. The starting data are “tropical manifolds“ rather than polytopes, and the result is a degeneration of varieties with effective anticanonical class. The construction is highly non-linear, for we first construct a formal scheme inductively order by order and then invoke the Grothendieck existence theorem. The induction step is governed by a scattering process on the tropical manifold involving the tropical vertex group. In the talk I will illustrate the construction by a series of increasingly complicated examples, following http://arxiv.org/abs/0808.2749 | 05.05.2009 | Alessandro Verra (Univ. Roma 3) |
| On the rationality of the moduli space of etale triple covers of genus four curves | |
| Abstract: The universal Picard variety Pic_0,4 over the moduli space of curves of genus 4 is strictly related to the moduli space of cubic surfaces S endowed with a birational morphism onto a plane. Using the geometry of cubic surfaces, and of the so called double six of lines on them, we describe the locus T in Pic_0,4 of 3-torsion elements. The rationality of the quotient of T by a certain involution is proved. | |
| Mittwoch | |
| 20.05.2009 | Remke Kloosterman (HU Berlin) |
| Point counting on singular hypersurfaces | |
| Abstract: Currently, one of the most efficient algorithms to compute the zeta function of a smooth hypersurface is Lauder's deformation method. In this talk we discuss several approaches to extend Lauder's deformation method to singular hypersurfaces. | |
| 26.05.2009 | Edoardo Sernesi (Univ. Roma 3) |
| The degree of the Luroth hypersurface according to Morley | |
| Abstract: Luroth quartics fill a hypersurface L in the space of all plane quartics. The degree of L has been computed for the first time by Morley in 1919 in a forgotten paper containing several beautiful geometrical constructions. I will explain Morley's proof, which has been reconstructed and rewritten in a joint work with G. Ottaviani. | |
| 02.06.2009 | Orsola Tommasi (Univ. Hannover) |
| On hypersurfaces containing too many lines | |
| Abstract: Let X be an n-dimensional hypersurface containing a family of lines of dimension larger than the expected one. If the degree of X is at most n+1, it was conjectured by Debarre and De Jong that X is necessarily singular. In this talk, we present a strategy for looking for singular points on X in the case in which the degree of X is at least n+1 and X contains an (n-1)-dimensional family of lines. This is a variant of the Debarre - De Jong conjecture proposed by Beheshti and Starr. (Joint work with J.M. Landsberg) | |
| 23.06.2009 | Leticia Brambila-Paz (CIMAT Guanajuato) |
| Endomorphisms and moduli for unstable bundles | |
| Abstract: For a smooth irreducible projective curve X, we describe the algebra of endomorphisms of indecomposable unstable vector bundles over X of rank 2. Fixing the Harder-Narasimhan type and the dimension of the algebra of endomorphisms, we construct the moduli scheme for such bundles. | |
| 30.06.2009 | Angela Ortega |
| Existence results on moduli spaces of coherent systems | |
| Abstract: The Brill-Noether problem for higher rank is concerned with describing the moduli space of stable vector bundles over a curve having a prescribed number of sections. One way of studying this problem is via coherent systems, that is, pairs (E,V) consisting of a vector bundle E and a subspace V of global sections subject to a stability condition. In this talk we will mention some generalities about the moduli space of coherent systems and present results on the nonemptiness of such spaces, including recent joint work with Brambila-Paz in the case when the number of sections is bigger than the rank. | |
| 07.07.2009 | Markus Brodmann (Univ. Zürich) |
| Projective varieties of low degree | |
| Abstract: Projective varieties of low degree play an important role in classical projective algebraic geometry. We study such varieties, admitting singularities and ground fields of arbitrary characteristic, unlike to the classical investigations. For an irreducible, non-degenerate variety in projective r-space one always has the inequality deg(X)>codim(X). If deg(X) = codim(X) + 1, the variety X is said to be of minimal degree. These varieties were classified already in the 19th century by Castelnuovo and Del Pezzo. Up to taking cones, they are just projective spaces, hyperquadrics, the Veronese surface in P5 or rational normal scrolls. Varieties of almost minimal degree are those with deg(X) = codim(X) + 2. Normal varieties of this type were classified by Fujita in 1990. We study a different class of varieties, which containes all non-normal varieties of almost minimal degree: simple linear projections of rational normal scrolls. The key invariant to understand these varieties is their arithmetic depth. A refined investigation leads to study the possible secant loci of a rational normal scroll, if the center of projection runs through the whole ambient space. We completely determine the corresponding secant stratification in this case. As an application, we classify the non-normal Del Pezzo varieties and the so called embedding scrolls of varieties of almost minimal degree. Finally, we we mention a few results about surfaces X with deg(X) = codim(X) + 3 and surfaces of maximal sectional regularity. (Joint work with E.Park, Seoul and P.Schenzel, Halle) | |
| 07.07.2009 | Ana-Maria Castravet (Univ. of Arizona) |
| Rational curves of minimal degree on higher Fano manifolds (joint work with Carolina Araujo) | |
| Abstract: We will discuss a notion of higher Fano variety introduced by de Jong/Starr. We will especially study what one can say about the families of minimal degree rational curves that sweep out such a higher Fano variety. Related questions: Can one classify higher Fanos? Is there an inductive structure on the collection of all higher Fano varieties? | |
| 14.07.2009 | Alessandro Chiodo (Universite Fourier Grenoble) |
| Landau-Ginzburg/Calabi-Yau correspondence for quintic three-folds via symplectic transformations | |
| Abstract: I will present work in collaboration with Yongbin Ruan (arXiv:0812.4660). We compute the recently introduced Fan-Jarvis-Ruan-Witten theory of W-curves in genus zero for quintic polynomials in five variables and we show that it matches the Gromov-Witten genus-zero theory of the quintic three-fold via a symplectic transformation. More specifically, we show that the J-function encoding the Fan-Jarvis-Ruan-Witten theory on the A-side equals via a mirror map the I-function embodying the period integrals at the Gepner point on the B-side. This identification inscribes the physical Landau-Ginzburg/Calabi-Yau correspondence within the enumerative geometry of moduli of curves, matches the genus-zero invariants computed by the physicists Huang, Klemm, and Quackenbush at the Gepner point, and yields via Givental's quantization a prediction on the relation between the full higher genus potential of the quintic three-fold and that of Fan-Jarvis-Ruan-Witten theory. |
Wintersemester 2008/09
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Rückfragen an: Christa Dobers
e-mail: dobers@math.hu-berlin.de