Algebraic Geometry Seminar

An arithmetic toric variety is a toric variety over a field k that splits over a Galois extension K/k. We classify these, up to toric isomorphism, using Galois cohomology with coefficients in the toric automorphism group. Cox's quotient construction helps to compute this cohomology set; we use this to study the different k-forms of projective space over a cyclic extension K/k. We also study k-forms of affine toric varieties and toric ideals. This is joint work in progress with Javier Elizondo, Paulo Lima-Filho, and Frank Sottile.

The Cox ring, or total coordinate ring, of an algebraic variety is the object of much recent work in both algebraic geometry and number theory. For example, the Cox rings of Del Pezzo surfaces, have been used to count points of bounded height on these surfaces and thus verify instances of a deep conjecture of Batyrev and Manin. Determining which varieties have a finitely generated Cox ring is a notoriously difficult problem, even in the case of surfaces. We will show that the class of smooth projective rational surfaces with big anticanonical class has a finitely generated Cox ring. We will also present some systematic collections of examples of these surfaces. This is joint work with D. Testa and M. Velasco.
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A tropical curve is a metric graph with possibly unbounded edges, and tropical rational functions are continuous piecewise linear functions with integer slopes. We define the complete linear system |D| of a divisor D on a tropical curve analogously to the classical counterpart. We investigate the structure of |D| as a cell complex and show that linear systems are quotients of tropical modules, finitely generated by vertices of the cell complex. Using a finite set of generators, |D| defines a map from the tropical curve to a tropical projective space, and the image can be extended to a parameterized tropical curve of degree equal to deg(D). The tropical convex hull of the image realizes the linear system |D| as an embedded polyhedral complex. We also show that curves for which the canonical divisor is not very ample are hyperelliptic. This is joint work with Christian Haase and Gregg Musiker.
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Mirror symmetry has developed into an important link between physics and mathematics and predicts that Calabi-Yau varieties come as pairs of mirrors.

In this talk I will describe an explicit mirror construction for monomial degenerations of Calabi-Yau varieties.

Using tropical geometry and Groebner basis techniques the construction is formulated in a Cox homogeneous toric setup and generalizes those for hypersurfaces and complete intersections by Batyrev and Borisov. I will also comment on the relations of the tropical construction to deformation theory.

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I will define what is a tropical inflection point of a tropical plane curve, and explain how this notion is related to inflection points of complex algebraic plane curves. As an application, I will show that there exist maximal plane real algebraic curves of any degree with the maximal number of real inflection points allowed by the Klein Formula.

A crucial tool in our approach is the use of tropical modifications. If time permits, I will give other examples of applications of this powerful tool.

(This is joint work with Lucia Lopez de Medrano.)

Abstract

Hypergeometric D-modules (also known as GKZ-systems) are systems of linear partial differential equations determinded by a matrix A with integer entries and a vector of complex parameters. We will describe how to construct their Gevrey solutions along coordinate subspaces in a combinatorial way. This type of formal solutions are closely related with the irregularity of a D-module.
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(joint work with A. Degtyarev and V. Kharlamov)

The main results of the talk concern complete intersections of three real quadrics. We prove that the maximal number $B^0_2(N)$ of connected components that a regular complete intersection of three real quadrics in $\Bbb{P}^N$ can have differs at most by one from the maximal number of ovals of the submaximal depth $[(N-1)/2]$ of a real plane projective curve of degree $d=N+1$. As a consequence, we obtain a lower bound $\frac14 N^2+O(N)$ and an upper bound $\frac38 N^2+O(N)$ for $B^0_2(N)$.

Abstract