Algebraic Geometry Seminar Mondays 3:00--3:50 PM Texas A&M University Milner 216

Related seminars: Algebra & Combinatorics Seminar Geometry Seminar Number Theory Seminar Several Complex Variables Seminar All Mathematics Seminars

Fall 2009 Schedule:

Spring 2010 Schedule:

Abstracts:

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.
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An orbitope is the convex hull of an orbit of a compact group G acting linearly on a vector space. Orbitopes are the simplest convex bodies which possess many symmetries. Some, particularly those in low-dimensional representations of G have very beautiful structure. Our interest is in whether or not these appealing convex bodies are spectrahedra, that is, if they are described by a system of linear matrix inequalities, preferably with coefficients in the field of definition of the orbitope.

In this talk, I will introduce orbitopes and discuss spectrahedra and the new field of convex algebraic geometry in which these questions lie. I will illustrate this with orbitopes for SO(2) and for the special orthogonal group acting on trace-free symmetric matrices.

This is joint work with Raman Sanyal and Bernd Sturmfels.
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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 subcanonical point p on a compact Riemann surface of genus g>1 is a point such that some holomorphic 1-form vanishes at p to order 2g-2 and nowhere else. These points are Weierstrass points, and we study their associated Weierstrass gap sequences.
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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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We study chains of toric ideals that are invariant under a symmetric group action. In our setting, the ambient rings for these ideals are polynomial rings which are increasing in (Krull) dimension. Thus, these chains will fail to stabilize in the traditional commutative algebra sense. However, we prove a general theorem which says that "up to the action of the group", these chains stabilize up to monomial localization. This gives a partial resolution to a conjecture of Aschenbrenner and Hillar.

This is a joint work with Chris Hillar.
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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 start by briefly introducing these objects in the complex algebraic hypersurfaces cases.

The purpose of my talk will be to explain the relation between complex and non-Archimedean coamoebas on one hand, and Newton polytope on the other hand in the case of hypersurfaces. Moreover, a brief survey of the further development of complex and non-Archimedean amoebas will be given, as well as a description of some new results.

However, the same circle of ideas used on amoebas, also shows that the coamoebas have a similar geometric and combinatorial structure. Part of this work is in preparation jointly with M. Passare in the complex case, and with F. Sottile in the non-Archimedean case in codimension greater than one. Many examples, with pictures, will be given.
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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.)
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This talk will start by introducing the cluster algebras of Fomin and Zelevinsky with principal coefficients, and work of Fomin, Shapiro, and Thurston to realize these from triangulated surfaces. Then I will explain our result giving combinatorial formulas for cluster variables in any cluster algebra arisng from a triangulated surface, include those with punctures. This proves the positivity conjecture of Fomin and Zelevinsky for all such cluster algebras (which, by Felikson-Shapiro-Tumarkin, comprise "almost all" of the skew-symmetric cluster algebras of finite mutation type). This is joint work with Lauren Williams and Ralf Schiffler.
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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 $B^0_2(N)$.

Previous semesters:

For more information, email Zach Teitler.