Algebraic Geometry Seminar Mondays 10:20--11:10 AM Texas A&M University Milner 317

Picture courtesy of Frank Sottile

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Fall 2007 Schedule:

Abstracts:

Gale duality for polynomial systems is an elementary reformulation of a system of polynomial equations as a system of equations involving rational master functions in the complement of a hyperplane arrangement. Some properties of the original system are easier to understand in the Gale dual system. In the first part of this talk, I will describe this Gale duality, look at some examples of this construction, and give some elementary consequences.
This is joint work with Frédéric Bihan, and is described more completely in this preprint.
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We compute a naturally defined measure of the size of the nef cone of a generalized Del Pezzo surface. The resulting number appears in a conjecture of Manin on the asymptotic behavior of the number of rational points of bounded height on the surface. The nef cone volume is computed using two elementary techniques: a simplicial decomposition of the cone, and the Weyl group of a root system associated to the configuration of (-2)-curves on the surface. Over a non-closed field this root system is not necessarily simply-laced.
This is joint work with Ulrich Derenthal and Michael Joyce and is described in this preprint.
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I will explain how problems in statistics and computer science, respectively the recognition problem for principal minors (see e.g. arXiv:math/0604374) and characterization of matchgate relations (see e.g. pages.cs.wisc.edu/~jyc/papers/icalp-talk-06.pdf), are related to a uniform construction of compact Hermitian symmetric spaces discovered with L. Manivel. (Selecta Math. (N.S.) 8 (2002), no. 1, 137--159, also see arXiv:math/0203260)
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We develop an 'integral Deligne cohomology theory' for real varieties which bears to Bredon cohomology the same relation that ordinary Deligne cohomology for complex varieties bears to singular cohomology. The theory has a wide range of connections ranging from equivariant topology (via Bredon equivariant cohomology), through complex differential geometry (via holomorphic line bundles with quadratic forms and holomorphic connections) to number theory (via Milnor K-theory of number fields and regulator maps). We will present --time permitting-- many examples. In a forthcoming work we show that the cycle map from motivic cohomology to Bredon cohomology factors through our Deligne cohomology groups.

In Esnault-Vieweg's survey on Deligne they outline a theory for real varieties, as well. Such a variant would be related to Liechtenbaum's etale motivic cohomology and the Borel version of equivariant cohomology. The difference between their version and ours stems from the difference between the etale and Nisnevich topologies.
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In recent years, techniques from semidefinite programming have produced numerical algorithms for finding representations of positive semidefinite polynomials as sums of squares. These algorithms have many applications in optimization, control theory, quadratic programming, and matrix analysis. One major drawback with these algorithms is that their output is, in general, numerical. For many applications, however, exact polynomial identities are needed. In this regard, Sturmfels has asked whether a representation with real coefficients implies one over the rationals. We discuss recent progress on this question; in particular, we outline a positive answer to this question for totally real number fields.
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Briefly, I will continue and finish my talk of September 3rd.

I will show how to compute the nef cone volume by decomposing the nef cone into pieces corresponding to chambers of a Weyl group associated to the generalized Del Pezzo surface. If the surface is defined over a non-algebraically closed field, then there is also a Galois action in the picture, and the Weyl group corresponds to a root system which is not necessarily simply-laced.
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We introduce and explore some of the similarities and differences between real and p-adic algorithmic algebraic geometry. After briefly reviewing some quantitative results on sparse systems of polynomial equations over the reals and p-adic rationals, we then focus on one variable: How hard is it to decide if one polynomial in one variable has a root?

We will see that the complexity of this question is still quite open over the real numbers, but admits a more decisive answer over the p-adic rationals. This leads us to some natural questions, in higher dimensions, over both the reals and the p-adics.

We assume no background in number theory or complexity theory.
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A-hypergeometric systems are parametric systems of PDE built from toric ideals. I will outline a proof that these systems are Weyl closed for very generic parameters, i.e. they are the differential annihilators of their solution spaces. No background is required for this talk, but a familiarity with toric ideals will help.
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A general hypersurface of degree d in CP^n will have a 2n-d-3 dimensional space of lines (embedded P^1's) on it. O. Debarre and J. de Jong independently conjectured that a hypersurface of degree d\leq n in CP^n having a larger family of lines on it cannot be smooth. I will explain joint work with O. Tommasi where we prove this conjecture in a slightly sharper form.

In the course of proving the conjecture, we developed new methods using projective differential geometry to locate singularities on varieties uniruled by lines which we expect to have significant applications.
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Let $R$ be a Noetherian local ring with infinite residue field $k$ and $I$ an $R$-ideal. The ideal $J$ is a \textit{reduction} of $I$ if $J \subset I$ and $I^{r+1}=JI^{r}$ for some positive integer $r$. A reduction can be thought of as a simplification of the ideal $I$. The notion of a reduction for an ideal was introduced by D. Northcott and D. Rees in order to study multiplicities. Reductions are connected to the study of blowup algebras such as the Rees ring $\mathcal{R}(I)=R[It]$ of $I$, and the associated graded ring ${\rm{gr}}_{I} (R)=R[It]/IR[It]$ of $I$.

In general minimal reductions are not unique. To remedy this lack of uniqueness, one considers the intersection of all reductions, namely the \textit{core} of the ideal, ${\rm{core}}(I)$. This object, that appears naturally in the context of the Brian\c con-Skoda theorem, encodes information about all possible reductions. We present some recent work on the shape of the core of ideals.
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A principal minor of a matrix is the determinant of a submatrix which has the same row and column set. Given an n by n symmetric matrix, one can calculate all of its principal minors and store this information in a vector of length 2^n. A natural question is: given a vector of length 2^n, can one find a matrix that has those principal minors? Does such a matrix always exist? If not, can we completely describe a set of conditions that will guarantee existence? In more geometric language, we ask, What are the minimal generators of the homogeneous ideal of the variety of principal minors of symmetric matrices?

There is a lot of underlying structure in this problem which leads to some beautiful geometry and representation theory and we'll explore some of it in this talk.
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For more information, email Zach Teitler.