UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

January 19, 2010

939 Evans Hall

5:00AM: Arf Rings and Strict Closure of Semigroups

Claudiu Raicu

3:45PM: On the set-theoretic versions of conjectures of Holtz-Sturmfels and Landsberg-Weyman

Luke Oeding

A principal minor of a matrix is the determinant of a submatrix centered about the main diagonal. A basic linear algebra question asks to what extent is it possible to prescribe the principal minors of a matrix. We focus on the case of symmetric matrices. The equations the algebraic variety of principal minors of symmetric matrices tell when it is possible for a given vector of length 2^n to be the principal minors of a symmetric n-by-n matrix. Holtz and Sturmfels found these equations in the cases n=3,4 and discovered an interesting connection to Cayley's hyperdeterminant. They conjectured that the hyperdeterminantal module generates the ideal of the variety. Starting with examples using basic linear algebra I will describe some of the questions that motivated the original question and show some of the beautiful symmetry and geometry of this variety. Next I will sketch a proof of the set-theoretic version of the Holtz-Sturmfels conjecture. Finally I will point out an unexpected connection to the tangential variety of the Segre product of projective spaces, and sketch a proof of a set-theoretic version of the Landsberg-Weyman conjecture.

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