Title: Chow forms and resultants, old and new
Speaker: Professor David Eisenbud
Speaker Info: MSRI, Berkeley
An algebraic plane curve is determined by the vector of coefficients of the equation that defines it; this gives a convenient method of parametrizing such curves. Cayley noticed that although the defining ideal of a curve in space may have many generators, one can still specify the curve with one equation, which is actually the equation satisfied by the lines in space that meet the curve. The idea was generalized to arbitrary varieties by Chow and van der Waerden; the equation is called (by many people) the Chow form. In the case of the rational curve t --> (t, t^2, t^3, ... ,t^d) in d-space, this equation is the familiar resultant of two polynomials of degree d.Date: Tuesday, February 20, 2001
any people, also starting with Cayley, have given formulas for resultants that are alternating products of determinants of matrices. In the 1960's Grothendieck gave a general framework for such formulas for resultants (and more generally for Chow forms), expressing them as determinants of certain complexes, defined up to quasi-isomorphism. The ideas were used, for exmaple, in the Deligne-Mumford construction of the moduli space of curves. They have also been used to study resultants, but in this setting they are made more complicated by the difficulty of finding a complex of the desired type.
I will explain the background of Chow forms and resultants and show how canonical representatives of Grothendieck's complexes can be constructed using free resolutions over exterior algebras. Applications include explicit formulas for resultants in cases where no formula was known.
This is part of ongoing work of mine with Frank-Olaf Schreyer.