Enumerative geometry of degeneracy loci

Degeneracy Loci in CR Geometry

Starting on the next page is the 16-page Chapter II excerpted from my dissertation, “Enumeration and Normal Forms for Singularities in CauchyRiemann Structures,” 1997, University of Chicago. I have made a few minor editorial modifications so that Chapter II can be read without referring to the other Chapters. Section 5 of Chapter II is a summary of results appearing in my unpublished lecture no...

Enumerative Geometry

Given a complex projective variety V (as defined in [1]), we wish to count the curves in V that satisfy certain prescribed conditions. Let f C denote complex projective -dimensional space. In our first example, V = f C2, the complex projective plane; in the second and third, V is a general hypersurface in f C of degree 2 − 3. Call such V a cubic twofold when  = 3 and a quintic threefold wh...

Enumerative Real Algebraic Geometry

Two well-defined classes of structured polynomial systems have been studied from this point of view—sparse systems, where the structure is encoded by the monomials in the polynomials fi—and geometric systems, where the structure comes from geometry. This second class consists of polynomial formulations of enumerative geometric problems, and in this case Question 1.1 is the motivating question o...

Enumerative Real Algebrai Geometry

Nonemptiness of Symmetric Degeneracy Loci

Let V be a renk N vector bundle on a d-dimensional complex projective scheme X; assume that V is equipped with a quadratic form with values in a line bundle L and that SV ∗ ⊗ L is ample. Suppose that the maximum rank of the quadratic form at any point of X is r > 0. The main result of this paper is that if d > N − r, the locus of points where the rank of the quadratic form is at most r − 1 is n...