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  • Print publication year: 2016
  • Online publication date: March 2016

Chapter 10 - Segre classes and varieties of linear spaces

Summary

Keynote Questions

(a) Let v1, …, v2n be general tangent vector fields on ℙn. At how many points of ℙn is there a nonzero cotangent vector annihilated by all the vi? (Answer on page 366.)

(b) If f is a general polynomial of degree d = 2m - 1 in one variable over a field of characteristic 0, then there is a unique way to write f as a sum of m d-th powers of linear forms (Proposition 10.15). If f and g are general polynomials of degree d = 2m in one variable, how many linear combinations of f and g are expressible as a sum of m d-th powers of linear forms? (Answer on page 377.)

(c) If C ⊂ ℙ4 is a general rational curve of degree d, how many 3-secant lines does C have? (Answer on page 379.)

(d) If C ⊂ ℙ3 is a general rational curve of degree d, what is the degree of the surface swept out by the 3-secant lines to C? (Answer on page 380.)

Segre classes

Our understanding of the Chow rings of projective bundles makes accessible the computation of the classes of another natural series of loci associated to a vector bundle.

We start with a naive question. Suppose that ε is a vector bundle on a scheme X and that ε is generated by global sections. How many global sections does it actually take to generate ε? More generally, what sort of locus is it where a given number of general global sections fail to generate ε locally?

We can get a feeling for these questions as follows. First, consider the case where ε is a line bundle. In this case, each regular section corresponds to a divisor of class c1(ε). If ε is generated by its global sections, the linear series of these divisors is base point free, so a general collection of i of them will intersect in a codimension-i locus of class c1(ε)i. That is, the locus where i general sections of ε fail to generate ε has “expected” codimension i and class c1(ε)i.