Calculating discriminants by higher direct images

Abstract

The author uses the homological algebra to construct for any line bundle L \mathcal {L} on a nonsingular projective variety X the complex F ( L ) \mathbb {F}(\mathcal {L}) whose determinant is equal to the equation of the dual variety X V {X^{\text {V}}} . This generalizes the Cayley-Koszul complexes defined by Gelfand, Kapranov and Zelevinski. The formulas for the codimension and degree of X V {X^{\text {V}}} in terms of complexes F ( L ) \mathbb {F}(\mathcal {L}) are given. In the second part of the article the general technique is applied to classical discriminants and hyperdeterminants.