Numerical Polar calculus and cohomology of line bundles

Abstract

Let $L_1,\dots ,L_s$ be line bundles on a smooth variety $X\subset {\mathbb{P}}^r$ and let $D_1,\dots ,D_s$ be divisors on $X$ such that $D_i$ represents $L_i$. We give a probabilistic algorithm for computing the degree of intersections of polar classes which are in turn used for computing the Euler characteristic of linear combinations of $L_1,\dots ,L_s$. The input consists of generators for the homogeneous ideals $I_X,I_{D_i}\subset \mathbb{C}[x_0,\dots ,x_r]$ defining $X$ and $D_i$.