Numerical Polar calculus and cohomology of line bundles

TL;DR

This paper introduces a probabilistic algorithm to compute intersection degrees of polar classes and the Euler characteristic of linear combinations of line bundles on smooth projective varieties.

Contribution

It presents a novel probabilistic method for calculating intersection degrees and Euler characteristics using generators of homogeneous ideals.

Findings

Algorithm effectively computes degrees of polar class intersections.

Method enables calculation of Euler characteristics of line bundle combinations.

Applicable to smooth varieties defined by homogeneous ideals.

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 $I_X, I_{D_i} \subset \mathbb{C}[x_0,\ldots,x_r]$ defining $X$ and $D_i$.