Algorithms And Programming On The Minimal Combinations Of Weights Of Projective Hypersurfaces

TL;DR

This paper introduces an algorithm for computing minimal weight combinations of finite sets in Euclidean spaces, applied to study moment maps and stability of projective hypersurfaces, including cubic curves and surfaces.

Contribution

The paper presents a novel algorithm that determines minimal combinations and unstable points for hypersurfaces, extending to cubic surfaces and analyzing affine dependencies.

Findings

Algorithm successfully computes minimal combinations for cubic curves and surfaces.

Identifies unique unstable points in Morse strata when they exist.

Provides insights into affine dependencies of monomial sets.

Abstract

This paper designs an alogrithm to compute the minimal combinations of finite sets in Euclidean spaces, and applys the algorithm of study the moment maps and geometric invariant stability of hypersurfaces. The classical example of cubic curves is repeated by the algorithm. Furhtermore the alogrithm works for cubic surfaces. For given affinely indepdent subsets of monomials, the algorithm can output the unique unstable points of the Morse strata if it exists. Also there is a discussion on the affinely dependent sets of monomials.