Mixed multiplicities, Hilbert polynomials and homaloidal surfaces

TL;DR

This paper explores the connections between numerical invariants of free projective hypersurfaces, providing explicit equations for certain homaloidal surfaces in projective 3-space.

Contribution

It establishes relationships among mixed multiplicities, Hilbert polynomials, and exponents of free hypersurfaces, and derives explicit equations for specific homaloidal surfaces.

Findings

Relationships among invariants of free hypersurfaces are established.

Explicit equations for some homaloidal surfaces are obtained.

Connections between algebraic invariants and geometric properties are demonstrated.

Abstract

We investigate the relationship among several numerical invariants associated to a (free) projective hypersurface $V$: the sequence of mixed multiplicities of its Jacobian ideal, the Hilbert polynomial of its Milnor algebra, and the sequence of exponents when $V$ is free. As a byproduct, we obtain explicit equations for some of the homaloidal surfaces in the projective 3-dimensional space constructed by C. Ciliberto, F. Russo and A. Simis.