Associated form morphism

TL;DR

This paper investigates a geometric morphism between moduli spaces of hypersurfaces, providing a compactification that often contracts the discriminant divisor, enhancing understanding of hypersurface moduli.

Contribution

It introduces a new compactification of the hypersurface moduli space via an associated form morphism, revealing contraction properties of the discriminant divisor.

Findings

The morphism induces a compactification of the moduli space.

The rational map often contracts the discriminant divisor.

Provides geometric insights into hypersurface moduli spaces.

Abstract

We study the geometry of the morphism between moduli spaces of hypersurfaces in $\mathbb P^{n-1}$ that sends a smooth hypersurface of degree $d+1$ to its associated hypersurface of degree $n(d-1)$. As a result, we obtain a compactification of the moduli space of smooth hypersurfaces such that the induced rational map from the standard GIT compactification often contracts the discriminant divisor.