Upgraded methods for the effective computation of marked schemes on a strongly stable ideal

TL;DR

This paper introduces improved computational methods for marked schemes associated with strongly stable ideals, enabling explicit calculations and embeddings into affine spaces, with applications to Hilbert schemes.

Contribution

It develops a new reduction relation called superminimal reduction to compute equations of marked schemes more efficiently and characterizes when certain embeddings are isomorphisms.

Findings

New superminimal reduction simplifies equations of marked schemes.

Explicit embeddings of marked schemes into low-dimensional affine spaces.

Characterization of the minimal m for isomorphisms between marked schemes.

Abstract

Let $J\subset S=K[x_0,...,x_n]$ be a monomial strongly stable ideal. The collection $\Mf(J)$ of the homogeneous polynomial ideals $I$, such that the monomials outside $J$ form a $K$-vector basis of $S/I$, is called a {\em $J$-marked family}. It can be endowed with a structure of affine scheme, called a {\em $J$-marked scheme}. For special ideals $J$, $J$-marked schemes provide an open cover of the Hilbert scheme $\hilbp$, where $p(t)$ is the Hilbert polynomial of $S/J$. Those ideals more suitable to this aim are the $m$-truncation ideals $\underline{J}_{\geq m}$ generated by the monomials of degree $\geq m$ in a saturated strongly stable monomial ideal $\underline{J}$. Exploiting a characterization of the ideals in $\Mf(\underline{J}_{\geq m})$ in terms of a Buchberger-like criterion, we compute the equations defining the $\underline{J}_{\geq m}$-marked scheme by a new reduction relation, called {\em superminimal reduction}, and obtain an embedding of $\Mf(\underline{J}_{\geq m})$ in an affine space of low dimension. In this setting, explicit computations are achievable in many non-trivial cases. Moreover, for every $m$, we give a closed embedding $\phi_m: \Mf(\underline{J}_{\geq m})\hookrightarrow \Mf(\underline{J}_{\geq m+1})$, characterize those $\phi_m$ that are isomorphisms in terms of the monomial basis of $\underline{J}$, especially we characterize the minimum integer $m_0$ such that $\phi_m$ is an isomorphism for every $m\geq m_0$.