Generically log smooth families via generators and relations

TL;DR

This paper develops algorithmic tools to analyze log-geometric properties of flat morphisms using generators and relations, with implementations in Macaulay2, and explores log smooth structures on toroidal crossing schemes.

Contribution

It introduces new algorithms for studying log schemes via presentations of coordinate rings and provides computational implementations, advancing the analysis of log smooth families.

Findings

Algorithms for log-geometric properties using generators and relations

Implementation of algorithms in Macaulay2

Results on sheaf of log smooth structures on toroidal crossing schemes

Abstract

Let $f\colon X \to \mathbb{A}^1_t$ be an affine flat morphism of finite type, and let $V = f^{-1}(0)$. Then, we obtain a morphism of log schemes $f\colon (X|V) \to (\mathbb{A}^1_t|0)$. In this article, we develop algorithmic tools to study the log-geometric properties of $f$ by means of a presentation \[\Gamma(X,\mathcal{O}_X) = \Bbbk[t,x_1,\ldots,x_n]/(f_1,\ldots,f_r).\] We obtain similar tools for projective flat morphisms when the homogeneous coordinate ring is given by generators and relations. We provide an implementation of our algorithms in Macaulay2. In a slightly different direction, we give some results on the sheaf $\mathcal{LS}_V$ of log smooth structures on a toroidal crossing scheme $(V,\mathcal{P},\bar\rho)$.