Generically log smooth families via generators and relations

Abstract

Let $f:X\to {\mathbb{A}}_t^1$ be an affine flat morphism of finite type, and let $V=f^{-1}(0)$. Then, we obtain a morphism of log schemes $f:(X|V)\to ({\mathbb{A}}_t^1|0)$. In this article, we develop algorithmic tools to study the log-geometric properties of $f$ by means of a presentation \[Γ(X,\mathcal{O}_X) = \Bbbk[t,x_1,\ldots,x_n]/(f_1,\ldots,f_r).\] 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 $(V,\mathcal{P},\bar{\rho })$.

arXiv comment57 pages, 2 figures