Generically log smooth families via generators and relations

Abstract

Abstract Let $$f:X \rightarrow \mathbb {A}^1_t$$ f : X → A t 1 be an affine flat morphism of finite type, and let $$V = f^{-1}(0)$$ V = f - 1 ( 0 ) . Then, we obtain a morphism of log schemes $$f:(X|V) \rightarrow (\mathbb {A}^1_t|0)$$ f : ( X | V ) → ( A t 1 | 0 ) . In this article, we develop algorithmic tools to study the log-geometric properties of f by means of a presentation $$\begin{aligned} \Gamma (X,\mathcal {O}_X) = \Bbbk [\hspace{1.111pt}t,x_1,\ldots ,x_n]/(f_1,\ldots ,f_r). \end{aligned}$$ Γ ( X , O X ) = k [ t , x 1 , … , x n ] / ( f 1 , … , 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 $$\mathcal {L}\mathcal {S}_V$$ L S V of log smooth structures on a toroidal crossing scheme $$(V,\mathcal {P},\bar{\rho })$$ ( V , P , ρ ¯ ) .