Generically log smooth families via generators and relations

Let f:X → A¹ₜ 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) → (A¹ₜ|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 aligned Γ (X,OX) = [1.111ptt,x₁,… ,xₙ]/(f₁,… ,fᵣ). 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 LSV L S V of log smooth structures on a toroidal crossing scheme (V,P,ρ ̄) ( V , P , ρ ¯ ) .