We study how to lift Markov bases and Grobner bases along linear maps of lattices. We give a lifting algorithm that allows to compute such bases iteratively provided a certain associated semigroup is normal. Our main application is the toric fiber product of toric ideals, where lifting gives Markov bases of the factor ideals that satisfy the compatible projection property. We illustrate the technique by computing Markov bases of various infinite families of hierarchical models. The methodology also implies new finiteness results for iterated toric fiber products.
We observe that there is an equivalence between the singularity category of an affine complete intersection and the homotopy category of matrix factorizations over a related scheme. This relies in part on a theorem of Orlov. Using this equivalence, we give a geometric construction of the ring of cohomology operators, and a generalization of the theory of support varieties, which we call stable support sets. We settle a question of Avramov about which stable support sets can arise for a given complete intersection ring. We also use the equivalence to construct a projective resolution of a module over a complete intersection ring from a matrix factorization, generalizing the well-known result in the hypersurface case.
Minimal Markov bases of configurations of integer vectors correspond to minimal binomial generating sets of the assocciated lattice ideal. We give necessary and sufficient conditions for the elements of a minimal Markov basis to be (a) inside the universal Gr{ o}bner basis and (b) inside the Graver basis. We study properties of Markov bases of generalized Lawrence liftings for arbitrary matrices $Ainmathcal{M}_{mtimes n}(Bbb{Z})$ and $Binmathcal{M}_{ptimes n}(Bbb{Z})$ and show that in cases of interest the {em complexity} of any two Markov bases is the same.
We show that a non-trivial fiber product $Stimes_k T$ of commutative noetherian local rings $S,T$ with a common residue field $k$ is Gorenstein if and only if it is a hypersurface of dimension 1. In this case, both $S$ and $T$ are regular rings of dimension 1. We also give some applications of this result.
Let $R=oplus_{igeq 0} R_i$ be an Artinian standard graded $K$-algebra defined by quadrics. Assume that $dim R_2leq 3$ and that $K$ is algebraically closed of characteristic $ eq 2$. We show that $R$ is defined by a Grobner basis of quadrics with, essentially, one exception. The exception is given by $K[x,y,z]/I$ where $I$ is a complete intersection of 3 quadrics not containing the square of a linear form.
The Eisenbud--Goto conjecture states that $operatorname{reg} Xleoperatorname{deg} X -operatorname{codim} X+1$ for a nondegenerate irreducible projective variety $X$ over an algebraically closed field. While this conjecture is known to be false in general, it has been proven in several special cases, including when $X$ is a projective toric variety of codimension $2$. We classify the projective toric varieties of codimension $2$ having maximal regularity, that is, for which equality holds in the Eisenbud--Goto bound. We also give combinatorial characterizations of the arithmetically Cohen--Macaulay toric varieties of maximal regularity in characteristic $0$.