Related papers: Complete intersections in simplicial toric varieti…

Let $X$ be a complete $n$-dimensional simplicial toric variety with homogeneous coordinate ring $S$. We study the multigraded Hilbert function $H_Y$ of reduced $0$-dimensional subschemes $Y$ in $X$. We provide explicit formulas and prove…

Let $\mathcal A$ be a simple, $\sigma$-unital, non-unital, non-elementary C*-algebra and let $I_{min}$ be the intersection of all the ideals of $\mathcal M(\mathcal A)$ that properly contain $\mathcal A$. $I_{min}$ coincides with the ideal…

Let $\mathcal{A}=\{{\bf a}_1,\ldots,{\bf a}_n\}\subset\Bbb{N}^m$. We give an algebraic characterization of the universal Markov basis of the toric ideal $I_{\mathcal{A}}$. We show that the Markov complexity of $\mathcal{A}=\{n_1,n_2,n_3\}$…

We define a variant of intersection space theory that applies to many compact complex and real analytic spaces $X$, including all complex projective varieties; this is a significant extension to a theory which has so far only been shown to…

In this paper, we characterize the positive integers $n$ for which intersection graph of ideals of $\mathbb{Z}_n$ is perfect.

To any toric ideal $I_A$, encoded by an integer matrix $A$, we associate a matroid structure called {\em the bouquet graph} of $A$ and introduce another toric ideal called {\em the bouquet ideal} of $A$. We show how these objects capture…

Let $G$ be a simple graph on the vertex set $\{v_{1},\ldots,v_{n}\}$. An algebraic object attached to $G$ is the toric ideal $I_G$. We say that $I_G$ is subgraph splittable if there exist subgraphs $G_1$ and $G_2$ of $G$ such that…

Let $\mathcal A=\{A_1,\ldots,A_n\}$ be a family of sets in the plane. For $0 \leq i < n$, denote by $f_i$ the number of subsets $\sigma$ of $\{1,\ldots,n\}$ of cardinality $i+1$ that satisfy $\bigcap_{i \in \sigma} A_i \neq \emptyset$. Let…

We give a proof of the $p$-adic weight monodromy conjecture for scheme-theoretic complete intersections in projective smooth toric varieties. The strategy is based on Scholze's proof in the $\ell$-adic setting, which we adapt using…

We classify all unmixed monomial ideals I of codimension 2 which are generically a complete intersection and which have the property that the symbolic power algebra A(I) is standard graded. We give a lower bound for the highest degree of a…

We show that for all $n\geq 3$ and all primes $p$ there are infinitely many simplicial toric varieties of codimension $n$ in the $2n$-dimensional affine space whose minimum number of defining equations is equal to $n$ in characteristic $p$,…

This paper is a continuation of a previous work by the author and G. Puninski where iterated intersections of powers of ideals were studied in rings of iterated differential polynomials. We present a method which can be used to show that…

We show that the Stanley-Reisner ideal of the one-dimensional simplicial complex whose diagram is an $n$-gon is always a set-theoretic complete intersection in any positive characteristic.

An equivariant linear system on a toric variety is a linear system invariant under the torus action. We study the number of irreducible components of the complete intersection of general divisors from a fixed collection of equivariant…

Let $R$ be a smooth affine algebra over an infinite perfect field $k$. Let $I\subset R$ be an ideal, $\omega_I:(R/I)^n\to I/I^2$ a surjective homomorphism and $Q_{2n}\subset \mathbb{A}^{2n+1}$ be the smooth quadric defined by the equation…

We examine the ideal $I=(x_1^2, \dots, x_n^2, (x_1+\dots+x_n)^2)$ in the polynomial ring $Q=k[x_1, \dots, x_n]$, where $k$ is a field of characteristic zero or greater than $n$. We also study the Gorenstein ideal $G$ linked to $I$ via the…

We give an introduction to Tropical Geometry and prove some results in Tropical Intersection Theory. The first part of this paper is an introduction to tropical geometry aimed at researchers in Algebraic Geometry from the point of view of…

In this paper, we provide a combinatorial characterization of those collections of cells whose inner $2$-minor ideals are complete intersections. More precisely, given a collection of cells $\mathcal C$ and its associated inner $2$-minor…

The toric ideal $I_A$ is splittable if it has a toric splitting; namely, if there exist toric ideals $I_{A_1}, I_{A_2}$ such that $I_A=I_{A_1}+I_{A_2}$ and $I_{A_i}\not =I_{A}$ for all $1 \leq i \leq 2$. We provide a necessary and…

Associated to any graph is a toric ideal whose generators record relations among the cuts of the graph. We study these ideals and the geometry of the corresponding toric varieties. Our theorems and conjectures relate the combinatorial…