相关论文: A Step in Castelnuovo theory via Grobner bases

We construct new explicit examples of nonsmoothable Gorenstein algebras with Hilbert function $(1,n,n,1)$. This gives a new infinite family of elementary components in the Gorenstein locus of the Hilbert scheme of points and solves the…

In his work on singularities, expanders and topology of maps, Gromov showed, using isoperimetric inequalities in graded algebras, that every real valued map on the $n$-torus admits a fibre whose homological size is bounded below by some…

McCullough and Peeva found sequences of counterexamples to the Eisenbud--Goto conjecture on the Castelnuovo--Mumford regularity by using Rees-like algebras, where entries of each sequence have increasing dimensions and codimensions. In this…

Let $K$ be an algebraically closed field of null characteristic and $p(z)$ a Hilbert polynomial. We look for the minimal Castelnuovo-Mumford regularity $m_{p(z)}$ of closed subschemes of projective spaces over $K$ with Hilbert polynomial…

We develop the theory of Gr\"obner bases for ideals in a polynomial ring with countably infinite variables over a field. As an application we reconstruct some of the one-one correspondences among various sets of partitions by using division…

In this article we present two new algorithms to compute the Groebner basis of an ideal that is invariant under certain permutations of the ring variables and which are both implemented in SINGULAR (cf. [DGPS12]). The first and major…

Given a parametric polynomial ideal I, the algorithm DISPGB, introduced by the author in 2002, builds up a binary tree describing a dichotomic discussion of the different reduced Groebner bases depending on the values of the parameters,…

We give a bound on the Castelnuovo-Mumford regularity of a homogeneous ideals I, in a polynomial ring A, in terms of number of variables and the degrees of generators, when the dimension of A/I is at most two. This bound improves the one…

Let $X$ be a reduced closed subscheme in $\mathbb P^n$. As a slight generalization of property $\textbf{N}_p$ due to Green-Lazarsfeld, we can say that $X$ satisfies property $\textbf{N}_{2,p}$ scheme-theoretically if there is an ideal $I$…

We estimate the Castelnuovo-Mumford regularity of ideals in a polynomial ring over a field by studying the regularity of certain modules generated in degree zero and with linear relations. In dimension one, this process gives a new type of…

We show that, given an absolutely continuous horizontal curve $\gamma$ in the Heisenberg group, there is a $C^1$ horizontal curve $\Gamma$ such that $\Gamma=\gamma$ and $\Gamma'=\gamma'$ outside a set of small measure. Conversely, we…

Here we study the problem of generalizing one of the main tools of Groebner basis theory, namely the flat deformation to the leading term ideal, to the border basis setting. After showing that the straightforward approach based on the…

Let $\Gamma \subset \mathbb{P}^n$ be a nondegenerate finite subscheme of degree $d$. Then the Castelnuovo-Mumford regularity ${\rm reg} ({\Gamma})$ of $\Gamma$ is at most $\left\lceil \frac{d-n-1}{t(\Gamma)} \right\rceil +2$ where…

In this paper, we give new explicit representations of the Hilbert scheme of $\mu$ points in $\PP^{r}$ as a projective subvariety of a Grassmanniann variety. This new explicit description of the Hilbert scheme is simpler than the previous…

We calculate the genus 1 Gromov-Witten theory of the Hilbert scheme $\mathsf{Hilb}^n(\mathbb{C}^2)$ of points in the plane. The fundamental 1-point invariant (with a divisor insertion) is calculated using a correspondence with the families…

We extend the circle of ideas from a previous paper on hypersurfaces to functions $f \colon (\mathbb C^n, 0) \to (\mathbb C^k, 0)$ with an isolated singularity in a stratified sense on an arbitrary, but fixed complex analytic germ $(X, 0)$.…

Castelnuovo-Mumford regularity is an important invariant of projective algebraic varieties. A well known conjecture due to Eisenbud and Goto gives a bound for regularity in terms of the codimension and degree,i.e., Castelnuovo-Mumford…

In this thesis we consider the geometry of the Hilbert scheme of points in P^n, concentrating on the locus of points corresponding to the Gorenstein subschemes of P^n. New results are given, most importantly we provide tools for…

We develop numerical homotopy algorithms for solving systems of polynomial equations arising from the classical Schubert calculus. These homotopies are optimal in that generically no paths diverge. For problems defined by hypersurface…

In this paper we introduce a new and large family of configurations whose toric ideals possess quadratic Groebner bases. As an application, a generalization of algebras of Segre-Veronese type will be studied.