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Register a new userComputing minimal Gorenstein covers
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We analyze and present an effective solution to the minimal Gorenstein cover problem: given a local Artin k-algebra $A = k[[x 1 ,. .. x n ]]/I$, compute an Artin Gorenstein $k$-algebra $G = k[[x 1 ,. .. x n ]]/J$ such that $ell(G)--ell(A)$ is minimal. We approach the problem by using Macaulays inverse systems and a modification of the integration method for inverse systems to compute Gorenstein covers. We propose new characterizations of the minimal Gorenstein cover and present a new algorithm for the effective computation of the variety of all minimal Gorenstein covers of A for low Gorenstein colength. Experimentation illustrates the practical behavior of the method.
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Using a classical result of Avramov-Golod we strengthen a recent result of Gorodentsev, Khoroshkin and Rudakov on syzygies of highest weight orbit closure.
A projectively normal Calabi-Yau threefold $X subseteq mathbb{P}^n$ has an ideal $I_X$ which is arithmetically Gorenstein, of Castelnuovo-Mumford regularity four. Such ideals have been intensively studied when $I_X$ is a complete intersection, as well as in the case where $X$ is codimension three. In the latter case, the Buchsbaum-Eisenbud theorem shows that $I_X$ is given by the Pfaffians of a skew-symmetric matrix. A number of recent papers study the situation when $I_X$ has codimension four. We prove there are 16 possible betti tables for an arithmetically Gorenstein ideal $I$ with $mathrm{codim}(I)=4=mathrm{reg}(I)$, and that exactly 8 of these occur for smooth irreducible nondegenerate threefolds. We investigate the situation in codimension five or more, obtaining examples of $X$ with $h^{p,q}(X)$ not among those appearing for $I_X$ of lower codimension or as complete intersections in toric Fano varieties. A key tool in our approach is the use of inverse systems to identify possible betti tables for $X$.
We prove the Conjecture of Catenese--Chen--Zhang: the inequality $K_X^3geq frac{4}{3}p_g(X)-frac{10}{3}$ holds for all projective Gorenstein minimal 3-folds $X$ of general type.
The main results of this paper are already known (V.V. Shokurov, the non-vanishing theorem, 1985). Moreover, the non-$mathbb{Q}$-factorial MMP was more recently considered by O~Fujino, in the case of toric varieties (Equivariant completions of toric contraction morphisms, 2006), for klt pairs (Special termination and reduction to pl flips, 2007) and more generally for log-canonical pairs (Foundation of the minimal model program, 2014). Here we rewrite the proofs of some of these results, by following the proofs given by Y. Kawamata, K. Matsuda, and K. Matsuki (Introduction to the minimal model problem, 1985) of the same results in $mathbb{Q}$-factorial MMP. And, in the family of $mathbb{Q}$-Gorenstein spherical varieties, we answer positively to the questions of existence of flips and of finiteness of sequences of flips. I apologize for the first version of this paper, which I wrote without knowing that these results already exist.
Let $rho_C$ be the regularity of the Hilbert function of a projective curve $C$ in $mathbb P^n_K$ over an algebraically closed field $K$ and $alpha_1,...,alpha_{n-1}$ be minimal degrees for which there exists a complete intersection of type $(alpha_1,...,alpha_{n-1})$ containing the curve $C$. Then the Castelnuovo-Mumford regularity of $C$ is upper bounded by $max{rho_C+1,alpha_1+...+alpha_{n-1}-(n-2)}$. We study and, for space curves, refine the above bound providing several examples.
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