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Register a new userProjective toric varieties of codimension 2 with maximal Castelnuovo--Mumford regularity
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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 $0$.
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This note has two goals. The first is to give a short and self contained introduction to the Castelnuovo-Mumford regularity for standard graded ring $R$ over a general base ring. The second is to present a simple and concise proof of a classical result due to Cutkosky, Herzog and Trung and, independently, to Kodiyalam asserting that the regularity of powers of an homogeneous ideal $I$ of $R$ is eventually a linear function in $v$. Finally we show how the flexibility of the definition of the Castelnuovo-Mumford regularity over general base rings can be used to give a simple characterization of the ideals whose powers have a linear resolution in terms of the regularity of the Rees ring.
Let $A$ be a Noetherian standard $mathbb{N}$-graded algebra over an Artinian local ring $A_0$. Let $I_1,ldots,I_t$ be homogeneous ideals of $A$ and $M$ a finitely generated $mathbb{N}$-graded $A$-module. We prove that there exist two integers $k$ and $k$ such that [ mathrm{reg}(I_1^{n_1} cdots I_t^{n_t} M) leq (n_1 + cdots + n_t) k + k quadmbox{for all }~n_1,ldots,n_t in mathbb{N}. ]
Set $ A := Q/({bf z}) $, where $ Q $ is a polynomial ring over a field, and $ {bf z} = z_1,ldots,z_c $ is a homogeneous $ Q $-regular sequence. Let $ M $ and $ N $ be finitely generated graded $ A $-modules, and $ I $ be a homogeneous ideal of $ A $. We show that (1) $ mathrm{reg}left( mathrm{Ext}_A^{i}(M, I^nN) right) le rho_N(I) cdot n - f cdot leftlfloor frac{i}{2} rightrfloor + b mbox{ for all } i, n ge 0 $, (2) $ mathrm{reg}left( mathrm{Ext}_A^{i}(M,N/I^nN) right) le rho_N(I) cdot n - f cdot leftlfloor frac{i}{2} rightrfloor + b mbox{ for all } i, n ge 0 $, where $ b $ and $ b $ are some constants, $ f := mathrm{min}{ mathrm{deg}(z_j) : 1 le j le c } $, and $ rho_N(I) $ is an invariant defined in terms of reduction ideals of $ I $ with respect to $ N $. There are explicit examples which show that these inequalities are sharp.
Let $A$ be a connected graded $k$-algebra with a balanced dualizing complex. We prove that $A$ is a Koszul AS-regular algebra if and only if that the Castelnuovo-Mumford regularity and the Ext-regularity coincide for all finitely generated $A$-modules. This can be viewed as a non-commutative version of cite[Theorem 1.3]{ro}. By using Castelnuovo-Mumford regularity, we prove that any Koszul standard AS-Gorenstein algebra is AS-regular. As a preparation to prove the main result, we also prove the following statements are equivalent: (1) $A$ is AS-Gorenstein; (2) $A$ has finite left injective dimension; (3) the dualizing complex has finite left projective dimension. This generalizes cite[Corollary 5.9]{mori}.
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 $p(z)$. Experimental evidences led us to consider the idea that $m_{p(z)}$ could be achieved by schemes having a suitable minimal Hilbert function. We give a constructive proof of this fact. Moreover, we are able to compute the minimal Castelnuovo-Mumford regularity $m_p(z)^{varrho}$ of schemes with Hilbert polynomial $p(z)$ and given regularity $varrho$ of the Hilbert function, and also the minimal Castelnuovo-Mumford regularity $m_u$ of schemes with Hilbert function $u$. These results find applications in the study of Hilbert schemes. They are obtained by means of minimal Hilbert functions and of two new constructive methods which are based on the notion of growth-height-lexicographic Borel set and called ideal graft and extended lifting.
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