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Register a new userRegularity bounds for curves by minimal generators and Hilbert function
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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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For a toric pair $(X, D)$, where $X$ is a projective toric variety of dimension $d-1geq 1$ and $D$ is a very ample $T$-Cartier divisor, we show that the Hilbert-Kunz density function $HKd(X, D)(lambda)$ is the $d-1$ dimensional volume of ${overline {mathcal P}}_D cap {z= lambda}$, where ${overline {mathcal P}}_Dsubset {mathbb R}^d$ is a compact $d$-dimensional set (which is a finite union of convex polytopes). We also show that, for $kgeq 1$, the function $HKd(X, kD)$ can be replaced by another compactly supported continuous function $varphi_{kD}$ which is `linear in $k$. This gives the formula for the associated coordinate ring $(R, {bf m})$: $$lim_{kto infty}frac{e_{HK}(R, {bf m}^k) - e_0(R, {bf m}^k)/d!}{k^{d-1}} = frac{e_0(R, {bf m})}{(d-1)!}int_0^inftyvarphi_D(lambda)dlambda, $$ where $varphi_D$ (see Proposition~1.2) is solely determined by the shape of the polytope $P_D$, associated to the toric pair $(X, D)$. Moreover $varphi_D$ is a multiplicative function for Segre products. This yields explicit computation of $varphi_D$ (and hence the limit), for smooth Fano toric surfaces with respect to anticanonical divisor. In general, due to this formulation in terms of the polytope $P_D$, one can explicitly compute the limit for two dimensional toric pairs and their Segre products. We further show that (Theorem~6.3) the renormailzed limit takes the minimum value if and only if the polytope $P_D$ tiles the space $M_{mathbb R} = {mathbb R}^{d-1}$ (with the lattice $M = {mathbb Z}^{d-1}$). As a consequence, one gets an algebraic formulation of the tiling property of any rational convex polytope.
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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Hara Charalambous
, Kostas Karagiannis
, Sotiris Karanikolopoulos andn Aristides Kontogeorgis
2021
We consider a Fermat curve $F_n:x^n+y^n+z^n=1$ over an algebraically closed field $k$ of characteristic $pgeq0$ and study the action of the automorphism group $G=left(mathbb{Z}/nmathbb{Z}timesmathbb{Z}/nmathbb{Z}right)rtimes S_3$ on the canonical ring $R=bigoplus H^0(F_n,Omega_{F_n}^{otimes m})$ when $p>3$, $p mid n$ and $n-1$ is not a power of $p$. In particular, we explicitly determine the classes $[H^0(F_n,Omega_{F_n}^{otimes m})]$ in the Grothendieck group $K_0(G,k)$ of finitely generated $k[G]$-modules, describe the respective equivariant Hilbert series $H_{R,G}(t)$ as a rational function, and use our results to write a program in Sage that computes $H_{R,G}(t)$ for an arbitrary Fermat curve.
We reformulate a fundamental result due to Cook, Harbourne, Migliore and Nagel on the existence and irreduciblity of unexpected plane curves of a set of points $Z$ in $mathbb{P}^2$, using the minimal degree of a Jacobian syzygy of the defining equation for the dual line arrangement $mathcal A_Z$. Several applications of this new approach are given. In particular, we show that the irreducible unexpected quintics may occur only when the set $Z$ has the cardinality equal to 11 or 12, and describe five cases where this happens.
Progress on the problem whether the Hilbert schemes of locally Cohen-Macaulay curves in projective 3 space are connected has been hampered by the lack of an answer to a question that was raised by Robin Hartshorne in his paper On the connectedness of the Hilbert scheme of curves in projective 3 space Comm. Algebra 28 (2000) and more recently in the open problems list of the 2010 AIM workshop Components of Hilbert Schemes available at http://aimpl.org/hilbertschemes: does there exist a flat irreducible family of curves whose general member is a union of d disjoint lines on a smooth quadric surface and whose special member is a locally Cohen-Macaulay curve in a double plane? In this paper we give a positive answer to this question: for every d, we construct a family with the required properties, whose special fiber is an extremal curve in the sense of Martin-Deschamps and Perrin. From this we conclude that every effective divisor in a smooth quadric surface is in the connected component of its Hilbert scheme that contains extremal curves.
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