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We give a notion of combinatorial proximity among strongly stable ideals in a given polynomial ring with a fixed Hilbert polynomial. We show that this notion guarantees geometric proximity of the corresponding points in the Hilbert scheme. We define a graph whose vertices correspond to strongly stable ideals and whose edges correspond to pairs of adjacent ideals. Every term order induces an orientation of the edges of the graph. This directed graph describes the behavior of the points of the Hilbert scheme under Grobner degenerations with respect to the given term order. Then, we introduce a polyhedral fan that we call Grobner fan of the Hilbert scheme. Each cone of maximal dimension corresponds to a different directed graph induced by a term order. This fan encodes several properties of the Hilbert scheme. We use these tools to present a new proof of the connectedness of the Hilbert scheme. Finally, we improve the technique introduced in the paper Double-generic initial ideal and Hilbert scheme by Bertone, Cioffi and Roggero to give a lower bound on the number of irreducible components of the Hilbert scheme.
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The Hilbert scheme $mathbf{Hilb}_{p(t)}^{n}$ parametrizes closed subschemes and families of closed subschemes in the projective space $mathbb{P}^n$ with a fixed Hilbert polynomial $p(t)$. It is classically realized as a closed subscheme of a Grassmannian or a product of Grassmannians. In this paper we consider schemes over a field $k$ of characteristic zero and we present a new proof of the existence of the Hilbert scheme as a subscheme of the Grassmannian $mathbf{Gr}_{p(r)}^{N(r)}$, where $N(r)= h^0 (mathcal{O}_{mathbb{P}^n}(r))$. Moreover, we exhibit explicit equations defining it in the Plucker coordinates of the Plucker embedding of $mathbf{Gr}_{p(r)}^{N(r)}$. Our proof of existence does not need some of the classical tools used in previous proofs, as flattening stratifications and Gotzmanns Persistence Theorem. The degree of our equations is $text{deg} p(t)+2$, lower than the degree of the equations given by Iarrobino and Kleiman in 1999 and also lower (except for the case of hypersurfaces) than the degree of those proved by Haiman and Sturmfels in 2004 after Bayers conjecture in 1982. The novelty of our approach mainly relies on the deeper attention to the intrinsic symmetries of the Hilbert scheme and on some results about Grassmannian based on the notion of extensors.
Following the approach in the book Commutative Algebra, by D. Eisenbud, where the author describes the generic initial ideal by means of a suitable total order on the terms of an exterior power, we introduce first the generic initial extensor of a subset of a Grassmannian and then the double-generic initial ideal of a so-called GL-stable subset of a Hilbert scheme. We discuss the features of these new notions and introduce also a partial order which gives another useful description of them. The double-generic initial ideals turn out to be the appropriate points to understand some geometric properties of a Hilbert scheme: they provide a necessary condition for a Borel ideal to correspond to a point of a given irreducible component, lower bounds for the number of irreducible components in a Hilbert scheme and the maximal Hilbert function in every irreducible component. Moreover, we prove that every isolated component having a smooth double-generic initial ideal is rational. As a byproduct, we prove that the Cohen-Macaulay locus of the Hilbert scheme parameterizing subschemes of codimension 2 is the union of open subsets isomorphic to affine spaces. This improves results by J. Fogarty (1968) and R. Treger (1989).
We study the locus of the liftings of a homogeneous ideal $H$ in a polynomial ring over any field. We prove that this locus can be endowed with a structure of scheme $mathrm L_H$ by applying the constructive methods of Grobner bases, for any given term order. Indeed, this structure does not depend on the term order, since it can be defined as the scheme representing the functor of liftings of $H$. We also provide an explicit isomorphism between the schemes corresponding to two different term orders. Our approach allows to embed $mathrm L_H$ in a Hilbert scheme as a locally closed subscheme, and, over an infinite field, leads to find interesting topological properties, as for instance that $mathrm L_H$ is connected and that its locus of radical liftings is open. Moreover, we show that every ideal defining an arithmetically Cohen-Macaulay scheme of codimension two has a radical lifting, giving in particular an answer to an open question posed by L. G. Roberts in 1989.
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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.
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.
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