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Register a new userLocal cohomology of Du Bois singularities and applications to families
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In this paper we study the local cohomology modules of Du Bois singularities. Let $(R,m)$ be a local ring, we prove that if $R_{red}$ is Du Bois, then $H_m^i(R)to H_m^i(R_{red})$ is surjective for every $i$. We find many applications of this result. For example we answer a question of Kovacs and the second author on the Cohen-Macaulay property of Du Bois singularities. We obtain results on the injectivity of $Ext$ that provide substantial partial answers of questions of Eisenbud-Mustata-Stillman in characteristic $0$, and these results can be viewed as generalizations of the Kodaira vanishing theorem for Cohen-Macaulay Du Bois varieties. We prove results on the set-theoretic Cohen-Macaulayness of the defining ideal of Du Bois singularities, which are characteristic $0$ analog of results of Singh-Walther and answer some of their questions. We extend results of Hochster-Roberts on the relation between Koszul cohomology and local cohomology for $F$-injective and Du Bois singularities, see Hochster-Roberts. We also prove that singularities of dense $F$-injective type deform.
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We study the Hodge filtration on the local cohomology sheaves of a smooth complex algebraic variety along a closed subscheme Z in terms of log resolutions, and derive applications regarding the local cohomological dimension, the Du Bois complex, local vanishing, and reflexive differentials associated to Z.
The ACC conjecture for local volumes predicts that the set of local volumes of klt singularities $xin (X,Delta)$ satisfies the ACC if the coefficients of $Delta$ belong to a DCC set. In this paper, we prove the ACC conjecture for local volumes under the assumption that the ambient germ is analytically bounded. We introduce another related conjecture, which predicts the existence of $delta$-plt blow-ups of a klt singularity whose local volume has a positive lower bound. We show that the latter conjecture also holds when the ambient germ is analytically bounded. Moreover, we prove that both conjectures hold in dimension 2 as well as for 3-dimensional terminal singularities.
We consider a series of four subexceptional representations coming from the third line of the Freudenthal-Tits magic square; using Bourbaki notation, these are fundamental representations $(G,X)$ corresponding to $(C_3, omega_3),, (A_5, omega_3), , (D_6, omega_5)$ and $(E_7, omega_6)$. In each of these four cases, the group $G=Gtimes mathbb{C}^*$ acts on $X$ with five orbits, and many invariants display a uniform behavior, e.g. dimension of orbits, their defining ideals and the character of their coordinate rings as $G$-modules. In this paper, we determine some more subtle invariants and analyze their uniformity within the series. We describe the category of $G$-equivariant coherent $mathcal{D}_X$-modules as the category of representations of a quiver with relations. We construct explicitly the simple $G$-equivariant $mathcal{D}_X$-modules and compute the characters of their underlying $G$-structures. We determine the local cohomology groups with supports given by orbit closures, determining their precise $mathcal{D}_X$-module structure. As a consequence, we calculate the intersection cohomology groups and Lyubeznik numbers of the orbit closures. While our results for the cases $(A_5, omega_3), , (D_6, omega_5)$ and $(E_7, omega_6)$ are still completely uniform, the case $(C_3, omega_3)$ displays a surprisingly different behavior. We give two explanations for this phenomenon: one topological, as the middle orbit of $(C_3, omega_3)$ is not simply-connected; one geometric, as the closure of the orbit is not Gorenstein.
We study the coherent cohomology of generalized flag supervarieties. Our main observation is that these groups are closely related to the free resolutions of (certain generalizations of) determinantal ideals. In the case of super Grassmannians, we completely compute the cohomology of the structure sheaf: it is composed of the singular cohomology of a Grassmannian and the syzygies of a determinantal variety. The majority of the work involves studying the geometry of an analog of the Grothendieck-Springer resolution associated to the super Grassmannian; this takes place in the world of ordinary (non-super) algebraic geometry. Our work gives a conceptual explanation of the result of Pragacz-Weyman that the syzygies of determinantal ideals admit an action of the general linear supergroup. In a subsequent paper, we will treat other flag supervarieties in detail.
The application of methods of computational algebra has recently introduced new tools for the study of Hilbert schemes. The key idea is to define flat families of ideals endowed with a scheme structure whose defining equations can be determined by algorithmic procedures. For this reason, several authors developed new methods, based on the combinatorial properties of Borel-fixed ideals, that allow to associate to each ideal $J$ of this type a scheme $mathbf{Mf}_{J}$, called $J$-marked scheme. In this paper we provide a solid functorial foundation to marked schemes and show that the algorithmic procedures introduced in previous papers do not depend on the ring of coefficients. We prove that for all strongly stable ideals $J$, the marked schemes $mathbf{Mf}_{J}$ can be embedded in a Hilbert scheme as locally closed subschemes, and that they are open under suitable conditions on $J$. Finally, we generalize Lederers result about Grobner strata of zero-dimensional ideals, proving that Grobner strata of any ideals are locally closed subschemes of Hilbert schemes.
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