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Anurag K. Singh

Publications and source records attributed to Anurag K. Singh.

At least 37 records · Page 2Linked to original sources
arXiv2021

Stabilization of the cohomology of thickenings

For a local complete intersection subvariety $X=V({\mathcal I})$ in ${\mathbb P}^n$ over a field of characteristic zero, we show that, in cohomological degrees smaller than the codimension of the singular locus of $X$, the cohomology of vector bundles on the formal completion of ${\mathbb P}^n$ along $X$ can be effectively computed as the cohomology on any sufficiently high thickening $X_t=V({\mathcal I^t})$; the main ingredient here is a positivity result for the normal bundle of $X$. Furthermore, we show that the Kodaira vanishing theorem holds for all thickenings $X_t$ in the same range of cohomological degrees; this extends the known version of Kodaira vanishing on $X$, and the main new ingredient is a version of the Kodaira-Akizuki-Nakano vanishing theorem for $X$, formulated in terms of the cotangent complex.

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arXiv2021

An asymptotic vanishing theorem for the cohomology of thickenings

Let $X$ be a closed equidimensional local complete intersection subscheme of a smooth projective scheme $Y$ over a field, and let $X_t$ denote the $t$-th thickening of $X$ in $Y$. Fix an ample line bundle $\mathcal{O}_Y(1)$ on $Y$. We prove the following asymptotic formulation of the Kodaira vanishing theorem: there exists an integer $c$, such that for all integers $t \geqslant 1$, the cohomology group $H^k(X_t,\mathcal{O}_{X_t}(j))$ vanishes for $k < \dim X$ and $j < -ct$. Note that there are no restrictions on the characteristic of the field, or on the singular locus of $X$. We also construct examples illustrating that a linear bound is indeed the best possible, and that the constant $c$ is unbounded, even in a fixed dimension.

math.AG↗
arXiv2020

Koszul and local cohomology, and a question of Dutta

For a local ring $(A,\mathfrak{m})$ of dimension $n$, we study the natural map from the Koszul cohomology module $H^n(\mathfrak{m}; A)$ to the local cohomology module $H^n_\mathfrak{m}(A)$. We prove that the injectivity of this map characterizes the Cohen-Macaulay property of the ring $A$. We also answer a question of Dutta by constructing normal rings $A$ for which this map is zero.

math.AC↗
arXiv2019

On a conjecture of Lynch

This concerns a conjecture of Lynch on annihilators of local cohomology modules; we present a counterexample that has dimension three.

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arXiv2018

Hankel determinantal rings have rational singularities

Hankel determinantal rings, i.e., determinantal rings defined by minors of Hankel matrices of indeterminates, arise as homogeneous coordinate rings of higher order secant varieties of rational normal curves; they may also be viewed as linear specializations of generic determinantal rings. We prove that, over fields of characteristic zero, Hankel determinantal rings have rational singularities; in the case of positive prime characteristic, we prove that they are F-pure. Independent of the characteristic, we give a complete description of the divisor class groups of these rings, and show that each divisor class group element is the class of a maximal Cohen-Macaulay module.

math.AC↗
arXiv2017

Local cohomology modules supported at determinantal ideals

We provide new results on the vanishing of local cohomology modules supported at ideals of minors of matrices over arbitrary commutative Noetherian rings. In the process, we compute the local cohomology of rings of polynomials with integer coefficients---supported at generic determinantal ideals---and also obtain results on F-modules and D-modules that are likely to be of independent interest.

math.AC↗
arXiv2015

The F-pure threshold of a Calabi-Yau hypersurface

The F-pure threshold is a positive characteristic numerical invariant, analogous to the log canonical threshold in characteristic zero. We compute the F-pure threshold of the affine cone over a Calabi-Yau hypersurface, and relate it to the order of vanishing of the Hasse invariant on the versal deformation space of the hypersurface.

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arXiv2013

The $F$-pure threshold of a determinantal ideal

The $F$-pure threshold is a numerical invariant of prime characteristic singularities, that constitutes an analogue of the log canonical thresholds in characteristic zero. We compute the $F$-pure thresholds of determinantal ideals, i.e., of ideals generated by the minors of a generic matrix.

math.AC↗
arXiv2013

Local cohomology modules of a smooth Z-algebra have finitely many associated primes

Let $R$ be a commutative Noetherian ring that is a smooth $\mathbb Z$-algebra. For each ideal $I$ of $R$ and integer $k$, we prove that the local cohomology module $H^k_I(R)$ has finitely many associated prime ideals. This settles a crucial outstanding case of a conjecture of Lyubeznik asserting this finiteness for local cohomology modules of all regular rings.

math.AC↗
arXiv2013

Rings of Frobenius operators

Let R be a local ring of prime characteristic. We study the ring of Frobenius operators F(E), where E is the injective hull of the residue field of R. In particular, we examine the finite generation of F(E) over its degree zero component, and show that F(E) need not be finitely generated when R is a determinantal ring; nonetheless, we obtain concrete descriptions of F(E) in good generality that we use, for example, to prove the discreteness of F-jumping numbers for arbitrary ideals in determinantal rings.

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arXiv2011

Galois extensions, plus closure, and maps on local cohomology

Given a local domain $(R,m)$ of prime characteristic that is a homomorphic image of a Gorenstein ring, Huneke and Lyubeznik proved that there exists a module-finite extension domain $S$ such that the induced map on local cohomology modules $H^i_m(R)\to H^i_m(S)$ is zero for each $i<\dim R$. We prove that the extension $S$ may be chosen to be generically Galois, and analyze the Galois groups that arise.

math.AC↗
arXiv2009

Bockstein homomorphisms in local cohomology

Let $R$ be a polynomial ring in finitely many variables over the integers, and fix an ideal $I$ of $R$. We prove that for all but finitely prime integers $p$, the Bockstein homomorphisms on local cohomology, $H^k_I(R/pR)\to H^{k+1}_I(R/pR)$, are zero. This provides strong evidence for Lyubeznik's conjecture which states that the modules $H^k_I(R)$ have a finite number of associated prime ideals.

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arXiv2009

Divisor class groups of graded hypersurfaces

We demonstrate how some classical computations of divisor class groups can be obtained using the theory of rational coefficient Weil divisors and related results of Watanabe.

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arXiv2009

Multigraded rings, diagonal subalgebras, and rational singularities

We study the properties of F-rationality and F-regularity in multigraded rings and their diagonal subalgebras. The main focus is on diagonal subalgebras of bigraded rings: these constitute an interesting class of rings since they arise naturally as homogeneous coordinate rings of blow-ups of projective varieties. As a consequence of some of the results obtained here, it is shown that there exist standard bigraded hypersurfaces whose rings of invariants under torus actions have rational singularities, but are not of F-regular type. Another application is the construction of families of rings with divisor class groups that are finitely generated, but not discrete in the sense of Danilov.

math.AC↗
arXiv2007

Annihilators of Local Cohomology in Characteristic Zero

This paper discusses the problem of whether it is possible to annihilate elements of local cohomology modules by elements of arbitrarily small order under a fixed valuation. We first discuss the general problem and its relationship to the Direct Summand Conjecture, and next present two concrete examples where annihilators with small order are shown to exist. We then prove a more general theorem, where the existence of such annihilators is established in some cases using results on abelian varieties and the Albanese map.

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