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Manoj Kummini

Publications and source records attributed to Manoj Kummini.

At least 19 recordsLinked to original sources
arXiv2025

On codimension-two subcanonical varieties inside $\mathbb{P}^n$

Let $X \subseteq \mathbb{P}^n, n \geq 4$ be a codimension-two subcanonical local complete intersection variety with ideal sheaf $\mathcal{I}_X$. Let $a_X \in \mathbb{Z}$ be such that $\omega_X = \mathscr{O}_X(a_X)$. Assume that there exists $\displaystyle j \leq \frac{a_X+n+2}{2}$ such that $\Gamma(\mathcal{I}_X(j)) \neq 0$. We prove some sufficient conditions on the first deficiency module $\mathrm{H}^1_*(\mathcal{I}_X)$ that ensures that $X$ is a complete intersection. We also show that smooth codimension-two $3$-Buchsbaum varieties inside $\mathbb{P}^n, n \geq 6$ are complete intersections.

math.AC↗
arXiv2025

Maximal minors of $1$-generic matrices have rational singularities

We show that the quotient ring by the ideal of maximal minors of a $1$-generic matrix has rational singularities. This answers a conjecture of Eisenbud (1988) that such rings are normal, and generalizes a result of Conca, Mostafazadehfard, Singh and Varbaro (2018) that generic Hankel determinantal rings have rational singularities in characteristic zero.

math.AC↗
arXiv2025

Ramification in modular invariant rings

Let $p$ be a prime number, $\Bbbk$ a field of characteristic $p$ and $G$ a finite $p$-group acting on a standard graded polynomial ring $S = \Bbbk[x_1, \ldots, x_n]$ as degree-preserving $\Bbbk$-algebra automorphisms. Assume that $G$ is generated by pseudo-reflections. In our earlier work (\emph{J. Pure Appl. Algebra}, vol. 228, no. 12, 2024) we introduced a composition series of $G$. In this note, we study the height-one ramification for the invariant rings at the consecutive stages of this composition series. We prove a condition for the extension $S^{G}\subseteq S^{G'}$ to split in terms of the Dedekind different $\mathscr{D}_D(S^{G'}/S^G)$. We construct an example illustrating that $\mathscr{D}_D(S^{G'}/S^G)$ need not have `expected' generators.

math.AC↗
arXiv2025

Generators of top cohomology

Let $R$ be a commutative noetherian ring and $f: X \to \mathrm{Spec} R$ a proper smooth morphism, of relative dimension $n$. From Hartshorne, Residues and Duality, Springer, 1966, one knows that the trace map $\mathrm{Tr}_f : \mathrm{H}^n(X, \omega_{X/R}) \to R$ is an isomorphism when $f$ has geometrically connected fibres. We construct an exact sequence that generates $\mathrm{Ext}_X^n(\mathcal{O}_X, \omega_{X/R}) = \mathrm{H}^n(X, \omega_{X/R})$ as an $R$-module in the following cases: (1) when $R$ is a DVR and $f$ has a section; (2) when $R=\mathbb{Z}$ and $X$ is the Grassmannian $G_{2,m}$ for some $m \geq 4$. This partially answers a question raised by Lipman.

math.AC↗
arXiv2023

Blow-up rings and $F$-rationality

In this paper, we prove some sufficient conditions for Cohen-Macaulay normal Rees algebras to be $F$-rational. Let $(R,\mathfrak{m})$ be a Gorenstein normal local domain of dimension $d\geq 2$ and of characteristic $p > 0$. Let $I$ be a $\mathfrak{m}$-primary ideal. Our first set of results give conditions on the test ideals $\tau(I^n)$, $n \geq 1$ which would imply that the normalization of the Rees algebra $R[It]$ is $F$-rational. Another sufficient condition is that the socle of $\mathrm{H}_{\overline{G}_+}^d(\overline{G})$ (where $\overline{G}$ is the associated graded ring for the integral closure filtration) is entirely in degree $-1$, if $R$ is $F$-rational (but not necessarily Gorenstein). Then we show that if $R$ is a hypersurface of degree $2$ or is three-dimensional and $F$-rational, and $\mathrm{Proj} (R[\mathfrak{m} t ])$ is $F$-rational, then $R[\mathfrak{m} t ]$ is $F$-rational.

math.AC↗
arXiv2022

On polynomial invariant rings in modular invariant theory

Let $\Bbbk$ be a field of characteristic $p>0$, $V$ a finite-dimensional $\Bbbk$-vector-space, and $G$ a finite $p$-group acting $\Bbbk$-linearly on $V$. Let $S = \Sym V^*$. We show that $S^G$ is a polynomial ring if and only if the dimension of its singular locus is less than $\rank_\Bbbk V^G$. Confirming a conjecture of Shank-Wehlau-Broer, we show that if $S^G$ is a direct summand of $S$, then $S^G$ is a polynomial ring, in the following cases: \begin{enumerate} \item $\Bbbk = \bbF_p$ and $\rank_\Bbbk V^G = 4$; or \item $|G| = p^3$. \end{enumerate} In order to prove the above result, we also show that if $\rank_\Bbbk V^G \geq \rank_\Bbbk V - 2$, then the Hilbert ideal $\hilbertIdeal_{G,S}$ is a complete intersection.

math.AC↗
arXiv2022

The Charney-Davis conjecture for simple thin polyominoes

Let $\mathcal{P}$ be a simple thin polyomino and $\Bbbk$ a field. Let $R$ be the toric $\Bbbk$-algebra associated to $\mathcal{P}$. Write the Hilbert series of $R$ as $h_{R}(t)/(1-t)^{\dim(R)}$. We show that $$(-1)^{\left\lfloor{\frac{\mathrm{deg} h_R(t)}{2}}\right\rfloor}h_{R}(-1) \geq 0$$ if $R$ is Gorenstein. This shows that the Gorenstein rings associated to simple thin polyominoes satisfy the Charney-Davis conjecture.

math.AC↗
arXiv2021

The $h$-polynomial and the rook polynomial of some polyominoes

Let $X$ be a convex polyomino such that its vertex set is a sublattice of $\mathbb{N}^2$. Let $\Bbbk[X]$ be the toric ring (over a field $\Bbbk$) associated to $X$ in the sense of Qureshi, \emph{J. Algebra}, 2012. Write the Hilbert series of $\Bbbk[X]$ as $(1 + h_1 t + h_2 t^2 + \cdots )/(1-t)^{\dim(\Bbbk[X])}$. For $k \in \mathbb{N}$, let $r_k$ be the number of configurations in $X$ with $k$ pairwise non-attacking rooks. We show that $h_2 < r_2$ if $X$ is not a thin polyomino. This partially confirms a conjectured characterization of thin polyominoes by Rinaldo and Romeo, \emph{J. Algebraic Combin.}, 2021.

math.AC↗
arXiv2021

On Hilbert ideals for a class of $p$-groups in characteristic $p$

Let $p$ be a prime number, $\Bbbk$ a field of characteristic $p$ and $G$ a finite $p$-group. Let $V$ be a finite-dimensional linear representation of $G$ over $\Bbbk$. Write $S = \mathrm{Sym} V^*$. For a class of $p$-groups which we call generalised Nakajima groups, we prove the following: \begin{enumerate} \item The Hilbert ideal is a complete intersection. As a consequence, for the case of generalised Nakajima groups, we prove a conjecture of Shank and Wehlau (reformulated by Broer) that asserts that if the invariant subring $S^G$ is a direct summand of $S$ as $S^G$-modules then $S^G$ is a polynomial ring. \item The Hilbert ideal has a generating set with elements of degree at most $|G |$. This bound is conjectured by Derksen and Kemper. \end{enumerate}

math.AC↗
arXiv2020

An Upper Bound for the First Hilbert Coefficient of Gorenstein Algebras and Modules

Let $R$ be a polynomial ring over a field and $M= \bigoplus_n M_n$ a finitely generated graded $R$-module, minimally generated by homogeneous elements of degree zero with a graded $R$-minimal free resolution $\mathbf{F}$. A Cohen-Macaulay module $M$ is Gorenstein when the graded resolution is symmetric. We give an upper bound for the first Hilbert coefficient, $e_1$ in terms of the shifts in the graded resolution of $M$. When $M = R/I$, a Gorenstein algebra, this bound agrees with the bound obtained in \cite{ES} in Gorenstein algebras with quasi-pure resolution. We conjecture a similar bound for the higher coefficients.

math.AC↗
arXiv2018

$F$-rationality of Rees algebras

In this paper, we study the $F$-rationality of the Rees algebra and the extended Rees algebra of $\mathfrak{m}$-primary ideals in excellent local rings $(R, \mathfrak{m})$ of prime characteristic. We partially answer some conjectures and questions raised by N. Hara, K.-i. Watanabe and K.-i. Yoshida (J. Algebra, pp.153--190, vol 247, 2002).

math.AC↗
arXiv2018

Depth and regularity modulo a principal ideal

We study the relationship between depth and regularity of a homogeneous ideal I and those of (I,f) and I:f, where f is a linear form or a monomial. Our results has several interesting consequences on depth and regularity of edge ideals of hypegraphs and of powers of ideals.

math.AC↗
arXiv2015

On conjectures of Itoh and of Lipman on the cohomology of normalized blow-ups

Let $(R, \mathfrak{m}, \Bbbk)$ be a Noetherian three-dimensional Cohen-Macaulay analytically unramified ring and $I$ an $\mathfrak{m}$-primary $R$-ideal. Write $X = \mathrm{Proj}\left(\oplus_{n \in \mathbb{N}} \overline{I^n}t^n\right)$. We prove some consequences of the vanishing of $\mathrm{H}^2(X, \mathscr{O}_X)$, whose length equals the the constant term $\bar e_3(I)$ of the normal Hilbert polynomial of $I$. Firstly, $X$ is Cohen-Macaulay. Secondly, if the extended Rees ring $A := \oplus_{n \in \mathbb{Z}} \overline{I^n}t^n$ is not Cohen-Macaulay, and either $R$ is equicharacteristic or $\overline{I} = \mathfrak{m}$, then $\bar e_2(I) - \mathrm{length}_R\left(\frac{\overline{I^2}}{I\overline{I}}\right) \geq 3$; this estimate is proved using Boij-Söderberg theory of coherent sheaves on $\mathbb{P}^2_\Bbbk$. The two results above are related to a conjecture of S. Itoh (J. Algebra, 1992). Thirdly, $\mathrm{H}^2_E(X, I^m\mathscr{O}_X) = 0$ for all integers $m$, where $E$ is the exceptional divisor in $X$. Finally, if additionally $R$ is regular and $X$ is pseudo-rational, then the adjoint ideals $\widetilde{I^n}, n \geq 1$ satisfy $\widetilde{I^n} = I\widetilde{I^{n-1}}$ for all $n \geq 3$. The last two results are related to conjectures of J. Lipman (Math. Res. Lett., 1994).

math.AC↗
arXiv2015

Free resolutions of some Schubert singularities

In this paper we construct free resolutions of certain class of closed subvarieties of affine spaces (the so-called "opposite big cells" of Grassmannians). Our class covers the determinantal varieties, whose resolutions were first constructed by A. Lascoux (Adv. Math., 1978). Our approach uses the geometry of Schubert varieties. An interesting aspect of our work is its connection to the computation of the cohomology of homogeneous bundles (that are not necessarily completely reducible) on partial flag varieties.

math.AG↗
arXiv2014

Poset Embeddings of Hilbert functions and Betti numbers

We study inequalities between graded Betti numbers of ideals in a standard graded algebra over a field and their images under embedding maps, defined earlier by us in [Math. Z. 274, (2013), no. 3-4, pp. 809-819; arXiv:1009.4488]. We show that if graded Betti numbers do not decrease when we replace ideals in an algebra by their embedded versions, then the same behaviour is carried over to ring extensions. As a corollary we give alternative inductive proofs of earlier results of Bigatti, Hulett, Pardue, Mermin-Peeva-Stillman and Murai. We extend a hypersurface restriction theorem of Herzog-Popescu to the situation of embeddings. We show that we can obtain the Betti table of an ideal in the extension ring from the Betti table of its embedded version by a sequence of consecutive cancellations. We further show that the lex-plus-powers conjecture of Evans reduces to the Artinian situation.

math.AC↗
arXiv2013

Tensor complexes: Multilinear free resolutions constructed from higher tensors

The most fundamental complexes of free modules over a commutative ring are the Koszul complex, which is constructed from a vector (i.e., a 1-tensor), and the Eagon-Northcott and the Buchsbaum-Rim complexes, which are constructed from a matrix (i.e., a 2-tensor). The subject of this paper is a multilinear analogue of these complexes, which we construct from an arbitrary higher tensor. Our construction provides detailed new examples of minimal free resolutions, as well as a unifying view on a wide variety of complexes including: the Eagon-Northcott, Buchsbaum-Rim and similar complexes, the Eisenbud-Schreyer pure resolutions, and the complexes used by Gelfand-Kapranov-Zelevinsky and Weyman to compute hyperdeterminants. In addition, we provide applications to the study of pure resolutions and Boij-Soederberg theory, including the construction of infinitely many new families of pure resolutions and the first explicit description of the differentials of the Eisenbud-Schreyer pure resolutions.

math.AC↗