arXiv subjects

Suchitra Pande

Publications and source records attributed to Suchitra Pande.

8 recordsLinked to original sources

arXiv2026

On positivity of the limit F-signature

We study a conjecture of Carvajal-Rojas, Schwede and Tucker which states that for a complex KLT singularity $(R, \mathfrak{m})$, the F-signatures of the reductions of $R$ to characteristic $p \gg 0$ remain bounded away from zero as $p \to \infty$. We prove that this conjecture holds for three-dimensional non-weakly exceptional singularities by an inductive argument. We also prove that the conjecture holds for smooth hypersurfaces of very low degree by constructing isotrivial normal toric degenerations. By considering the version of this conjecture for the Frobenius-alpha invariant, our techniques are inspired by K-stability theory and involve using degenerations and birational geometry.

math.AG↗

arXiv2025

Hilbert-Kunz multiplicity and $F$-signature can disagree

We compute the $F$-signature function of the ample cone of any nontrivial ruled surface over $\mathbb{P}^1_k$ where $k$ is an algebraically closed field of prime characteristic. As an application, we construct a Noetherian $F$-finite strongly $F$-regular ring $R$ of prime characteristic admitting two maximal ideals $\mathfrak{n}_1,\mathfrak{n}_2\in \mathrm{Spec} R$ at which the Hilbert-Kunz multiplicity and $F$-signature measure different singularities; that is, $\operatorname{e}_{\operatorname{HK}}(R_{\mathfrak{n}_1})<\operatorname{e}_{\operatorname{HK}}(R_{\mathfrak{n}_2})$ and $s(R_{\mathfrak{n}_1})<s(R_{\mathfrak{n}_2})$. Our calculation of the $F$-signature for the Hirzebruch surfaces also corrects an inaccuracy in a preprint by different authors.

math.AC↗

arXiv2025

The F-pure threshold versus the a-invariant for standard graded rings

Hirose, Watanabe and Yoshida conjectured a criterion for a standard graded strongly $F$-regular ring to be Gorenstein in terms of the $F$-pure threshold. We complete the proof of this conjecture. We also prove natural extensions of the conjecture to section rings of normal, $F$-split projective varieties with respect to globally generated ample divisors. Our proof exploits the geometry of the Proj of the graded ring.

math.AC↗

arXiv2025

Limit $F$-signature functions of two-variable binomial hypersurfaces

The $F$-signature is a fundamental numerical invariant of singularities in positive characteristic. Its positivity detects strong $F$-regularity, an important class of singularities related to KLT singularities in characteristic zero. In this paper, we compute the limiting $F$-signature function of binomial and other related hypersurfaces in two variables as the characteristic $p \to \infty$. In particular, we show it is a piecewise polynomial function, and relate it to the normalized volume.

math.AC↗

arXiv2025

Plus-pure thresholds of some cusp-like singularities in mixed characteristic

Log-canonical and $F$-pure thresholds of pairs in equal characteristic admit an analog in the recent theory of singularities in mixed characteristic, which is known as the plus-pure threshold. In this paper we study plus-pure thresholds for singularities of the form $p^a + x^b \in {\bf Z}_p [[ x ]]$, showing that in a number of cases this plus-pure threshold agrees with the $F$-pure threshold of the singularity $t^a + x^b \in {\bf F}_p [[ t, x ]]$. We also discuss a few other sporadic examples.

math.AG↗

arXiv2023

A Frobenius Version of Tians Alpha-Invariant

For a pair (X,L) consisting of a projective variety X over a perfect field of characteristic p>0 and an ample line bundle L on X, we introduce and study a positive characteristic analog of the $\alpha$-invariant introduced by Tian, which we call the $\alpha_F$-invariant. We utilize the theory of F-singularities in positive characteristics, and our approach is based on replacing klt singularities with the closely related notion of global F-regularity. We show that the $\alpha_F$-invariant of a pair (X,L) can be understood in terms of the global Frobenius splittings of the linear systems |mL|, for m>0. We establish inequalities relating the $\alpha_F$-invariant with the F- signature, and use that to prove the positivity of the $\alpha_F$-invariant for all globally F-regular projective varieties (with respect to any ample L on X). When X is a Fano variety and L is $-K_X$, we prove that the $\alpha_F$-invariant of X is always bounded above by 1/2 and establish tighter comparisons with the F-signature. We also show that for toric Fano varieties, the $\alpha_F$-invariant matches with the usual (complex) $\alpha$-invariant.

math.AG↗

arXiv2022

The F-signature Function on the Ample Cone

For any fixed globally F-regular projective variety X over an algebraically closed field of positive characteristic, we study the F-signature of section rings of X with respect to the ample Cartier divisors on X. In particular, we define an F-signature function on the ample cone of X and show that it is locally Lipschitz continuous. We further prove that the F-signature function extends to the boundary of the ample cone. We also establish an effective comparison between the F-signature function and the volume function on the ample cone. As a consequence, we show that for divisors that are nef but not big, the extension of the F-signature is zero.

math.AG↗

arXiv2021

Multiplicities of Jumping Numbers

We study multiplicities of jumping numbers of multiplier ideals in a smooth variety of arbitrary dimension. We prove that the multiplicity function is a quasi-polynomial, hence proving that the Poincar\'e series is a rational function. We further study when the various components of the quasi-polynomial have the highest possible degree and relate it to jumping numbers contributed by Rees valuations. Finally, we study the special case of monomial ideals.

math.AG↗