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Yevgeniya Tarasova

Publications and source records attributed to Yevgeniya Tarasova.

6 recordsLinked to original sources
arXiv2026

The F-signature of Determinantal Rings

Let $R_{n,p}$ be the determinantal hypersurface ring defined by the determinant of a generic $n \times n$ matrix in characteristic $p$ where $n \geq 2$. In this paper, we obtain explicit bounds for the $F$-signature of $R_{n,p}$. For the upper bound, we prove that this $F$-signature is decreasing in $n$, so the $F$-signature of $R_{2,p}$ serves as an upper bound, which is $2/3$. For the lower bound, we find a toric degeneration $B_n$ of $R_{n,p}$, so the $F$-signature of $B_n$ serves as a lower bound. This $F$-signature can be computed using either Gröbner basis or Han-Monsky machinery, and its value is $\frac{(n!)^2}{(2n-1)!}$.

math.AC↗
arXiv2024

On the natural nullcones of the symplectic and general linear groups

Consider a group acting on a polynomial ring S over a field K by degree-preserving K-algebra automorphisms. Several key properties of the invariant ring can be deduced by studying the nullcone of the action, that is, the vanishing locus of all non-constant homogeneous invariant polynomials. These properties include the finite generation of the invariant ring and the purity of its embedding in S. In this article, we study the nullcones arising from the natural actions of the symplectic and general linear groups. For the natural representation of the symplectic group (via copies of the regular representation), the invariant ring is a generic Pfaffian ring. We show that the nullcone of this embedding is F-regular in positive characteristic. Independent of characteristic, we give a complete description of the divisor class group of the nullcone and determine precisely when it is Gorenstein. For the natural representation of the general linear group (via copies of the regular representation and copies of its dual), the invariant ring is a generic determinantal ring. The nullcone of this embedding is typically non-equidimensional; its irreducible components are the varieties of complexes introduced by Buchsbaum and Eisenbud. We show that each of these irreducible components are F-regular in positive characteristic. We also show that the Frobenius splittings of the varieties of complexes may be chosen compatibly so that the nullcone is F-pure.

math.AC↗
arXiv2024

Linkage and $F$-Regularity of Determinantal Rings

In this paper, we prove that the generic link of a generic determinantal ring defined by maximal minors is strongly $F$-regular. In the process, we strengthen a result of Chardin and Ulrich in the graded setting. They showed that the generic residual intersections of a complete intersection ring with rational singularities again have rational singularities. We show that if the said complete intersection is defined by homogeneous elements and is $F$-rational, then in fact, its generic residual intersections are strongly $F$-regular in positive prime characteristic. Hochster and Huneke showed that determinantal rings are strongly $F$-regular; however, their proof is quite involved. Our techniques allow us to give a new and simple proof of the strong $F$-regularity of determinantal rings defined by maximal minors.

math.AC↗
arXiv2022

Generating Residual Intersections of Determinantal Ideals

If $I$ is a perfect ideal in a local Cohen-Macaulay ring, the generators of ideals linked to $I$ are well understood. However, the generators of the residual intersections of $I$ have only been computed in a few special cases. In this paper, we show that the $n$-residual intersections of determinantal ideals of generic $2\times n$ matrices are sums of links.

math.AC↗
arXiv2021

Generators of Residual Intersections

Under suitable technical assumptions, a description is given for the generators of $s$-residual intersections of an ideal $I$ in terms of lower residual intersections, if $s \geq μ(I)-2$. This implies that $s$-residual intersections can be expressed in terms of links, if the deviation of $I$ is less than or equal to 3 and some other hypotheses are satisfied.

math.AC↗