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1. Stable behavior of Frobenius powers over a general hypersurface

   https://doi.org/10.1090/btran/199  

   Miller, Claudia ; Rahmati, Hamidreza ; R_G, Rebecca ( November 2024 , Transactions of the American Mathematical Society, Series B)

   The main goal of this paper is to prove, in positive characteristic$p$, stability behavior for the graded Betti numbers in the periodic tails of the minimal resolutions of Frobenius powers of the homogeneous maximal ideals for very general choices of hypersurface in three variables whose degree has the opposite parity to that of$p$. We also find some of the structure of the matrix factorization giving the resolution. We achieve this by developing a method for obtaining the degrees of the generators of the defining ideal of an$\mathfrak {c}$-compressed Gorenstein Artinian graded algebra from its socle degree, where$\mathfrak {c}$is a Frobenius power of the homogeneous maximal ideal. As an application, we also obtain the Hilbert–Kunz function of the hypersurface ring, as well as the Castelnuovo–Mumford regularity of the quotients by Frobenius powers of the homogeneous maximal ideal.

    

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2. Expected Resurgence of Ideals Defining Gorenstein Rings

   https://doi.org/10.1307/mmj/20206004  

   Grifo, Eloísa ; Huneke, Craig ; Mukundan, Vivek ( August 2023 , Michigan Mathematical Journal)

   Building on previous work by the same authors, we show that certain ideals defining Gorenstein rings have expected resurgence and thus satisfy the stable Harbourne conjecture. In prime characteristic, we can take any radical ideal defining a Gorenstein ring in a regular ring, provided that its symbolic powers are given by saturations with the maximal ideal. Although this property is not suitable for reduction to characteristic p, we show that a similar result holds in equicharacteristic 0 under the additional hypothesis that the symbolic Rees algebra of I is Noetherian. 

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3. Residual Intersections and linear powers

   EISENBUD, David ; HUNEKE, CRAIG ; ULRICH, BERND ( July 2024 , Transactions of the American Mathematical Society)

   If I is an ideal in a Gorenstein ring S, and S/I is Cohen-Macaulay, then the same is true for any linked ideal I ; but such statements hold for residual intersections of higher codimension only under restrictive hypotheses, not satisfied even by ideals as simple as the ideal Ln of minors of a generic 2 × n matrix when n > 3. In this paper we initiate the study of a different sort of Cohen-Macaulay property that holds for certain general residual intersections of the maximal (interesting) codimension, one less than the analytic spread of I. For example, suppose that K is the residual intersection of Ln by 2n − 4 general quadratic forms in Ln. In this situation we analyze S/K and show that In−3(S/K) is a self-dual maximal Cohen-Macaulay S/K-module with linear free resolution over S. The technical heart of the paper is a result about ideals of analytic spread 1 whose high powers are linearly presented. 

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4. Frobenius powers

   https://doi.org/10.1007/s00209-019-02442-2  

   Hernández, Daniel J. ; Teixeira, Pedro ; Witt, Emily E. ( January 2020 , Mathematische Zeitschrift)

   null (Ed.)

   This article extends the notion of a Frobenius power of an ideal in prime characteristic to allow arbitrary nonnegative real exponents. These generalized Frobenius powers are closely related to test ideals in prime characteristic, and multiplier ideals over fields of characteristic zero. For instance, like these well-known families of ideals, Frobenius powers also give rise to jumping exponents that we call critical Frobenius exponents. In fact, the Frobenius powers of a principal ideal coincide with its test ideals, but Frobenius powers appear to be a more refined measure of singularities than test ideals in general. Herein, we develop the theory of Frobenius powers in regular domains, and apply it to study singularities, especially those of generic hypersurfaces. These applications illustrate one way in which multiplier ideals behave more like Frobenius powers than like test ideals. 

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5. Axial constants and sectional regularity of homogeneous ideals

   https://doi.org/10.1090/proc/16244  

   DeBellevue, Michael ; Lebovitz, Audric ; Li, Yik ; Lotfi, Mohamed ; Mohite, Shivam ; Pan, Xin ; Pathak, Mrigank ; Roshan Zamir, Shah ; Seceleanu, Alexandra ; Zhang, Sindy ( January 2023 , Proceedings of the American Mathematical Society)

   We introduce a notion of sectional regularity for a homogeneous ideal I, which measures the regularity of its general sections with respect to linear spaces of various dimensions. It is related to axial constants defined as the intercepts on the coordinate axes of the set of exponents of monomials in the reverse lexicographic generic initial ideal of I. We show the equivalence of these notions and several other homological and ideal-theoretic invariants. We also establish that these equivalent invariants grow linearly for the family of powers of a given ideal. 

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