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1. -SIGNATURE UNDER BIRATIONAL MORPHISMS

   https://doi.org/10.1017/fms.2019.6  

   MA, LINQUAN; POLSTRA, THOMAS; SCHWEDE, KARL; TUCKER, KEVIN (January 2019, Forum of Mathematics, Sigma)

   We study $$F$$ -signature under proper birational morphisms $$\unicode[STIX]{x1D70B}:Y\rightarrow X$$ , showing that $$F$$ -signature strictly increases for small morphisms or if $$K_{Y}\leqslant \unicode[STIX]{x1D70B}^{\ast }K_{X}$$ . In certain cases, we can even show that the $$F$$ -signature of $$Y$$ is at least twice as that of  $$X$$ . We also provide examples of $$F$$ -signature dropping and Hilbert–Kunz multiplicity increasing under birational maps without these hypotheses. 

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2. CONTINUITY OF HILBERT–KUNZ MULTIPLICITY AND F-SIGNATURE

   https://doi.org/10.1017/nmj.2018.43  

   POLSTRA, THOMAS; SMIRNOV, ILYA (September 2020, Nagoya Mathematical Journal)

   We establish the continuity of Hilbert–Kunz multiplicity and F-signature as functions from a Cohen–Macaulay local ring $$(R,\mathfrak{m},k)$$ of prime characteristic to the real numbers at reduced parameter elements with respect to the $$\mathfrak{m}$$ -adic topology. 

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3. A Volume = Multiplicity formula for p-families of ideals

   https://doi.org/10.1090/proc/16451  

   Das, Sudipta (October 2023, Proceedings of the American Mathematical Society)

   In this article, we work with certain families of ideals called $p$ -families in rings of prime characteristic. This family of ideals is present in the theories of tight closure, Hilbert-Kunz multiplicity, and $F$ -signature. For each $p$ -family of ideals, we attach an Euclidean object called $p$ -body, which is analogous to the Newton Okounkov body associated with a graded family of ideals. Using the combinatorial properties of $p$ -bodies and algebraic properties of the Hilbert-Kunz multiplicity, we establish a Volume = Multiplicity formula for $p$ -families of ${\mathfrak{m}}_R$ -primary ideals in a Noetherian local ring $R$ . 

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4. Bernstein’s Inequality and Holonomicity for Certain Singular Rings

   https://doi.org/10.1093/imrn/rnad121  

   Montaner, Josep Àlvarez; Hernández, Daniel J; Jeffries, Jack; Núñez-Betancourt, Luis; Teixeira, Pedro; Witt, Emily E (June 2023, International Mathematics Research Notices)

   Abstract In this manuscript, we prove the Bernstein inequality and develop the theory of holonomic $$D$$-modules for rings of invariants of finite groups in characteristic zero, and for strongly $$F$$-regular finitely generated graded algebras with finite $$F$$-representation type in prime characteristic. In each of these cases, the ring itself, its localizations, and its local cohomology modules are holonomic. We also show that holonomic $$D$$-modules, in this context, have finite length, and we prove the existence of Bernstein–Sato polynomials in characteristic zero. We obtain these results using a more general version of Bernstein filtrations. 

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5. NUMERICAL SEMIGROUPS FROM RATIONAL MATRICES II: MATRICIAL DIMENSION DOES NOT EXCEED MULTIPLICITY

   https://doi.org/10.1017/S0004972724001035  

   CHHABRA, ARSH; GARCIA, STEPHAN RAMON; O’NEILL, CHRISTOPHER (December 2024, Bulletin of the Australian Mathematical Society)

   Abstract We continue our study of exponent semigroups of rational matrices. Our main result is that the matricial dimension of a numerical semigroup is at most its multiplicity (the least generator), greatly improving upon the previous upper bound (the conductor). For many numerical semigroups, including all symmetric numerical semigroups, our upper bound is tight. Our construction uses combinatorially structured matrices and is parametrised by Kunz coordinates, which are central to enumerative problems in the study of numerical semigroups. 

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