We give an explicit formula to count the number of geometric branches of a curve in positive characteristic using the theory of tight closure. This formula readily shows that the property of having a single geometric branch characterizes F-nilpotent curves. Further, we show that a reduced, local F-nilpotent ring has a single geometric branch; in particular, it is a domain. Finally, we study inequalities of Frobenius test exponents along purely inseparable ring extensions with applications to F-nilpotent affine semigroup rings.

This paper extends the results of Boij, Eisenbud, Erman, Schreyer and Söderberg on the structure of Betti cones of finitely generated graded modules and finite free complexes over polynomial rings, to all finitely generated graded rings admitting linear Noether normalizations. The key new input is the existence of lim Ulrich sequences of graded modules over such rings.

There are several ways to convert a closure or interior operation to a different operation that has particular desirable properties. In this paper, we axiomatize three ways to do so, drawing on disparate examples from the literature, including tight closure, basically full closure, and various versions of integral closure. In doing so, we explore several such desirable properties, including hereditary, residual, and cofunctorial, and see how they interact with other properties such as the finitistic property.

We investigate the geometry of codimension one foliations on smooth projective varieties defined over fields of positive characteristic with an eye toward applications to the structure of codimension one holomorphic foliations on projective manifolds.

In this article, we prove that a complete Noetherian local domain of mixed characteristic $p>0$ with perfect residue field has an integral extension that is an integrally closed, almost Cohen–Macaulay domain such that the Frobenius map is surjective modulo p. This result is seen as a mixed characteristic analog of the fact that the perfect closure of a complete local domain in positive characteristic is almost Cohen–Macaulay. To this aim, we carry out a detailed study of decompletion of perfectoid rings and establish the Witt-perfect (decompleted) version of André’s perfectoid Abhyankar’s lemma and Riemann’s extension theorem.

Consider a reductive linear algebraic group G acting linearly on a polynomial ring S over an infinite field; key examples are the general linear group, the symplectic group, the orthogonal group, and the special linear group, with the classical representations as in Weyl’s book: For the general linear group, consider a direct sum of copies of the standard representation and copies of the dual; in the other cases, take copies of the standard representation. The invariant rings in the respective cases are determinantal rings, rings defined by Pfaffians of alternating matrices, symmetric determinantal rings and the Plücker coordinate rings of Grassmannians; these are the classical invariant rings of the title, with $S^G\subseteq S$ being the natural embedding.

Over a field of characteristic zero, a reductive group is linearly reductive, and it follows that the invariant ring $S^G$ is a pure subring of S, equivalently, $S^G$ is a direct summand of S as an $S^G$-module. Over fields of positive characteristic, reductive groups are typically no longer linearly reductive. We determine, in the positive characteristic case, precisely when the inclusion $S^G\subseteq S$ is pure. It turns out that if $S^G\subseteq S$ is pure, then either the invariant ring $S^G$ is regular or the group G is linearly reductive.

Hilbert–Kunz multiplicity and F-signature are numerical invariants of commutative rings in positive characteristic that measure severity of singularities: for a regular ring both invariants are equal to one and the converse holds under mild assumptions. A natural question is for what singular rings these invariants are closest to one. For Hilbert–Kunz multiplicity this question was first considered by the last two authors and attracted significant attention. In this paper, we study this question, i.e., an upper bound, for F-signature and revisit lower bounds on Hilbert–Kunz multiplicity.

Let $f\colon Y \to X$ be a proper flat morphism of locally noetherian schemes. Then the locus in $X$ over which $f$ is smooth is stable under generization. We prove that, under suitable assumptions on the formal fibers of $X$, the same property holds for other local properties of morphisms, even if $f$ is only closed and flat. Our proof of this statement reduces to a purely local question known as Grothendieck's localization problem. To answer Grothendieck's problem, we provide a general framework that gives a uniform treatment of previously known cases of this problem, and also solves this problem in new cases, namely for weak normality, seminormality, $F$-rationality, and the ‘Cohen–Macaulay and $F$-injective’ property. For the weak normality statement, we prove that weak normality always lifts from Cartier divisors. We also solve Grothendieck's localization problem for terminal, canonical, and rational singularities in equal characteristic zero.

Unfortunately, there is a mistake in [PS, Lemma 3.10] which invalidates [PS, Theorem 3.12]. We show that the theorem still holds if the ring is assumed to be Gorenstein.

Following the work of Mustaţă and Bitoun, we recently developed a notion of Bernstein–Sato roots for arbitrary ideals, which is a prime characteristic analogue for the roots of the Bernstein–Sato polynomial. Here, we prove that for monomial ideals the roots of the Bernstein–Sato polynomial (over $\mathbb{C}$) agree with the Bernstein–Sato roots of the mod $p$ reductions of the ideal for $p$ large enough. We regard this as evidence that the characteristic-$p$ notion of Bernstein–Sato root is reasonable.

We prove that the essential dimension of central simple algebras of degree $p^{\ell m}$ and exponent $p^{m}$ over fields $F$ containing a base-field $k$ of characteristic $p$ is at least $\ell +1$ when $k$ is perfect. We do this by observing that the $p$-rank of $F$ bounds the symbol length in $\text{Br}_{p^{m}}(F)$ and that there exist indecomposable $p$-algebras of degree $p^{\ell m}$ and exponent $p^{m}$. We also prove that the symbol length of the Kato-Milne cohomology group $\text{H}_{p^{m}}^{n+1}(F)$ is bounded from above by $\binom{r}{n}$ where $r$ is the $p$-rank of the field, and provide upper and lower bounds for the essential dimension of Brauer classes of a given symbol length.

We extend results on asymptotic invariants of line bundles on complex projective varieties to projective varieties over arbitrary fields. To do so over imperfect fields, we prove a scheme-theoretic version of the gamma construction of Hochster and Huneke to reduce to the setting where the ground field is $F$-finite. Our main result uses the gamma construction to extend the ampleness criterion of de Fernex, Küronya, and Lazarsfeld using asymptotic cohomological functions to projective varieties over arbitrary fields, which was previously known only for complex projective varieties. We also extend Nakayama’s description of the restricted base locus to klt or strongly $F$-regular varieties over arbitrary fields.

Let (R, ) be an analytically unramified local ring of positive prime characteristic p. For an ideal I, let I* denote its tight closure. We introduce the tight Hilbert function $$H_I^*\left( n \right) = \Im \left( {R/\left( {{I^n}} \right)*} \right)$$ and the corresponding tight Hilbert polynomial $$P_I^*\left( n \right)$$, where I is an m-primary ideal. It is proved that F-rationality can be detected by the vanishing of the first coefficient of $$P_I^*\left( n \right)$$. We find the tight Hilbert polynomial of certain parameter ideals in hypersurface rings and Stanley-Reisner rings of simplicial complexes.

In this paper, we prove that the set of all $F$-pure thresholds on a fixed germ of a strongly $F$-regular pair satisfies the ascending chain condition. As a corollary, we verify the ascending chain condition for the set of all $F$-pure thresholds on smooth varieties or, more generally, on varieties with tame quotient singularities, which is an affirmative answer to a conjecture given by Blickle, Mustaţǎ and Smith.

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.

We develop the analog of crystalline Dieudonné theory for $p$-divisible groups in the arithmetic of function fields. In our theory $p$-divisible groups are replaced by divisible local Anderson modules, and Dieudonné modules are replaced by local shtukas. We show that the categories of divisible local Anderson modules and of effective local shtukas are anti-equivalent over arbitrary base schemes. We also clarify their relation with formal Lie groups and with global objects like Drinfeld modules, Anderson’s abelian $t$-modules and $t$-motives, and Drinfeld shtukas. Moreover, we discuss the existence of a Verschiebung map and apply it to deformations of local shtukas and divisible local Anderson modules. As a tool we use Faltings’s and Abrashkin’s theories of strict modules, which we review briefly.

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.

In this paper, we study the singularities of a general hyperplane section $H$ of a three-dimensional quasi-projective variety $X$ over an algebraically closed field of characteristic $p>0$. We prove that if $X$ has only canonical singularities, then $H$ has only rational double points. We also prove, under the assumption that $p>5$, that if $X$ has only klt singularities, then so does $H$.

Let $R$ be a commutative Noetherian ring of prime characteristic $p$. In this paper, we give a short proof using filter regular sequences that the set of associated prime ideals of $H_{I}^{t}(R)$ is finite for any ideal $I$ and for any $t\geqslant 0$ when $R$ has finite $F$-representation type or finite singular locus. This extends a previous result by Takagi–Takahashi and gives affirmative answers for a problem of Huneke in many new classes of rings in positive characteristic. We also give a criterion about the singularities of $R$ (in any characteristic) to guarantee that the set $\operatorname{Ass}H_{I}^{2}(R)$ is always finite.

Let $(R,\mathfrak{m})$ be a Noetherian local ring of characteristic $p>0$. We introduce and study $F$-full and $F$-anti-nilpotent singularities, both are defined in terms of the Frobenius actions on the local cohomology modules of $R$ supported at the maximal ideal. We prove that if $R/(x)$ is $F$-full or $F$-anti-nilpotent for a nonzero divisor $x\in R$, then so is $R$. We use these results to obtain new cases on the deformation of $F$-injectivity.