arXiv subjects
Publications and source records attributed to Kevin Tucker.
For every integer $d \geq 2$, we show that there exists an irreducible, reduced curve which embeds in $\mathbb{P}^d$ but in no projective normal toric variety of dimension less than $d$. In particular, there exist reduced irreducible curves that embed in $\mathbb{P}^d$ but do not embed in any weighted projective space of dimension less than $d$.
In this paper we show that any Noetherian $F$-finite scheme has a dualizing complex $\omega^{\bullet}_{X}$ with the property that for all finite type maps $f \colon X \to Y$ between $F$-finite Noetherian schemes there is a canonical isomorphism $\omega^{\bullet}_{X} \xrightarrow{\cong} f^!\omega^{\bullet}_{Y}$ in $D^b_{coh}(X)$. This, in particular, applies to the Frobenius morphism $F \colon X \to X$ so that we obtain a canonical isomorphism $\omega^{\bullet}_{X} \xrightarrow{\cong} F^!\omega^{\bullet}_{X}$. To prove this, we rely on a result of Gabber that every Noetherian $F$-finite ring is a quotient of a regular ring, from which it follows that every $F$-finite Noetherian scheme has a (potentially non-canonical) dualizing complex. To make this canonical, we identify the dualizing complex of any $F$-finite Noetherian scheme as a unit of an alternate symmetric monoidal structure on $D^b_{coh}(X)$ we call the $!$-tensor product. We also sketch an alternate approach to finding this canonical dualizing complex following the more classical approach to Grothendieck duality.
We prove that globally $+$-regular varieties are rationally chain connected in dimension three and mixed characteristic with residue field characteristic $p>5$. We also introduce a notion of strongly globally $+$-regular, and show that varieties of arbitrary dimension which are strongly globally $+$-regular over a dense open subset of $\mathrm{Spec}(\mathbb{Z})$ are rationally chain connected.
We show, using the techniques developed in arXiv:2504.06444 and arXiv:2305.11139, that dagger algebras and Tate algebras in the sense of Berkovich in prime characteristic $p > 0$ have intersection flat Frobenius. Equivalently, if $S$ is such a ring, then $S^{1/p}$ is a flat and Mittag-Leffler $S$-module. As a consequence, we deduce that any ideal-adic completion of a reduced ring that is essentially of finite type over a dagger algebra or a Berkovich Tate algebra in prime characteristic has big test elements from tight closure theory.
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.
The theory of singularities defined by Frobenius has been extensively developed for $F$-finite rings and for rings that are essentially of finite type over excellent local rings. However, important classes of non-local excellent rings, such as Tate algebras and their quotients (affinoid algebras) do not fit into either setting. We investigate here a framework for moving beyond the $F$-finite setting, developing the theory of three related classes of regular rings defined by properties of Frobenius. In increasing order of strength, these are Frobenius Ohm-Rush (FOR), Frobenius intersection flat, and Frobenius Ohm-Rush trace (FORT). We show that Tate algebras are Frobenius intersection flat, from which it follows that reduced affinoid algebras have test elements using a result of Sharp. We also deduce new cases of the openness of the $F$-pure locus.
We compute a number of invariants of singularities defined via the Frobenius morphism for seminormal affine toric varieties over fields of characteristic p > 0. Our main technical tool is a combinatorial description of the potential splittings of iterates of Frobenius for seminormal monoid algebras. This allows us to give an easy formula for the F-splitting ratio of such rings as well as to compute the ideals stable under the Cartier algebra, including the test ideal.
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.
Fix a prime number $p$. Inspired by the notion of $F$-pure or $F$-split singularities, we study the condition that a Noetherian ring with $p$ in its Jacobson radical is pure inside some perfectoid (classical) ring, a condition we call perfectoid pure. We also study a related a priori weaker condition which asks that $R$ is pure in its absolute perfectoidization, a condition we call lim-perfectoid pure. We show that both these notions coincide when $R$ is LCI. Mixed characteristic analogs of $F$-injective and Du Bois singularities are also explored. We study these notions of singularity, proving that they are weakly normal and that they are Du Bois after inverting $p$. We also explore the behavior of \claperfdpure singularities under finite covers and their relation to log canonical singularities. Finally, we prove an inversion of adjunction result in the LCI setting, and use it to prove that many common examples are perfectoid pure.
Let $f$ be a diagonal hypersurface in $A_p=\mathbb{F}_p[[x_1,\dots,x_n]]$. We study the behavior of the function $\phi_{f,p}({a}/{p^e})=p^{-ne}\dim_{\mathbb{F}_p}\big(A_p/(x_1^{p^e},\dots,x_n^{p^e},f^a)\big)$ which encodes information about the F-threshold, the Hilbert-Kunz, and the F-signature functions. We prove that when $p$ goes to infinity $\phi_{f,p}$ converges to a piecewise polynomial function $\phi_f$ and the left and right derivatives of $\phi_{f,p}$ converge to $\phi'_f$. We use this fact to prove the existence of the limit F-signature and limit Hilbert-Kunz multiplicity for diagonal hypersurfaces. When $f$ is a Fermat hypersurface, we investigate the shape of the F-signature function of $f$ and provide an explicit formula for the limit F-signature and, in some cases, also for the F-signature for fixed $p$. This allows us to answer negatively to a question of Watanabe and Yoshida.
Let $S$ be a submonoid of a free Abelian group of finite rank. We show that if $k$ is a field of prime characteristic such that the monoid $k$-algebra $k[S]$ is split $F$-regular, then $k[S]$ is a finitely generated $k$-algebra, or equivalently, that $S$ is a finitely generated monoid. Split $F$-regular rings are possibly non-Noetherian or non-$F$-finite rings that satisfy the defining property of strongly $F$-regular rings from the theories of tight closure and $F$-singularities. Our finite generation result provides evidence in favor of the conjecture that split $F$-regular rings in function fields over $k$ have to be Noetherian. The key tool is Diophantine approximation from convex geometry.
Let $X$ be an integral scheme of finite type over a complete DVR of mixed characteristic. We provide a definition of a test ideal which agrees with the multiplier ideal after inverting $p$, is computed from a sufficiently large alteration, agrees with previous mixed characteristic BCM test ideals after completing at any point of residue characteristic $p$ (up to small perturbation), and which satisfies the full suite of expected properties of a multiplier or test ideal. This object is obtained via the $p$-adic Riemann-Hilbert functor.
We analyze adjunction and inversion of adjunction for the $F$-purity of divisor pairs in characteristic $p > 0$. In this vein, we give a complete answer for principal divisors under $\mathbb{Q}$-Gorenstein assumptions but without divisibility restrictions on the index. We also give a detailed analysis relating the $F$-purity of the pairs $(R,\Delta + D)$ and that of $(R_D, \text{Diff}_D(\Delta))$ motivated by Kawakita's log canonical inversion of adjunction via reduction to prime characteristic.
We systematically study the intersection flatness and Ohm-Rush properties for modules over a commutative ring, drawing inspiration from the work of Ohm and Rush and of Hochster and Jeffries. We establish new structural results for modules that are intersection flat/Ohm-Rush by exhibiting intimate connections between these notions and the seminal work of Raynaud and Gruson on Mittag-Leffler modules. In particular, we develop a theory of Ohm-Rush modules that is parallel to the theory of Mittag-Leffler modules. We also obtain descent and local-to-global results for intersection flat/Ohm-Rush modules. Our investigations reveal a particularly pleasing picture for flat modules over a complete local ring, in which case many otherwise distinct properties coincide.
We define a (perfectoid) mixed characteristic version of $F$-signature and Hilbert-Kunz multiplicity by utilizing the perfectoidization functor of Bhatt-Scholze and Faltings' normalized length (also developed in the work of Gabber-Ramero). We show that these definitions coincide with the classical theory in equal characteristic $p > 0$. We prove that a ring is regular if and only if either its perfectoid signature or perfectoid Hilbert-Kunz multiplicity is 1 and we show that perfectoid Hilbert-Kunz multiplicity characterizes BCM closure and extended plus closure of $m$-primary ideals. We demonstrate that perfectoid signature detects BCM-regularity and transforms similarly to $F$-signature or normalized volume under quasi-\'etale maps. As a consequence, we prove that BCM-regular rings have finite local \'etale fundamental group and also finite torsion part of their divisor class groups. Finally, we also define a mixed characteristic version of relative rational signature, and show it characterizes BCM-rational singularities.
We show that Noetherian splinters ascend under essentially étale homomorphisms. Along the way, we also prove that the henselization of a Noetherian local splinter is always a splinter and that the completion of a local splinter with geometrically regular formal fibers is a splinter. Finally, we give an example of a (non-excellent) Gorenstein local splinter with mild singularities whose completion is not a splinter. Our results provide evidence for a strengthening of the direct summand theorem, namely that regular maps preserve the splinter property.
A splinter is a notion of singularity that has seen numerous recent applications, especially in connection with the direct summand theorem, the mixed characteristic minimal model program, Cohen-Macaulayness of absolute integral closures and cohomology vanishing theorems. Nevertheless, many basic questions about these singularities remain elusive. One outstanding problem is whether the splinter property spreads from a point to an open neighborhood of a noetherian scheme. Our paper addresses this problem in prime characteristic, where we show that a locally noetherian scheme that has finite Frobenius or that is locally essentially of finite type over a quasi-excellent local ring has an open splinter locus. In particular, all varieties over fields of positive characteristic have open splinter loci. Intimate connections are established between the openness of splinter loci and $F$-compatible ideals, which are prime characteristic analogues of log canonical centers. We prove the surprising fact that for a large class of noetherian rings with pure (aka universally injective) Frobenius, the splinter condition is detected by the splitting of a single generically \'etale finite extension. We also show that for a noetherian $\textbf{N}$-graded ring over a field, the homogeneous maximal ideal detects the splinter property.
We establish the Minimal Model Program for arithmetic threefolds whose residue characteristics are greater than five. In doing this, we generalize the theory of global $F$-regularity to mixed characteristic and identify certain stable sections of adjoint line bundles. Finally, by passing to graded rings, we generalize a special case of Fujita's conjecture to mixed characteristic.