arXiv subjects
Publications and source records attributed to Karen E. Smith.
We prove that a smooth projective surface of degree $d$ in $\mathbb P^3$ contains at most $d^2(d^2-3d+3)$ lines. We characterize the surfaces containing exactly $d^2(d^2-3d+3)$ lines: these occur only in prime characterize $p$ and, up to choice of projective coordinates, are cut out by equations of the form $x^{p^{e}+1}+y^{p^{e}+1}+z^{p^{e}+1}+ w^{p^{e}+1} = 0.$
Let $k$ be an arbitrary field. We construct examples of regular local $k$-algebras $R$ (of positive dimension) for which the ring of differential operators $D_k(R)$ is trivial in the sense that it contains {\it no} operators of positive order. The examples are excellent in characteristic zero but not in positive characteristic. These rings can be viewed as being non-singular but they are not simple as $D$-modules, laying to rest speculation that $D$-simplicity might characterize a nice class of singularities in general. In prime characteristic, the construction also provides examples of {\it regular} local rings $R$ (with fraction field a function field) whose Frobenius push-forward $F_*^eR$ is {\it indecomposable} as an $R$-module for all $e\in \mathbb N$. Along the way, we investigate hypotheses on a local ring $(R, m)$ under which $D$-simplicity for $R$ is equivalent to $D$-simplicity for its $m$-adic completion, and give examples of rings for which the differential operators do not behave well under completion. We also generalize a characterization of $D$-simplicity due to Jeffries in the $\mathbb N$-graded case: for a Noetherian local $k$-algebra $(R, m, k)$, $D$-simplicity of $R$ is equivalent to surjectivity of the natural map $D_k(R)\to D_k(R, k)$.
We find a formula, in terms of n, d and p, for the value of the F-pure threshold for the generic homogeneous polynomial of degree d in n variables over an algebraically closed field of characteristic p. We also show that, in every characteristic p and for all d (greater than 3) not divisible by p, there always exist reduced polynomials of degree d whose F-pure threshold is a truncation of the base p expansion of 2/d at some place; in particular, there always exist reduced polynomials whose F-pure threshold is strictly less than 2/d. We provide an example to resolve, negatively, a question proposed by Hernandez, Núñez-Betancourt, Witt and Zhang, as to whether a list of necessary restrictions they prove on the F-pure threshold of reduced forms are "minimal" for large p. On the other hand, we also provide evidence supporting and refining their ideas, including identifying specific truncations of the base p expansion of 2/d that are always F-pure thresholds for reduced forms of degree d, and computations that show their conditions suffice (in every characteristic) for degrees up to eight and several other situations. Finally, we point out a lower bound on the F-pure threshold of a reduced form in terms of its degree and the characteristic p.
We prove that if $f$ is a reduced homogenous polynomial of degree $d$, then its $F$-pure threshold at the unique homogeneous maximal ideal is at least $\frac{1}{d-1}$. We show, furthermore, that its $F$-pure threshold equals $\frac{1}{d-1}$ if and only if $f\in \mathfrak m^{[q]}$ and $d=q+1$, where $q$ is a power of $p$. Up to linear changes of coordinates (over a fixed algebraically closed field), we classify such "extremal singularities," and show that there is at most one with isolated singularity. Finally, we indicate several ways in which the projective hypersurfaces defined by such forms are "extremal," for example, in terms of the configurations of lines they can contain.
We classify Frobenius forms, a special class of homogeneous polynomials in characteristic $p>0$, in up to five variables over an algebraically closed field. We also point out some of the similarities with quadratic forms.
Cubic surfaces in characteristic two are investigated from the point of view of prime characteristic commutative algebra. In particular, we prove that, the non-Frobenius split cubic surfaces form a linear subspace of codimension four in the 19-dimensional space of all cubics, and that up to projective equivalence, there are finitely many non-Frobenius split cubic surfaces. We explicitly describe defining equations for each and characterize them as extremal in terms of configurations of lines on them. In particular, a (possibly singular) cubic surface in characteristic two fails to be Frobenius split if and only if no three lines on it form a "triangle".
We study the geometry of the smooth projective surfaces that are defined by Frobenius forms, a class of homogenous polynomials in prime characteristic recently shown to have minimal possible F-pure threshold among forms of the same degree. We call these surfaces $\textit{extremal surfaces}$, and show that their geometry is reminiscent of the geometry of smooth cubic surfaces, especially non-Frobenius split cubic surfaces of characteristic two, which are examples of extremal surfaces. For example, we show that an extremal surface $X$ contains $d^2(d^2-3d+3)$ lines where $d$ is the degree, which is notable since the number of lines on a complex surface is bounded above by a quadratic function in $d$. Whenever two of those lines meet, they determine a $d$-tangent plane to $X$ which consists of a union of $d$ lines meeting in one point; we count the precise number of such "star points" on $X$, showing that it is quintic in the degree, which recovers the fact that there are exactly 45 Eckardt points on an extremal cubic surface. Finally, we generalize the classical notion of a double six for cubic surfaces to a double $2d$ on an extremal surface of degree $d$. We show that, asymptotically in $d$, smooth extremal surfaces have at least $\frac{1}{16}d^{14}$ double $2d$'s. A key element of the proofs is using the large automorphism group of extremal surfaces which we show acts transitively on many sets, such as the set of (triples of skew) lines on the extremal surface. Extremal surfaces are closely related to finite Hermitian geometries, which we recover as the $\mathbb F_{q^2}$-rational points of special extremal surfaces defined by Hermitian forms over $\mathbb F_{q^2}$.
An $R$-algebra $S$ is $R$-solid if there exists a nonzero $R$-linear map $S \rightarrow R$. In characteristic $p$, the study of $F$-singularities such as Frobenius splittings implicitly rely on the $R$-solidity of $R^{1/p}$. Following recent results of the first two authors on the Frobenius non-splitting of certain excellent $F$-pure rings, in this paper we use the notion of solidity to systematically study the notion of excellence, with an emphasis on $F$-singularities. We show that for rings $R$ essentially of finite type over complete local rings of characteristic $p$, reducedness implies the $R$-solidity of $R^{1/p}$, $F$-purity implies Frobenius splitting, and $F$-pure regularity implies split $F$-regularity. We demonstrate that Henselizations and completions are not solid, providing obstructions for the $R$-solidity of $R^{1/p}$ for arbitrary excellent rings. This also has negative consequences for the solidity of big Cohen-Macaulay algebras, an important example of which are absolute integral closures of excellent local rings in prime characteristic. We establish a close relationship between the solidity of absolute integral closures and the notion of Japanese rings. Analyzing the Japanese property reveals that Dedekind domains $R$ for which $R^{1/p}$ is $R$-solid are excellent, despite our recent examples of excellent Euclidean domains with no nonzero $p^{-1}$-linear maps. Additionally, we show that while perfect closures are often solid in algebro-geometric situations, there exist locally excellent domains with solid perfect closures whose absolute integral closures are not solid. In an appendix, Karen E. Smith uses the solidity of absolute integral closures to characterize the test ideal for a large class of Gorenstein domains of prime characteristic.
We prove that under suitable graded and local hypothesis, a formally unramified algebra over a field must be reduced. We detail examples, including one due to Gabber, to show that it is not possible to generalize these results further.
Let $R$ be the coordinate ring of an affine toric variety. We show that the endomorphism ring $End_R(\mathbb A),$ where $\mathbb A$ is the (finite) direct sum of all (isomorphism classes of) conic $R$-modules, has finite global dimension. Furthermore, we show that $End_R(\mathbb A)$ is a non-commutative crepant resolution if and only if the toric variety is simplicial. For toric varieties over a perfect field $k$ of prime characteristic, we show that the ring of differential operators $D_\mathsf{k}(R)$ has finite global dimension.
Fix any field $K$ of characteristic $p$ such that $[K:K^p]$ is finite. We discuss excellence for Noetherian domains whose fraction field is $K$, showing for example, that $R$ is excellent if and only if the Frobenius map is finite on $R$. Furthermore, we show $R$ is excellent if and only if it admits some non-zero $p^{-e}$-linear map for $R$ or equivalently, that $R$ is a solid $R$-algebra under Frobenius. In particular, this means that Frobenius split Noetherian domains that are generically $F$-finite are always excellent. We also show that non-excellent rings are abundant and easy to construct in prime characteristic, even within the world of regular local rings of dimension one in function fields. This paper is mostly expository in nature.
The behavior of the Frobenius map is investigated for valuation rings of prime characteristic. We show that valuation rings are always F-pure. We introduce a generalization of the notion of strong F-regularity, which we call F-pure regularity, and show that a valuation ring is F-pure regular if and only if it is Noetherian. For valuations on function fields, we show that the Frobenius map is finite if and only if the valuation is divisorial; in this case the valuation ring is Frobenius split. For Noetherian valuation rings in function fields, we show that the valuation ring is Frobenius split if and only if Frobenius is finite, or equivalently, if and only if the valuation ring is excellent.
This is a survey of Frobenius splitting techniques in commutative algebra, based on the first author's lectures at the introductory workshop for the special year in commutative algebra at MSRI in fall 2012.
We show that locally acyclic cluster algebras have (at worst) canonical singularities. In fact, we prove that locally acyclic cluster algebras of positive characteristic are strongly F-regular. In addition, we show that upper cluster algebras are always Frobenius split by a canonically defined splitting, and that they have a free canonical module of rank one. We also give examples to show that not all upper cluster algebras are F-regular if the local acyclicity is dropped.
Consider a polynomial $f$ defined over a field $k$, the multiplicity is perhaps the most naive measurement of the singularities of $f$. This paper describes the first steps toward understanding a much more subtle measure of singularities which arises naturally in three different contexts-- analytic, algebro-geometric, and finally, algebraic. Miraculously, all three approaches lead to essentially the same measurement of singularities: the log canonical threshold (in characteristic zero) and the closely related $F$-pure threshold (in characteristic $p$). In this paper we present only the first steps in understanding these invariants, with an emphasis on the prime characteristic setting.
We prove that a Cohen-Macaulay normal variety $X$ has Du Bois singularities if and only if $π_*ω_{X'}(G) \simeq ω_X$ for a log resolution $π: X' \to X$, where $G$ is the reduced exceptional divisor of $π$. Many basic theorems about Du Bois singularities become transparent using this characterization (including the fact that Cohen-Macaulay log canonical singularities are Du Bois). We also give a straightforward and self-contained proof that (generalizations of) semi-log-canonical singularities are Du Bois, in the Cohen-Macaulay case. It also follows that the Kodaira vanishing theorem holds for semi-log-canonical varieties and that Cohen-Macaulay semi-log-canonical singularities are cohomologically insignificant in the sense of Dolgachev.
We prove that every globally $F$-regular variety is log Fano. In other words, if a prime characteristic variety $X$ is globally $F$-regular, then it admits an effective $\bQ$-divisor $Δ$ such that $-K_X - Δ$ is ample and $(X, Δ)$ has controlled (Kawamata log terminal, in fact globally $F$-regular) singularities. A weak form of this result can be viewed as a prime characteristic analog of de Fernex and Hacon's new point of view on Kawamata log terminal singularities in the non-$\bQ$-Gorenstein case. We also prove a converse statement in characteristic zero: every log Fano variety has globally $F$-regular type. Our techniques apply also to $F$-split varieties, which we show to satisfy a "log Calabi-Yau" condition. We also prove a Kawamata-Viehweg vanishing theorem for globally $F$-regular pairs.
It was recently established by the first two authors that multiplier ideals on a smooth variety satisfy some special syzygetic properties. The purpose of this note is to show how some of these can be extended to the singular setting.