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Kevin Tucker

Publications and source records attributed to Kevin Tucker.

43 records · Page 3Linked to original sources
arXiv2012

On the behavior of test ideals under finite morphisms

We derive transformation rules for test ideals and $F$-singularities under an arbitrary finite surjective morphism $π: Y \to X$ of normal varieties in prime characteristic $p > 0$. The main technique is to relate homomorphisms $F_{*} O_{X} \to O_{X}$, such as Frobenius splittings, to homomorphisms $F_{*} O_{Y} \to O_{Y}$. In the simplest cases, these rules mirror transformation rules for multiplier ideals in characteristic zero. As a corollary, we deduce sufficient conditions which imply that trace is surjective, i.e. $Tr_{Y/X}(π_{*}O_{Y}) = O_{X}$.

math.AG↗
arXiv2011

Test ideals via a single alteration and discreteness and rationality of $F$-jumping numbers

Suppose $(X, Δ)$ is a log-$\bQ$-Gorenstein pair. Recent work of M. Blickle and the first two authors gives a uniform description of the multiplier ideal $\mJ(X;Δ)$ (in characteristic zero) and the test ideal $τ(X;Δ)$ (in characteristic $p > 0$) via regular alterations. While in general the alteration required depends heavily on $Δ$, for a fixed Cartier divisor $D$ on $X$ it is straightforward to find a single alteration (e.g. a log resolution) computing $\mJ(X; Δ+ λD)$ for all $λ\geq 0$. In this paper, we show the analogous statement in positive characteristic: there exists a single regular alteration computing $τ(X; Δ+ λD)$ for all $λ\geq 0$. Along the way, we also prove the discreteness and rationality for the $F$-jumping numbers of $τ(X; Δ+ λD)$ for $λ\geq 0$ where the index of $K_X + Δ$ is arbitrary (and may be divisible by the characteristic).

math.AG↗
arXiv2011

A survey of test ideals

Test ideals were first introduced by Mel Hochster and Craig Huneke in their celebrated theory of tight closure, and since their invention have been closely tied to the theory of Frobenius splittings. Subsequently, test ideals have also found application far beyond their original scope to questions arising in complex analytic geometry. In this paper we give a contemporary survey of test ideals and their wide-ranging applications.

math.AG↗
arXiv2011

F-signature exists

Suppose R is a Noetherian local ring with prime characteristic p>0. In this article, we show the existence of a local numerical invariant, called the F-signature, which roughly characterizes the asymptotic growth of the number of splittings of the iterates of the Frobenius endomorphism of R. This invariant was first formally defined by C. Huneke and G. Leuschke and has previously been shown to exist only in special cases. The proof of our main result is based on the development of certain uniform Hilbert-Kunz estimates of independent interest. Additionally, we analyze the behavior of the F-signature under finite ring extensions and recover explicit formulae for the F-signatures of finite quotient singularities.

math.AC↗
arXiv2009

On the number of compatibly Frobenius split subvarieties, prime $F$-ideals, and log canonical centers

Let $X$ be a projective Frobenius split variety over an algebraically closed field with splitting $θ: F_* Ø_X \to Ø_X$. In this paper we give a sharp bound on the number of subvarieties of $X$ compatibly split by $θ$. In particular, suppose $\sL$ is a sufficiently ample line bundle on $X$ (for example, if $\sL$ induces a projectively normal embedding) with $n = \dim H^0(X, \sL)$. We show that the number of $d$-dimensional irreducible subvarieties of $X$ that are compatibly split by $θ$ is less than or equal to ${n \choose d+1}$. This generalizes a well known result on the number of closed points compatibly split by a fixed splitting $θ$. Similarly, we give a bound on the number of prime $F$-ideals of an $F$-finite $F$-pure local ring. Compatibly split subvarieties are closely related to log canonical centers. Our methods apply in any characteristic, and so we are also able to bound the possible number of log canonical centers of a log canonical pair $(X, Δ)$ passing through a closed point $x \in X$. Specifically, if $n$ is the embedding dimension of $X$ at $x$, then the number of $d$-dimensional log canonical centers of $(X, Δ)$ through $x$ is less than or equal to ${n \choose d}$.

math.AG↗
arXiv2008

Jumping Numbers on Algebraic Surfaces with Rational Singularities

In this article, we study the jumping numbers of an ideal in the local ring at rational singularity on a complex algebraic surface. By understanding the contributions of reduced divisors on a fixed resolution, we are able to present an algorithm for finding of the jumping numbers of the ideal. This shows, in particular, how to compute the jumping numbers of a plane curve from the numerical data of its minimal resolution. In addition, the jumping numbers of the maximal ideal at the singular point in a Du Val or toric surface singularity are computed, and applications to the smooth case are explored.

math.AG↗