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Sep 23, 2013
09/13

by
Neil Epstein; Karl Schwede

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We introduce an operation on modules over an $F$-finite ring of characteristic $p$. We call this operation \emph{tight interior}. While it exists more generally, in some cases this operation is equivalent to the Matlis dual of tight closure. Moreover, the interior of the ring itself is simply the big test ideal. We directly prove, without appeal to tight closure, results analogous to persistence, colon capturing, and working modulo minimal primes, and we begin to develop a theory dual to...

Source: http://arxiv.org/abs/1110.4647v3

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Jul 20, 2013
07/13

by
Karl Schwede; Kevin Tucker; Wenliang Zhang

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Suppose $(X, \Delta)$ is a log-$\bQ$-Gorenstein pair. Recent work of M. Blickle and the first two authors gives a uniform description of the multiplier ideal $\mJ(X;\Delta)$ (in characteristic zero) and the test ideal $\tau(X;\Delta)$ (in characteristic $p > 0$) via regular alterations. While in general the alteration required depends heavily on $\Delta$, for a fixed Cartier divisor $D$ on $X$ it is straightforward to find a single alteration (e.g. a log resolution) computing $\mJ(X; \Delta...

Source: http://arxiv.org/abs/1107.4059v2

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Sep 20, 2013
09/13

by
Zsolt Patakfalvi; Karl Schwede

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For a characteristic $p > 0$ variety $X$ with controlled $F$-singularities, we state conditions which imply that a divisorial sheaf is Cohen-Macaulay or at least has depth $\geq 3$ at certain points. This mirrors results of Koll\'ar for varieties in characteristic zero. As an application, we show that that relative canonical sheaves are compatible with arbitrary base change for certain families with sharply $F$-pure fibers.

Source: http://arxiv.org/abs/1207.1910v3

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Sep 17, 2013
09/13

by
Karl Schwede

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We point out that the usual argument used to prove that $R$ is strongly $F$-regular if and only if $R_{Q}$ is strongly $F$-regular for every prime ideal $Q \in \Spec R$, does not generalize to the case of pairs $(R, \ba^t)$. The author's definition of sharp $F$-purity for pairs $(R, \ba^t)$ suffers from the same defect. We therefore propose different definitions of sharply $F$-pure and strongly $F$-regular pairs. Our new definitions agree with the old definitions in several common contexts,...

Source: http://arxiv.org/abs/0912.5336v1

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Jul 20, 2013
07/13

by
Mircea Mustata; Karl Schwede

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We define and study a version of Seshadri constant for ample line bundles in positive characteristic. We prove that lower bounds for this constant imply the global generation or very ampleness of the corresponding adjoint line bundle. As a consequence, we deduce that the criterion for global generation and very ampleness of adjoint line bundles in terms of usual Seshadri constants holds also in positive characteristic.

Source: http://arxiv.org/abs/1203.1081v1

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Jul 20, 2013
07/13

by
Karl Schwede

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Suppose that $X$ is a projective variety over an algebraically closed field of characteristic $p > 0$. Further suppose that $L$ is an ample (or more generally in some sense positive) divisor. We study a natural linear system in $|K_X + L|$. We further generalize this to incorporate a boundary divisor $\Delta$. We show that these subsystems behave like the global sections associated to multiplier ideals, $H^0(X, \mJ(X, \Delta) \tensor L)$ in characteristic zero. In particular, we show that...

Source: http://arxiv.org/abs/1107.3833v4

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Sep 18, 2013
09/13

by
Karl Schwede; Kevin Tucker

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Suppose that $\pi \: Y \to X$ is a finite map of normal varieties over a perfect field of characteristic $p > 0$. Previous work of the authors gave a criterion for when Frobenius splittings on $X$ (or more generally any $p^{-e}$-linear map) extend to $Y$. In this paper we give an alternate and highly explicit proof of this criterion (checking term by term) when $\pi$ is tamely ramified in codimension 1. Some additional examples are also explored.

Source: http://arxiv.org/abs/1201.5973v1

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Jul 20, 2013
07/13

by
Manuel Blickle; Karl Schwede; Kevin Tucker

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We generalize $F$-signature to pairs $(R,D)$ where $D$ is a Cartier subalgebra on $R$ as defined by the first two authors. In particular, we show the existence and positivity of the $F$-signature for any strongly $F$-regular pair. In one application, we answer an open question of I. Aberbach and F. Enescu by showing that the $F$-splitting ratio of an arbitrary $F$-pure local ring is strictly positive. Furthermore, we derive effective methods for computing the $F$-signature and the $F$-splitting...

Source: http://arxiv.org/abs/1107.1082v2

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Sep 22, 2013
09/13

by
Osamu Fujino; Karl Schwede; Shunsuke Takagi

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We consider various definitions of non-lc ideal sheaves -- generalizations of the multiplier ideal sheaf which define the non-lc (non-log canonical) locus. We introduce the maximal non-lc ideal sheaf and intermediate non-lc ideal sheaves and consider the restriction theorem for these ideal sheaves. We also begin the development of the theory of a characteristic p>0 analog of maximal non-lc ideals, utilizing some recent work of Blickle.

Source: http://arxiv.org/abs/1004.5170v2

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Jun 29, 2018
06/18

by
Javier Carvajal-Rojas; Karl Schwede; Kevin Tucker

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We prove that the local etale fundamental group of a strongly $F$-regular singularity is finite (and likewise for the \'etale fundamental group of the complement of a codimension $\geq 2$ set), analogous to results of Xu and Greb-Kebekus-Peternell for KLT singularities in characteristic zero. In fact our result is effective, we show that the reciprocal of the $F$-signature of the singularity gives a bound on the size of this fundamental group. To prove these results and their corollaries, we...

Topics: Commutative Algebra, Algebraic Geometry, Mathematics

Source: http://arxiv.org/abs/1606.04088

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Jul 20, 2013
07/13

by
Mordechai Katzman; Karl Schwede; Anurag K. Singh; Wenliang Zhang

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Let R be a local ring of prime characteristic. We study the ring of Frobenius operators F(E), where E is the injective hull of the residue field of R. In particular, we examine the finite generation of F(E) over its degree zero component, and show that F(E) need not be finitely generated when R is a determinantal ring; nonetheless, we obtain concrete descriptions of F(E) in good generality that we use, for example, to prove the discreteness of F-jumping numbers for arbitrary ideals in...

Source: http://arxiv.org/abs/1304.6147v1

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Sep 21, 2013
09/13

by
Jen-Chieh Hsiao; Karl Schwede; Wenliang Zhang

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Assume that $X$ is an affine toric variety of characteristic $p > 0$. Let $\Delta$ be an effective toric $Q$-divisor such that $K_X+\Delta$ is $Q$-Cartier with index not divisible by $p$ and let $\phi_{\Delta}:F^e_* O_X \to O_X$ be the toric map corresponding to $\Delta$. We identify all ideals $I$ of $O_X$ with $\phi_{\Delta}(F^e_* I)=I$ combinatorially and also in terms of a log resolution (giving us a version of these ideals which can be defined in characteristic zero). Moreover, given a...

Source: http://arxiv.org/abs/1011.0804v3

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Jun 28, 2018
06/18

by
Omprokash Das; Karl Schwede

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We study the structure of Frobenius splittings (and generalizations thereof) induced on compatible subvarieties $W \subseteq X$. In particular, if the compatible splitting comes from a compatible splitting of a divisor on some birational model $E \subseteq X' \to X$ (ie, this is a log canonical center), then we show that the divisor corresponding to the splitting on $W$ is bounded below by the divisorial part of the different as studied by Kawamata, Shokurov, Ambro and others. We also show that...

Topics: Commutative Algebra, Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1508.07295

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Sep 17, 2013
09/13

by
Karl Schwede; Kevin Tucker

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We derive transformation rules for test ideals and $F$-singularities under an arbitrary finite surjective morphism $\pi : Y \to X$ 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...

Source: http://arxiv.org/abs/1003.4333v3

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39

Sep 18, 2013
09/13

by
Karl Schwede

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In this paper, we prove that singularities of $F$-injective type are Du Bois. This extends the correspondence between singularities associated to the minimal model program and singularities defined by the action of Frobenius in positive characteristic.

Source: http://arxiv.org/abs/0806.3298v2

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9.0

Jun 30, 2018
06/18

by
Zsolt Patakfalvi; Karl Schwede; Kevin Tucker

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These are notes for the Bootcamp volume for the 2015 AMS Summer Institute in Algebraic Geometry. They are based on earlier notes for the "Positive Characteristic Algebraic Geometry Workshop" held at University of Illinois at Chicago in March 2014.

Topics: Mathematics, Commutative Algebra, Algebraic Geometry

Source: http://arxiv.org/abs/1412.2203

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8.0

Jun 28, 2018
06/18

by
Eric Canton; Daniel Hernández; Karl Schwede; Emily Witt; Alessandro De Stefani; Jack Jeffries; Zhibek Kadyrsizova; Robert Walker; George Whelan

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We provide a family of examples where the $F$-pure threshold and the log canonical threshold of a polynomial are different, but where $p$ does not divide the denominator of the $F$-pure threshold (compare with an example of \mustata-Takagi-Watanabe). We then study the $F$-signature function in the case where either the $F$-pure threshold and log canonical threshold coincide or where $p$ does not divide the denominator of the $F$-pure threshold. We show that the $F$-signature function behaves...

Topics: Commutative Algebra, Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1508.05427

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Sep 23, 2013
09/13

by
Karl Schwede; Kevin Tucker

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Let $X$ be a projective Frobenius split variety over an algebraically closed field with splitting $\theta : F_* \O_X \to \O_X$. In this paper we give a sharp bound on the number of subvarieties of $X$ compatibly split by $\theta$. 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...

Source: http://arxiv.org/abs/0903.4112v3

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Sep 21, 2013
09/13

by
Karl Schwede; Kevin Tucker

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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.

Source: http://arxiv.org/abs/1104.2000v2

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50

Jul 20, 2013
07/13

by
Manuel Blickle; Karl Schwede; Kevin Tucker

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For a normal F-finite variety $X$ and a boundary divisor $\Delta$ we give a uniform description of an ideal which in characteristic zero yields the multiplier ideal, and in positive characteristic the test ideal of the pair $(X,\Delta)$. Our description is in terms of regular alterations over $X$, and one consequence of it is a common characterization of rational singularities (in characteristic zero) and F-rational singularities (in characteristic $p$) by the surjectivity of the trace map...

Source: http://arxiv.org/abs/1107.3807v3

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102

Jul 22, 2013
07/13

by
Karl Schwede

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In this paper, we study a positive characteristic analogue of the centers of log canonicity of a pair $(R, \Delta)$. We call these analogues centers of $F$-purity. We prove positive characteristic analogues of subadjunction-like results, prove new stronger subadjunction-like results, and in some cases, lift these new results to characteristic zero. Using a generalization of centers of $F$-purity which we call uniformly $F$-compatible ideals, we give a characterization of the test ideal (which...

Source: http://arxiv.org/abs/0807.1654v4

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46

Sep 23, 2013
09/13

by
Karl Schwede; Wenliang Zhang

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We prove that strongly F-regular and F-pure singularities satisfy Bertini-type theorems (including in the context of pairs) by building upon a framework of Cumino, Greco and Manaresi (compare with the work of Jouanolou and Spreafico). We also prove that F-injective singularities fail to satisfy even the most basic Bertini-type results.

Source: http://arxiv.org/abs/1112.2161v5

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Jul 20, 2013
07/13

by
Sándor J Kovács; Karl Schwede

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Let $X$ be a variety and $H$ a Cartier divisor on $X$. We prove that if $H$ has Du Bois (or DB) singularities, then $X$ has Du Bois singularities near $H$. As a consequence, if $X \to S$ is a family over a smooth curve $S$ whose special fiber has Du Bois singularities, then the nearby fibers also have Du Bois singularities. We prove this by obtaining an injectivity theorem for certain maps of canonical modules. As a consequence, we also obtain a restriction theorem for certain non-lc ideals.

Source: http://arxiv.org/abs/1107.2349v3

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129

Jun 30, 2018
06/18

by
Alberto Chiecchio; Florian Enescu; Lance Edward Miller; Karl Schwede

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Many results are known about test ideals and $F$-singularities for ${\bf Q}$-Gorenstein rings. In this paper we generalize many of these results to the case when the symbolic Rees algebra $O_X \oplus O_X(-K_X) \oplus O_X(-2K_X) \oplus ...$ is finitely generated (or more generally, in the log setting for $-K_X - \Delta$). In particular, we show that the $F$-jumping numbers of $\tau(X, a^t)$ are discrete and rational. We show that test ideals $\tau(X)$ can be described by alterations as in...

Topics: Mathematics, Commutative Algebra, Algebraic Geometry

Source: http://arxiv.org/abs/1412.6453

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47

Sep 22, 2013
09/13

by
Karl Schwede

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Suppose that $X = \Spec R$ is an $F$-finite normal variety in characteristic $p > 0$. In this paper we show that the big test ideal $\tau_b(R) = \tld \tau(R)$ is equal to $\sum_{\Delta} \tau(R; \Delta)$ where the sum is over $\Delta$ such that $K_X + \Delta$ is $\bQ$-Cartier. This affirmatively answers a question asked by various people, including Blickle, Lazarsfeld, K. Lee and K. Smith. Furthermore, we have a version of this result in the case that $R$ is not even necessarily normal.

Source: http://arxiv.org/abs/0906.4313v2

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Jul 20, 2013
07/13

by
Shrawan Kumar; Karl Schwede

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Let $X^v_w$ be a Richardson variety in the full flag variety $X$ associated to a symmetrizable Kac-Moody group $G$. Recall that $X^v_w$ is the intersection of the finite dimensional Schubert variety $X_w$ with the finite codimensional opposite Schubert variety $X^v$. We give an explicit $\bQ$-divisor $\Delta$ on $X^v_w$ and prove that the pair $(X^v_w, \Delta)$ has Kawamata log terminal singularities. In fact, $-K_{X^v_w} - \Delta$ is ample, which additionally proves that $(X^v_w, \Delta)$ is...

Source: http://arxiv.org/abs/1203.6126v2

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77

Sep 17, 2013
09/13

by
Karl Schwede

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Consider a pair $(R, \ba^t)$ where $R$ is a ring of positive characteristic, $\ba$ is an ideal such that $a \cap $R^{\circ} \neq \emptyset$, and $t > 0$ is a real number. In this situation we have the ideal $\tau_R(\ba^t)$, the generalized test ideal associated to $(R, a^t)$ as defined by Hara and Yoshida. We show that $\tau_R(a^t) \cap R^{\circ}$ is made up of appropriately defined generalized test elements which we call \emph{sharp test elements}. We also define a variant of $F$-purity for...

Source: http://arxiv.org/abs/0711.3380v2

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Sep 22, 2013
09/13

by
Karl Schwede

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Suppose that $R$ is a ring essentially of finite type over a perfect field of characteristic $p > 0$ and that $a \subseteq R$ is an ideal. We prove that the set of $F$-jumping numbers of $\tau_b(R; a^t)$ has no limit points under the assumption that $R$ is normal and $Q$-Gorenstein -- we do \emph{not} assume that the $Q$-Gorenstein index is not divisible by $p$. Furthermore, we also show that the $F$-jumping numbers of $\tau_b(R; \Delta, a^t)$ are discrete under the more general assumption...

Source: http://arxiv.org/abs/1004.1377v2

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Sep 23, 2013
09/13

by
Manuel Blickle; Karl Schwede; Kevin Tucker

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This paper contains a number of observations on the {$F$-signature} of triples $(R,\Delta,\ba^t)$ introduced in our previous joint work. We first show that the $F$-signature $s(R,\Delta,\ba^t)$ is continuous as a function of $t$, and for principal ideals $\ba$ even convex. We then further deduce, for fixed $t$, that the $F$-signature is lower semi-continuous as a function on $\Spec R$ when $R$ is regular and $\ba$ is principal. We also point out the close relationship of the signature function...

Source: http://arxiv.org/abs/1111.2762v2

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39

Sep 21, 2013
09/13

by
Karl Schwede

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In this paper we study singularities defined by the action of Frobenius in characteristic $p > 0$. We prove results analogous to inversion of adjunction along a center of log canonicity. For example, we show that if $X$ is a Gorenstein normal variety then to every normal center of sharp $F$-purity $W \subseteq X$ such that $X$ is $F$-pure at the generic point of $W$, there exists a canonically defined $\bQ$-divisor $\Delta_{W}$ on $W$ satisfying $(K_X)|_W \sim_{\bQ} K_{W} + \Delta_{W}$....

Source: http://arxiv.org/abs/0901.1154v4

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Jun 30, 2018
06/18

by
Paolo Cascini; Yoshinori Gongyo; Karl Schwede

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We show that if $(X,B)$ is a two dimensional Kawamata log terminal pair defined over an algebraically closed field of characteristic $p$, and $p$ is sufficiently large, depending only on the coefficients of $B$, then $(X,B)$ is also strongly $F$-regular.

Topics: Mathematics, Commutative Algebra, Algebraic Geometry

Source: http://arxiv.org/abs/1402.0027

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Sep 23, 2013
09/13

by
Karl Schwede

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We prove the following theorem characterizing Du Bois singularities. Suppose that $Y$ is smooth and that $X$ is a reduced closed subscheme. Let $\pi : \tld Y \to Y$ be a log resolution of $X$ in $Y$ that is an isomorphism outside of $X$. If $E$ is the reduced pre-image of $X$ in $\tld Y$, then $X$ has Du Bois singularities if and only if the natural map $\O_X \to R \pi_* \O_E$ is a quasi-isomorphism. We also deduce Koll\'ar's conjecture that log canonical singularities are Du Bois in the...

Source: http://arxiv.org/abs/0903.4125v1

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Sep 22, 2013
09/13

by
Manuel Blickle; Karl Schwede; Shunsuke Takagi; Wenliang Zhang

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We prove that the $F$-jumping numbers of the test ideal $\tau(X; \Delta, \ba^t)$ are discrete and rational under the assumptions that $X$ is a normal and $F$-finite variety over a field of positive characteristic $p$, $K_X+\Delta$ is $\bQ$-Cartier of index not divisible $p$, and either $X$ is essentially of finite type over a field or the sheaf of ideals $\ba$ is locally principal. This is the largest generality for which discreteness and rationality are known for the jumping numbers of...

Source: http://arxiv.org/abs/0906.4679v2

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43

Sep 20, 2013
09/13

by
Manuel Blickle; Karl Schwede

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In this article we survey the basic properties of $p^{-e}$-linear endomorphisms of coherent $\O_X$-modules, i.e. of $\O_X$-linear maps $F_* \sF \to \sG$ where $\sF,\sG$ are $\O_X$-modules and $F$ is the Frobenius of a variety of finite type over a perfect field of characteristic $p > 0$. We emphasize their relevance to commutative algebra, local cohomology and the theory of test ideals on the one hand, and global geometric applications to vanishing theorems and lifting of sections on the...

Source: http://arxiv.org/abs/1205.4577v2

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Sep 19, 2013
09/13

by
Karl Schwede; Shunsuke Takagi

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In this paper we introduce a notion of rational singularities associated to pairs $(X, \ba^t)$ where $X$ is a variety, $\ba$ is an ideal sheaf and $t$ is a nonnegative real number. We prove that most standard results about rational singularities extend to this context. We also show that some results commonly associated with log terminal pairs have analogs in this context, including results related to inversion of adjunction. A positive characteristic analogue of rational singularities of pairs...

Source: http://arxiv.org/abs/0708.1990v4

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Sep 19, 2013
09/13

by
Sándor J Kovács; Karl Schwede

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This is a survey of some recent developments in the study of singularities related to the classification theory of algebraic varieties. In particular, the definition and basic properties of Du Bois singularities and their connections to the more commonly known singularities of the minimal model program are reviewed and discussed.

Source: http://arxiv.org/abs/0909.0993v1

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Sep 22, 2013
09/13

by
Lance Edward Miller; Karl Schwede

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If $X$ is Frobenius split, then so is its normalization and we explore conditions which imply the converse. To do this, we recall that given an $\mathcal{O}_X$-linear map $\phi : F_* \mathcal{O}_X \to \mathcal{O}_X$, it always extends to a map $\bar{\phi}$ on the normalization of $X$. In this paper, we study when the surjectivity of $\bar{\phi}$ implies the surjectivity of $\phi$. While this doesn't occur generally, we show it always happens if certain tameness conditions are satisfied for the...

Source: http://arxiv.org/abs/1101.1033v3

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Sep 23, 2013
09/13

by
Karl Schwede; Kevin Tucker

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Given an ideal $a \subseteq R$ in a (log) $Q$-Gorenstein $F$-finite ring of characteristic $p > 0$, we study and provide a new perspective on the test ideal $\tau(R, a^t)$ for a real number $t > 0$. Generalizing a number of known results from the principal case, we show how to effectively compute the test ideal and also describe $\tau(R, a^t)$ using (regular) alterations with a formula analogous to that of multiplier ideals in characteristic zero. We further prove that the $F$-jumping...

Source: http://arxiv.org/abs/1212.6956v2

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Jun 29, 2018
06/18

by
Bhargav Bhatt; Javier Carvajal-Rojas; Patrick Graf; Karl Schwede; Kevin Tucker

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We prove that a strongly $F$-regular scheme $X$ admits a finite, generically Galois, and \'etale-in-codimension-one cover $\widetilde X \to X$ such that the \'etale fundamental groups of $\widetilde X$ and $\widetilde X_{reg}$ agree. Equivalently, every finite \'etale cover of $\widetilde X_{reg}$ extends to a finite \'etale cover of $\widetilde X$. This is analogous to a result for complex klt varieties by Greb, Kebekus and Peternell.

Topics: Commutative Algebra, Algebraic Geometry, Mathematics

Source: http://arxiv.org/abs/1611.03884

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Sep 21, 2013
09/13

by
Mordechai Katzman; Karl Schwede

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Let $R$ be a ring of prime characteristic $p$, and let $F^e_* R$ denote $R$ viewed as an $R$-module via the $e$th iterated Frobenius map. Given a surjective map $\phi : F^e_* R \to R$ (for example a Frobenius splitting), we exhibit an algorithm which produces all the $\phi$-compatible ideals. We also explore a variant of this algorithm under the hypothesis that $\phi$ is not necessarily a Frobenius splitting (or even surjective). This algorithm, and the original, have been implemented in...

Source: http://arxiv.org/abs/1104.1937v3