Research Papers:

Nearly all of the papers are available on the arXiv.

Title (arXiv link)Co-authorsJournalNotes
Seminormalization package for Macaulay2 Bernard Serbinowski Submitted. We describe an algorithm (implemented in Macaulay2) for computing seminormalization.
Singularities in mixed characteristic via Perfectoid big Cohen-Macaulay algebras Linquan Ma Submitted. We introduce mixed characteristic analogs of rational/F-rational singularities as well as log terminal/F-regular singularities and test/multiplier ideals. We also study their properties.
Bertini Theorems for F-signature Javier Carvajal-Rojas, Kevin Tucker We show how F-signature behaves under taking general hyperplanes.
Perfectoid multiplier/test ideals in regular rings and bounds on symbolic powers Linquan Ma To appear in Inventiones. We introduce a mixed characteristic test ideal analog and use this to generalize results on symbolic powers of Ein-Lazarsfeld-Smith and Hochster-Huneke to the mixed characteristic case.
Étale fundamental groups of strongly F$-regular schemes Bhargav Bhatt, Javier Carvajal-Rojas, Patrick Graf, Kevin Tucker To appear in IMRN. We show results analogous to the main technical results of Greb-Kebekus-Peternell.
Fundamental groups of F-regular singularities via F-signature Javier Carvajal-Rojas, Kevin Tucker To appear in Annales scientifiques de l'Ecole normale superieure. We show that etale fundamental groups of strongly F-regular singularities are finite
Discreteness of F-jumping numbers at isolated non-Q-Gorenstein points Patrick Graf To appear in Proceedings of the American Mathematical Society. We show that the F-jumping numbers at isolated Q-Gorenstein points have no accumulation points.
RationalMaps, A package for Macaulay2 C.J. Bott, Hamid Hassenzadeh, Daniel Smolkin Submitted. This paper documents the package RationalMaps.m2 in the Macaulay2 build tree.
Local cohomology of Du Bois singularities and applications to families Linquan Ma, Kazuma Shimomoto Compos. Math. 153 (2017), no. 10, 2147–2170.. We study deformations of Du Bois and F-injective singularities, and applications to local cohomology.
The dualizing complex of F-injective and Du Bois singularities Bhargav Bhatt, Linquan Ma To appear in Mathematische Zeitschrift. We show that Du Bois singularities with isolated non-CM points are Buchsbaum and more, and obtain analogous results for F-injective singularities.
The F-different and a canonical bundle formula Omprokash Das To appear in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze. We explore the geometric and arithmetic meaning of the F-different.
On the behavior of singularities at the F-pure threshold Eric Canton, Daniel Hernández, Emily Witt, with an appendix by Alessandro De Stefani, Jack Jeffries, Zhibek Kadyrsizova, Robert Walker, George Whelan. To appear in the Illinois Journal of Mathematics. We study how things like F-signature behave near the F-pure threshold.
Test ideals in rings with finitely generated anti-canonical algebras Alberto Chiecchio, Florian Enescu, Lance Edward Miller, To appear in Journal of the Institute of Mathematics of Jussieu. We generalize many results for Q-Gorenstein varieties to the setting where the anti-canonical algebra is finitely generated.
Positive characteristic algebraic geometry Zsolt Patakfalvi, Kevin Tucker To appear in the Proceedings of Symposia in Pure Mathematics volume for the 2015 AMS Algebraic Geometry Bootcamp These are based on notes from the March 2014 UIC workshop in positive characteristic algebraic geometry.
Divsior Package for Macaulay2 Zhaoning Yang Journal of Software for Algebra and Geoemtry, Vol. 8 (2018), 87–94. This is the paper associated to the Divisor.m2 package here.
Uniform bounds for strongly F-regular surfaces Paolo Cascini and Yoshinori Gongyo Trans. Amer. Math. Soc. 368 (2016), 5547-5563. We prove that for every finite set of coefficients there is a N > 0 such that if (X, D) is a log terminal and characteristic p > N, then (X, D) is also F-regular.
Inversion of adjunction for rational and Du Bois pairs Sándor Kovács Algebra Number Theory 10 (2016), no. 5, 969-1000. We prove a new inversion of adjunction result for Du Bois singularities, and prove many results for Du Bois pairs.
On Rational Connectedness of Globally F-Regular Threefolds Yoshinori Gongyo, Zhiyuan Li, Zsolt Patakfalvi, Hiromu Tanaka, Hong R. Zong Adv. Math. 280 (2015), 47-78 We prove rational connectivity of some globally F-regular threefolds.
The weak ordinarity conjecture and F-singularities Bhargav Bhatt, Shunsuke TakagiTo appear in a volume of Advanced Studies in Pure Mathematics We prove results linking singularities in characteristic zero and characteristic p, we generalize this paper to singular ambient spaces.
F-singularities in families Zsolt Patakfalvi and Wenliang Zhang To appear in Algebraic Geometry (Foundation Compositio Mathematica) We study rings of Frobenius operators with a particular eye towards finite generation and gauge boundedness.
Rings of Frobenius operators Mordechai Katzman, Anurag K. Singh and Wenliang Zhang Math. Proc. Cambridge Philos. Soc. 157 (2014), no. 1, 151-167. We study rings of Frobenius operators with a particular eye towards finite generation and gauge boundedness.
Test ideals of non-principal ideals: Computations, Jumping Numbers, Alterations and Division Theorems Kevin Tucker J. Math. Pures Appl. (9) 102 (2014), no. 5, 891-929. We generalize several theorems on test ideals of principal ideals to the non-principal case.
Appendix to Deformation of F-injectivity and local cohomology Anurag K. Singh
The paper was written by: Jun Horiuchi, Lance Edward Miller and Kazuma Shimomoto
Indiana Univ. Math. J. 63 (2014), no. 4, 1139-1157. We prove a characteristic p version of some results on Du Bois singularities proven in characteristic zero by János Kollár and Sándor Kovács.
Depth of F-singularities and base change of relative canonical sheaves Zsolt Patakfalvi Journal of the Institute of Mathematics Jussieu, vol 13, no. 1, (2014), 43-63. We prove a characteristic p version of THIS PAPER by János Kollár.
On the numerical dimension of pseudo-effective divisors in positive characteristic Paolo Cascini, Christopher Hacon, Mircea Mustaţă Amer. J. Math. 136 (2014), no. 6, 1609-1628. We obtain results of Nakayama in positive characteristic. We use Frobenius to replace the Kawamata-Viehweg vanishing theorem.
p-1-linear maps in algebra and geometry Manuel BlickleA volume dedicated to David Eisenbud on his 65th birthday We survey p-1-linear maps from Frobenius splittings to test ideals.
Richardson varieties have Kawamata log terminal singularities Shrawan Kumar Int. Math. Res. Not. IMRN 2014, no. 3, 842-864. As the title says.
A Frobenius variant of Seshadri constants Mircea Mustaţă Math. Ann. 358 (2014), no. 3-4, 861-878. We define and study a characteristic p > 0 variant of Seshadri constants.
Explicitly Extending Frobenius Splittings over Finite Maps Kevin Tucker Comm. Algebra 43 (2015), no. 10, 4070-4079. We give a more explicit proof of a result that appeared in "On the behavior of test ideals under finite morphisms" mentioned below.
Bertini theorems for F-singularities Wenliang Zhang Proc. Lond. Math. Soc. (3) 107 (2013), no. 4, 851-874. We prove Bertini-type theorems for F-pure and F-regular pairs. We also show that Bertini-type theorems can't hold for F-injective pairs.
F-signature of pairs: Continuity, p-fractals and minimal log discrepancies Manuel Blickle and Kevin Tucker J. Lond. Math. Soc. (2) 87 (2013), no. 3, 802-818. We continue our work on F-signature of pairs
A dual to tight closure theory Neil Epstein Nagoya Math. J. 213 (2014), 41-75. We study a dual to tight closure theory and test ideals
Test ideals via a single alteration and discreteness and rationality of F-jumping numbers Kevin Tucker and Wenliang Zhang Mathematical Research Letters, vol 19, (2012) no. 1 We show that the test ideal can be described via an alteration even as some coefficients vary.
A canonical linear system associated to adjoint divisors in characteristic p > 0   J. Reine Angew. Math. 696 (2014), 69-87. We show that certain twistings of test ideals are globally generated. This yields some statements similar those for multiplier ideals.
F-singularities via alterations Manuel Blickle and Kevin Tucker Amer. J. Math. 137 (2015), no. 1, 61-109. We show that de Jong's alterations can be used to describe test ideals and F-rationality in a way that mimics the descriptions of multiplier ideals. We also obtain Nadel-type vanishing results.
Du Bois singularities deform Sándor Kovács Advanced Studies in Pure Mathematics, Minimal Models and Extremal Rays (Kyoto, 2011), 70 (2016), 49--66 We prove that Du Bois (or DB) singularities deform.
F-signature of pairs and the asymptotic behavior of Frobenius splittings Manuel Blickle and Kevin Tucker Advances in Mathematics, Vol 231, Issue 6, Pages 3232-3258, 2012 We introduce F-signature of pairs proving that they exist and that their positivity characterizes F-regularity.
A survey of test ideals Kevin Tucker Progress in Commutative Algebra 2 published by de Gruyter. Pages 39--99. We survey test ideals and F-singularities.
An algorithm for computing compatibly Frobenius split subvarieties Mordechai Katzman Journal of Symbolic Computation, vol 47, issue 8, Aug. 2012, 996--1008. As the title says.
Semi-log canonical vs F-pure singularities Lance Edward Miller Journal of Algebra, vol 349, issue 1, Jan. 2012, 150--164. We study non-normal F-pure singularities.
Cartier modules on toric varieties Jen-Chieh Hsiao, and Wenliang Zhang Trans. Amer. Math. Soc. 366 (2014), no. 4, 1773-1795. We study Cartier submodules, in the sense of Blickle-Bockle, of the structure sheaf in the toric setting.
Supplements to non-lc ideal sheaves O. Fujino, and Shunsuke Takagi RIMS Kôkyûroku Bessatsu, B24, 2011-03, pp. 1-47 We study variants of Fujino's non-LC ideal sheaves.
A note on discreteness of F-jumping numbers   Proc. Amer. Math. Soc. 139 (2011), no. 11, 3895-3901 We prove that the F-jumping numbers of an ideal are discrete at least when the ambient ring is Q-Gorenstein, of any index.
On the behavior of test ideals under finite morphisms Kevin Tucker To appear in the Journal of Algebraic Geometry. We describe how the test ideal behaves under finite morphisms.
Hodge theory meets the minimal model program: a survey of log canonical and Du Bois singularities Sándor Kovács Topology of Stratified Spaces, Math. Sci. Res. Inst. Publ., 58, Cambridge Univ. Press, (2011), 51--94 This paper surveys recent results on log canonical and Du Bois singularities, several new proofs are and some minor new results are also included.
A refinement of sharply F-pure and strongly F-regular pairs   Journal of commutative algebra, Vol 2, no.1, 91--110. (2010) This corrects a problem in the definitions of strongly F-regular and sharply F-pure pairs.
Test ideals in non-Q-Gorenstein rings   Transactions of the American Mathematical Society 363 (2011), no. 11, 5925--5941. We prove that the big test ideal τb(R) is the sum of test ideals τ(R, Δ) where Δ runs over divisors that make the pair (R, Δ) log-Q-Gorenstein.
Discreteness and rationality of F-jumping numbers on rings with singularities Manuel Blickle,
Shunsuke Takagi,
and Wenliang Zhang
Mathematische Annalen, Volume 347, Number 4, 917-949, 2010.. We prove discreteness and rationality of F-jumping numbers of pairs (R, a^t) when R is Q-Gorenstein with index not divisible by the characteristic p.
On the number of compatibly Frobenius split subvarieties, prime F-ideals, and log canonical centers Kevin Tucker Annales de L'Institut Fourier (Grenoble)60 no. 5 (2010), p. 1515-1531. We give a bound on the number of subvarieties compatibly Frobenius split with a fixed splitting of the Frobenius.
F-adjunction   Algebra and Number Theory. Vol 3, no. 8, 907--950. (2009) We do a characteristic p > 0 analog of inversion of adjunction along a center of log canonicity (at least in terms of relating singularities). Some applications are also explored.
Globally F-regular and log Fano varieties Karen Smith Advances in Mathematics, Vol 224 Issue 3, 863--894, 2010. We study connections between globally F-regular and log Fano varieties
Centers of F-purity   Mathematische Zeitschrift Vol. 265, No 3, 687-714, 2010.. We introduce and discuss a positive characteristic analogue of a notion from characteristic zero, log canonical centers.
The canonical sheaf of Du Bois singularities Sándor Kovács and
Karen Smith
Advances in Mathematics, Volume 224, Issue 4, Pages 1618-1640, 2010. We characterize Cohen-Macaulay Du Bois singularities in terms of their canonical sheaves and a resolution of singularities. You can also view Karen Smith giving a talk on these results HERE back in 2007.
Generalized test ideals, sharp F-purity, and sharp test elements   Mathematical Research Letters, Vol 15, no 5-6, 1251--1262, (2008) We show that the test ideal of a pair is generated by test elements (as long as you define test elements appropriately).
Rational singularities associated to pairs Shunsuke Takagi Michigan Mathematical Journal, Vol 57, 625--658, (2008). We work out definitions of rational and F-rational pairs.
F-injective singularities are Du Bois   The American Journal of Mathematics, Vol 131, no. 2, 445--473, (2009) Based off my thesis. We prove that singularities of F-injective type are Du Bois.
A simple characterization of Du Bois singularities   Composito Mathematica, 143 : 813-828 (2007) Also based off my thesis.
Gluing schemes and a scheme without closed points   Recent Progress in Arithmetic and Algebraic Geometry, AMS Contemporary Mathematics Series (2005) This provides an explicit example that studying algebraic geometry can in fact be pointless (that joke was originally due to Paul Smith). I came up with this as a second year graduate student and managed to get it published.

Research papers in preparation

(Drafts available upon request)

  • A geometric algorithm for computing seminormalization, based off my thesis, in preparation. In principal, this algorithm could be implemented in Macaulay2 or a similar program (Macaulay2 needs a couple functions which have not yet been implemented). I described this algorithm in my Thesis, and if you CLICK HERE you can view the relevant portion of my thesis on the topic.