| Title (arXiv link) | Co-authors | Journal | Notes | |
| Variants on Frobenius Intersection Flatness and Applications to Tate Algebras | Rankeya Datta, Neil Epstein, Karl Schwede, Kevin Tucker | We show that Tate algebras are Frobenius intersection flat, and so are a good place to study F-singularities. We also show openness of the F-pure locus in some new settings. | ||
| Closure operations induced via resolutions of singularities in characteristic zero | Neil Epstein, Peter M. McDonald, Rebecca R.G., Karl Schwede | We use resolutions of singularities and vanishing theorems to explore some tight-closure-like operations in characteristic zero. | ||
| BCM-thresholds of non-principal ideals | Sandra Rodríguez-Villalobos, Karl Schwede | Submitted | We explore a characteristic free analog of the Frobenius threshold. | |
| Plus-pure thresholds of some cusp-like singularities in mixed characteristic | Hanlin Cai, Suchitra Pande, Eamon Quinlan-Gallego, Kevin Tucker | Submitted | We compute the +-pure threshold of some cusp-like equations, like pa + xb. | |
| Perfectoid pure singularities | Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker, Joe Waldron, Jakub Witaszek | Submitted | We explore a mixed characteristic analog of F-pure and F-injective, as well as log canonical and Du Bois, singularities. | |
| The Briançon-Skoda Theorem via weak functoriality of big Cohen-Macaulay algebras | Sandra Rodríguez-Villalobos, Karl Schwede | To appear in the Michigan Mathematical Journal | We prove that a closure-style Briançon-Skoda-type Theorem follows from weak functoriality of big Cohen-Macaulay algebras. | |
| Finite generation of split F-regular monoid algebras | Rankeya Datta, Karl Schwede, Kevin Tucker | under revision | We show that a split F-regular semi-group ring is automatically Noetherian. | |
| Test ideals in mixed characteristic: a unified theory up to perturbation | Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker, Joe Waldron, Jakub Witaszek | We show that the various mixed characteristic test ideals are the same. We also show they can be computed by a sufficiently large alteration. | ||
| Perfectoid signature, perfectoid Hilbert-Kunz multiplicity, and an application to local fundamental groups | Hanlin Cai, Seungsu Lee, Linquan Ma, Kevin Tucker | under revision | We define a (perfectoid) mixed characteristic version of F-signature and Hilbert-Kunz multiplicity. | |
| Global generation of test ideals in mixed characteristic and applications | Christopher Hacon, Alicia Lamarche | Algebraic Geometry (Compositio), Vol 11, Issue 5, 676-711 (2024). | We study mixed characteristic test ideals on quasi-projectives schemes and prove facts about the diminished base locus | |
| Maximal Cohen-Macaulay complexes and their uses: A partial survey | Srikanth B. Iyengar, Linquan Ma, Mark E. Walker | Part of Commutative Algebra (Expository Papers Dedicated to David Eisenbud on the Occasion of his 75th Birthday, Springer) | We survey Cohen-Macaulay complexes | |
| Finding points on varieties with Macaulay2 | Sankhaneel Bisui, Zhan Jiang, Sarasij Maitra, Thái Thành Nguyên | J. Softw. Algebra Geom., Vol. 13 (2023), 33-43. | We present a package for finding rational and geometric points on varieties over a field of characteristic p | |
| Globally +-regular varieties and the minimal model program for threefolds in mixed characteristic | Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Kevin Tucker, Joe Waldron, Jakub Witaszek | Publications mathématiques de l'IHÉS, 138, 69-227 (2023). | We prove a version of the minimal program for threefolds in mixed characteristic | |
| Compatible ideals in Q-Gorenstein rings | Thomas Polstra | Proc. Amer. Math. Soc. 151 (2023), 4099-4112. | We show that every compatibly F-split ideal is a trace image from a finite extension for (close-to) Gorenstein rings | |
| FastMinors package for Macaulay2 | Boyana Martinova, Marcus Robinson, Yuhui Yao | J. Softw. Algebra Geom. Vol. 13 (2023), 13-31. | We present tools in Macaulay2 for computing partial ideals of minors in ways that can speedup computations | |
| An analog of adjoint ideals and PLT singularities in mixed characteristic | Linquan Ma, Karl Schwede, Kevin Tucker, Joe Waldron, Jakub Witaszek | J. Alg. Geom., 31 (2022), 497-559. | We study a perfectoid-BCM variant of adjoint ideals and plt-singularities. | |
| Singularities in mixed characteristic via Perfectoid big Cohen-Macaulay algebras | Linquan Ma | Duke Math. J. 170 (2021), no. 13, 2815-2890. | 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. | |
| Symbolic power containments in singular rings in positive characteristic | Eloísa Grifo, Linquan Ma | Manuscripta Mathematica, Volume 170, pages 471-496, (2023). | We study symbolic powers in positive characteristic in singular rings, and also produce a Fedder-type result in non-regular ambient rings for ideals of finite projective dimension. | |
| The FrobeniusThresholds package for Macaulay2 | Daniel J. Hernández, Pedro Teixeira, Emily E. Witt | J. Softw. Algebra Geom. 11 (2021), no. 1, 25-39. | We present a package for computing F-pure thesholds. | |
| The TestIdeals package for Macaulay2 | Alberto F. Boix, Daniel J. Hernández, Zhibek Kadyrsizova, Mordechai Katzman, Sara Malec, Marcus Robinson, Daniel Smolkin, Pedro Teixeira, Emily E. Witt | J. Softw. Algebra Geom. 9 (2019), no. 2, 89-110. | We present a package for computing test ideals. | |
| F-signature under birational morphisms. | Linquan Ma, Thomas Polstra, Kevin Tucker | Forum Math. Sigma 7 (2019), Paper No. e11, 20 pp. | We show that F-signature goes up when you blow up. | |
| Recent applications of p-adic methods to commutative algebra. | Linquan Ma | Notices Amer. Math. Soc. 66 (2019), no. 6, 820-831. | We survey recent work of Y. André, B. Bhatt, O. Gabber and others. | |
| Covers of rational double points in mixed characteristic. | Javier Carvajal-Rojas, Linquan Ma, Thomas Polstra, Kevin Tucker | J. Singul. 23 (2021), 127-150. | We extend Artin's classification of 2-dimensional rational double points to those of mixed characteristic with residual characteristic bigger than 5. | |
| Seminormalization package for Macaulay2 | Bernard Serbinowski | J. Softw. Algebra Geom. 10 (2020), no. 1, 1-7. | We describe an algorithm (implemented in Macaulay2) for computing seminormalization. | |
| A Kunz-type characterization of regular rings via alterations. | Linquan Ma | J. Pure Appl. Algebra 224 (2020), no. 3, 1124-1131. | We characterize regular rings by the projective dimension of derived pushforwards of the structure sheaves of alterations. | |
| Bertini Theorems for F-signature and Hilbert-Kunz multiplicity | Javier Carvajal-Rojas, Kevin Tucker | Math. Z. 299 (2021), no. 1-2, 1131-1153. | We show how F-signature behaves under taking general hyperplanes. | |
| Perfectoid multiplier/test ideals in regular rings and bounds on symbolic powers | Linquan Ma | Invent. Math. 214 (2018), no. 2, 913-955. | 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 | Int. Math. Res. Not. IMRN 2019, no. 14, 4325-4339. | 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 | Ann. Sci. É c. Norm. Supér. (4) 51 (2018), no. 4, 993-1016. | 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 | Proc. Amer. Math. Soc. 146 (2018), no. 2, 473-487.. | 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 | Math. Z. 288 (2018), no. 3-4, 1143-1155. | 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 | Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 17 (2017), no. 3, 1173-1205. | 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. | Illinois J. Math. 60 (2016), no. 3-4, 669-685. | 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, | J. Inst. Math. Jussieu 17 (2018), no. 1, 171-206. | 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 | Surveys on recent developments in algebraic geometry, 33-80, Proc. Sympos. Pure Math., 95, Amer. Math. Soc., Providence, RI, 2017. | 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 Takagi | Higher dimensional algebraic geometry - in honour of Professor Yujiro Kawamata's sixtieth birthday, 11-39, Adv. Stud. Pure Math., 74, Math. Soc. Japan, Tokyo, 2017. | 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 | Algebr. Geom. 5 (2018), no. 3, 264-327. | 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 Blickle | A 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 | J. Algebraic Geom. 23 (2014), no. 3, 399-443. | 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. |