Research

Karen Smith’s research lies at the interface of commutative algebra and algebraic geometry. Algebraic geometry is the study of geometric shapes which are defined by polynomial equations; commutative algebra is the study of the rings of polynomial functions on such geometric objects. Specifically, one focus of Smith’s research the use of prime characteristic methods to prove results about complex projective varieties. For example, the singularities of varieties can be measured in various ways using reduction to characteristic p and then iteration of the Frobenius map, and Smith was one of the leaders in unravelling the connections with rational singularities and other singularities in birational geometry. Similarly, global properties of projective varieties, such as the ways in which they can embed in different projective spaces, can be understood by studying the splitting properties of the Frobenius map, and Smith was a leader in establishing connections with positivity. Karen Smith also has been involved with the development of asymptotic multiplier ideals (in char 0) and their characteristic p analog, test ideals, and invariants such a jumping numbers derived from them.

Karen E. Smith’s research is partially funded by the National Science Foundation, and has also been funded by the Alfred P. Sloan Foundation, by the Clay Foundation, and by a US government Fulbright fellowship.

Books

Pre-arxiv Preprints (before 2006)

You can also visit some of my collaborators’ homepages by clicking on their underlined names in the below list.

1. The jet schemes of a monomial scheme with Russell Goward.
2. Motivic Integration, a brief survey written by request for MSRI‘s Concepts page. A slightly longer version of the survey was written for MSRI’s Emissary newsletter, and appeared in the 2003 issue.
3. Jumping coefficients of multiplier ideals with Lawrence Ein, Rob Lazarsfeld and Dror Varolin. To appear in Duke Math Journal.
4. Core versus graded core and global sections of line bundles with Eero Hyry. To appear in Transactions of AMS.
5. On a non-vanishing conjecture of Kawamata and the core of an ideal with Eero Hyry. American Journal of Math. Volume 125. 2003.
6. Uniform approximation of Abhyankar valuation ideals in smooth function fields, with Lawrence Ein and Rob Lazarsfeld, American Journal of Math, 2003.
7. The Hodge Conjecture. This is an expository paper in Archimedes 2002, the journal of the Finnish Math and Physical Society. The figures are not included in this version.
8. Tight closure and vanishing theorems. (Lecture notes from summer school called “Vanishing theorems and effective results in algebraic geometry”, 2001) electronically published by the International Center for Theoretical Physics, Trieste Italy.
9. Behavior of the test ideal under smooth and etale homomorphisms, with Ana Bravo, Journal of Algebra,
10. Uniform Behavior of Symbolic Powers of Ideals, with Lawrence Ein and Rob Lazarsfeld, Inventiones Math, 2001
11. An algebraic proof of Zak’s inequality for the dimension of the Gauss image with Aron Simis and Bernd Ulrich, Math Z.
12. The strong test ideal with Nobuo Hara, Illinois Journal of Math.
13. Globally F-regular varieties: applications to vanishing theorems for quotients of Fano varieties, 2000 Fulton Volume of the Michigan Journal of Math.
14. The multiplier ideal is a universal test ideal. Hartshorne volume of Communications in Algebra (2000)
15. An introduction to tight closure. A survey in the proceedings of the Messina Conference on Commutative Algebra and Algebrais Geometry. (Note: this is essentially the same as the paper earlier listed here as notes from the 20-th annual Japanese commutative algebra conference.)
16. On the commutation of the test ideal under localization and completion, with Gennady Lyubeznik. Transactions of AMS
17. A Tight closure proof of Fujita’s freeness conjecture for very ample line bundles. Mathematische Annalen
18. Rational and non-rational Algebraic Varieties: Lectures of János Kollár. This is a detailed write up of János Kollár’s, Course At the EMS summer school in Algebraic Geometry in Eger, Hungary, August 1996. Also available in the Duke Archives paper number 9707013. We plan to eventually turn this into a book.
19. Weak and Strong F-regularity are equivalent in graded rings, with Gennady Lyubeznik. American Journal of Mathematics, 1999
20. F-regularity deforms for Q-Gorenstein rings, Just a brief note proving the above statement in characteristic zero (essentially just pointing out that Kollár has already proved this).
21. Tight Closure commutes with localization in binomial rings Proceedings of AMS
22. Sparse Systems of Parameters for Determinantal Varieties, with Donna Glassbrenner, Advances in Applied Mathematics 19, 529-558 (1997)
23. Fujita’s Freeness Conjecture in terms of Local Cohomology Journal of Algebraic Geometry,
24. Vanishing Theorems, Singularities, and Effective Bounds in Algebraic Geometry Via Prime Characteristic Local Algebra to appear in Proc. Symposia of Pure Mathematics, eds J. Kollár and R. Lazarsfeld. An expositional article surveying of applications of new ideas in characteristic p commutative algebra to algebraic geometry. Includes several new results as well. Important: Be sure to also download this Erratum as well.
25. Tight Closure in Graded Rings, Kyoto Journal of Mathematics, Vol 37 No. 1. (1997) 35-53.
26. Tight Closure and the Kodaira Vanishing Theorem, with Craig Huneke, Journal fur die riene und angewandte Mathematics, 484 (1997)
27. Linear Growth of Primary Decompositions of Monomial Ideals with Irena Swanson, Communications in Algebra, 25(10), 3071-3079 (1997)
28. Simplicity of Rings of Differential Operators in Prime Characteristic with Michel Van den Bergh, Proc. London Math. Society, (3) 75 (1997) 32-62.
29. The D-module structure of F-split rings, Mathematical Research Letters, 2 377-386 (1995)
30. A Tight Closure Approach to Arithmetic Macaulayfication with Ian Aberbach and Craig Huneke, Illinois Journal of Math. 40, 310-329 (1996)
31. F-Rational Rings have Rational Singularities, American Journal, 119, 159-180 (1997)
32. Tight Closure of Parameter Ideals, Inventiones Math. 115, 41-60, (1994)
33. Test ideals in local rings, Transactions AMS
34. Singularities of certain ladder determinantal varieties, with Donna Glassbrenner, Journal of Pure and Applied Algebra, 101, 59-75 (1995).
35. Tight Closure and Graded Integral Extensions, Journal of Algebra, 175, 568-574, (1995)