TOMMASO DE FERNEX

Associate Professor - University of Utah

MATH 7800 - ARC SPACES OF ALGEBRAIC VARIETIES - SPRING 2012

Meeting: MW 3:30-4:50PM, AEB 360

Office Hours: By appointment

Instructor: Tommaso de Fernex

Contact: Office JWB 322, email defernex@math.utah.edu

Course Webpage: http://www.math.utah.edu/~defernex/7800-S12.html

Course Content: The focus of the course will be on arc spaces and their role in the study of algebraic varieties and their singularities. Topics covered in this course will include:

  • Jet schemes and arc spaces of smooth and singular varieties: their structure, and the birational transformation formula.
  • Motivic integration, with applications to the study of stringy invariants, K-equivalence, and the McKay correspondence.
  • The Nash problem: proof in dimension two and counterexample in dimension four.
  • Invariants of singularities (discrepancies, log canonical thresholds, multiplier ideals) via arc spaces, with applications to inversion of adjunction and birational geometry.

Prerequisites The course is intended for students with a solid background in the basics of algebraic geometry. If you just took the first semester in Introduction to Algebraic Geometry and have no previous experience in the field, this class will moslt likely be too advanced for you. In case you are interested in the course but are not sure whether it might be too advanced for you, feel free to write me or stop by my office. If you believe that it might be too early for you to take this class but are interested in algebraic geometry, then I recomment you to check out the Baby Algebraic Geomerty Seminar.

References: The course will be based on personal notes. As the class proceeds, I will list below relevant references on the topics. Most papers are available on arXiv.

  • Nash, John F., Jr., Arc structure of singularities. A celebration of John F. Nash, Jr. Duke Math. J. 81 (1995), no. 1, 31–38 (1996).
  • Greenberg, Marvin J. Rational points in Henselian discrete valuation rings. Inst. Hautes Études Sci. Publ. Math. No. 31 1966 59–64.
  • Ein, Lawrence; Mustata, Mircea Jet schemes and singularities. Algebraic geometry—Seattle 2005. Part 2, 505–546, Proc. Sympos. Pure Math., 80, Part 2, Amer. Math. Soc., Providence, RI, 2009.
  • Ein, Lawrence; Lazarsfeld, Robert; Mustata, Mircea Contact loci in arc spaces. Compos. Math. 140 (2004), no. 5, 1229–1244.
  • de Fernex, Tommaso; Ein, Lawrence; Ishii, Shihoko Divisorial valuations via arcs. Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 425–448
  • Javier Fernandez de Bobadilla, Maria Pe Pereira. Nash problem for surfaces. arXiv:1102.2212
  • Javier Fernandez de Bobadilla, Maria Pe Pereira. Curve selection lemma in infinite dimensional algebraic geometry and arc spaces. arXiv:1201.6310
  • Ishii, Shihoko; Kollár, János The Nash problem on arc families of singularities. Duke Math. J. 120 (2003), no. 3, 601–620.