Pete  L. Clark
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 Math 8320 -- Algebraic Curves: MWF 1:25-2:15, 326 Boyd

Instructor: Assistant Professor Pete L. Clark, pete (at) math (at) uga (dot) edu

Course webpage: http://math.uga.edu/~pete/curves.html (i.e., right here)

Office Hours: During an hour long discussion section each week in which all students are strongly encouraged to attend. Otherwise by appointment.

I am quite amenable to booking ``extra'' office hours. The ground rules are: (i) please give me at least 24 hours notice. (ii) Please send me an email the night before a morning appointment or the morning of a later appointment to remind me that we are meeting. (iii) If I do not show for an appointment (empirically, the chance that I will fail to show seems to be about 5-10%), feel free to call me on my cell phone. Probably I'm not too far away. (iv) If we do book an appointment, please do show up or call or email to let me know you're not coming!

Course text: Algebraic Geometry and Arithmetic Curves, by Qing Liu.

Discussion Section: I would like to have an extra discussion section, one hour a week, for discussion of problems and presentation of projects.

How Your Grade is Computed: Based on attendance, class participation, and good efforts to work on homework problems, which may come from Liu's text or may be posted here.

Course Prerequisites: Some prior background in commutative algebra and algebraic geometry, including the language of schemes. Elliptic curves and elliptic surfaces make for nice motivating examples, so some prior familiarity with their arithmetic would be helpful. Seamless mastery of all these subjects is not required or expected -- I'm sure the seams of my own mastery of the subject will show during the course of the semester! -- rather students need to have some working knowledge and the ambition to improve upon it. Students are strongly encouraged to share their expertise with each other (and, of course, to ask me questions).

Course Content: The main topic of the course is one-parameter families of algebraic curves, or in other words arithmetic surfaces. This is a very beautiful and important subject and, especially, a meeting place for algebraic and arithmetic geometers. The notion of an arithmetic surface includes at the same time that of a surface over a field fibered over a curve (as every surface can be) and the theory of reduction modulo p of an algebraic curve over a number field.
More specifically, the breakdown is as follows:

Part 0: Review of scheme-theoretic algebraic geometry (Chapters 2 - 4 of Liu's book)

Part 1: Theory of "nice" algebraic curves over an arbitrary field (Chapter 7)

Part 2: Theory of singular algebraic curves over an arbitrary field (Chapter 7)

Part 3: Theory of arithmetic surfaces and models of curves; semi/stable reduction (Chapters 8 - 10)

Especially we hope to discuss the following results: let S be a Dedekind scheme with field of functions K = K(S), and C/K a nice curve over the generic fiber of S.

Theorem 1: C admits a regular model C/S, i.e., a regular arithmetic surface over S whose generic fiber is isomorphic to C/K.

Theorem 2: Assuming C has positive genus, there exists a unique minimal regular model.

Theorem 3 (semistable reduction theorem): It is possible to make a finite base change L/K such that the minimal model of C/L is a semistable curve. If the genus of C is at least two, then from this model we can construct the stable (normal, not necessarily regular) model.

Supplementary reading:

  • Pete L. Clark, On the indices of curves over local fields, Manuscripta Math. 124 (2007), 411-426, (pdf)

  • Pierre Deligne and David Mumford, The irreducibility of the space of curves of given genus, Publ. math. IHES 36 (1969), 75-109 (pdf)

  • Arnaldo Garcia, On higher-order Weierstrass points and the finiteness of the automorphism group, Manuscripta Math. 78 (1993), 413-416.

  • Dino Lorenzini, Qing Liu et Michel Raynaud, Néron models, Lie algebras, and reduction of curves of genus one, Invent. Math. 157 (2004), 455-518. (pdf)

  • Dino Lorenzini, Qing Liu et Michel Raynaud, On the Brauer group of a surface, Invent. Math. 159 (2005), 673-676. (pdf)

  • Dino Lorenzini et Qing Liu, The index of a variety over a discrete valuation field. (See Dino's talk on 8/27/08)

  • Catherine H. O'Neil, Models of some genus one curves with applications to descent, Journal of Number Theory 112 (2005), 369-385. (pdf)

  • Bjorn Poonen, Lectures on rational points on curves (pdf)

  • Bjorn Poonen, Rational points on varieties (pdf)

  • Bjorn Poonen, Varieties without extra automorphisms I: curves (pdf)

  • Jean-Pierre Serre, Algebraic Groups and Class Fields, Springer GTM. Chapter 4: Singular Algebraic Curves. Chapter 5: Generalized Jacobians.

  • Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Springer GTM. Chapter 3: Elliptic Surfaces. Chapter 4: Néron Models.

  • Karl-Otto Stohr and J. Felipe Voloch, Weierstrass points and curves over finite fields (pdf)

  • Richard G. Swan, On Munshi's Proof of the Nullstellensatz (pdf)

    Handouts/Lecture Notes: When possible, I will post my lecture notes here, with the understanding that they are not intended as polished, complete documents: CAVEAT EMPTOR.

    Lecture Notes 1: Introduction. (11 pages) (pdf)

    Lecture Notes 2: The category of affine k-varieties. (5 pages) (pdf)

    Lecture Notes 2bis: Function fields and geometric integrality. (3 pages) (pdf)

    Lecture Notes 3: Nullstellensatzen et al. (10 pages) (pdf)

    Lecture Notes 4: Introduction to the geometry of schemes. (8 pages) (pdf)

    Lecture Notes 5: Fiber products; separated and proper morphisms. (10 pages) (pdf)

    Lecture Notes 6: First steps in the geometry of curves. (21 pages) (pdf)

    Homework Problems (1-28): (pdf)

    Homework Problems (29-47): (pdf)

    Homework Problems (48-71): (pdf)