SB Student Algebraic Geometry Seminar, Spring 2010
The topic this semester is spaces of Kontsevich stable maps. The
first half of the semester will focus on using Kontsevich stable maps
to define a system of Gromov-Witten invariants satisfying the
Kontsevich-Manin axioms. Later we will study the geometry of
Kontsevich moduli spaces with a view to other applications.
Textbook:
We are not following a textbook this semester. But one useful
reference is Mirror symmetry and
algebraic geometry by D. Cox and S. Katz, published by the AMS.
For the most part, the lectures will be based on the original
articles.
Meetings:
The seminar will meet on Mondays, 12:50-2:20pm in Math Tower 5-127.
Here is a very tentative list of lectures (but we can easily change
this to suit the tastes of the seminar participants).
- February 1st. Jason Starr.
Overview of Gromov-Witten invariants.
The recursion for genus 0 invariants in the plane. The
Kontsevich-Manin axioms. Rough sketch of construction of
Gromov-Witten invariants.
- February 8th. Jason Starr.
The Kontsevich-Manin axioms imply the genus 0
recursion. Definition of stable maps and basic properties:
Deligne-Mumford stack, functoriality, properness, obstruction
theory, the Behrend-Manin spaces.
- February 15th. Jason Starr.
Perfect obstruction theories and virtual
fundamental classes after Behrend-Fantechi, part I.
- February 22nd. Yi Zhu.
Perfect obstruction theories and virtual
fundamental classes after Behrend-Fantechi, part II.
- March 1st. Jason Starr.
Verification of (some of) the Kontsevich-Manin
axioms.
- March 8th. AJ Tolland.
GW theory is a QFT: a useful mnenonic. I'll sketch the origins of
Gromov-Witten theory in physics, and explain in what sense it is a
topological quantum field theory. Then I'll explain what the "deep
inner meaning" of the Kontsevich-Manin axioms is
- March 15th. Rob Findley.
When are Gromov-Witten invariants
"enumerative". Kleiman-Bertini theorem. Irreducibility of some
Kontsevich moduli spaces.
- March 22nd. Matt DeLand.
The cohomology (especially the divisor
class group) of genus 0 Kontsevich moduli spaces.
- April 5th. Yi Zhu.
The ample and big cones of genus 0 Kontsevich
spaces. Application to "new" moduli spaces.
- April 12th. Jason Starr.
Singularities of genus 0 Kontsevich spaces.
- April 19th. Zhiyu Tian (Jason out of town).
Kodaira dimensions of genus
0 Kontsevich spaces.
- April 26th. Zhiyu Tian.
Uniruledness / rational connectedness of
Kontsevich spaces (rational simple connectedness). Non-uniruledness
of Kontsevich spaces.
- May 3rd. TBA.
Additional topics.
Office Hours:
Graduate student speakers are encouraged to talk with me about the
lecture beforehand (I am not an expert, but I have taught this
material once before). My
office hours are Tuesday 9-10am, Thursday 9-10am and 1-2pm in Math
Tower P-143. (But we can set up another appointment if these times do
not work.)
References:
For the most part, lectures will be based on the original articles.
Here are references and links to some of the original articles, as
well as links to
useful secondary sources. These will be updated as the semester goes
on.
MR1291244 (95i:14049)
Kontsevich, M.(D-MPI); Manin, Yu.(D-MPI)
Gromov-Witten classes, quantum cohomology, and enumerative
geometry. (English. English summary)
Comm. Math. Phys. 164 (1994), no. 3, 525--562.
14N10 (53C15 58D10 58F05)
(article,
preprint).
MR1363062 (97d:14077)
Kontsevich, Maxim(D-MPI)
Enumeration of rational curves via torus actions.
The moduli space of curves (Texel Island, 1994), 335--368,
Progr. Math., 129,
Birkhäuser Boston, Boston, MA, 1995.
14N10 (14D22 14L30)
(preprint).
MR1412436 (98i:14014)
Behrend, K.(3-BC); Manin, Yu.(D-MPI)
Stacks of stable maps and Gromov-Witten invariants.
Duke Math. J. 85 (1996), no. 1, 1--60.
14D20 (14C25 14D22)
(article,
preprint).
MR1437495 (98e:14022)
Behrend, K.(3-BC); Fantechi, B.(I-TRNT)
The intrinsic normal cone. (English. English summary)
Invent. Math. 128 (1997), no. 1, 45--88.
14F99 (14C15 14D20)
(article,
preprint).
MR1714820 (2001i:14074)
Behrend, K.(3-BC)
Algebraic Gromov-Witten invariants. (English. English summary)
New trends in algebraic geometry (Warwick, 1996), 19--70,
London Math. Soc. Lecture Note Ser., 264,
Cambridge Univ. Press, Cambridge, 1999.
14N35 (14C17 14D20)
(preprint).
MR1666787 (2000h:14005)
Graber, T.(1-HRV); Pandharipande, R.(1-CAIT)
Localization of virtual classes.
Invent. Math. 135 (1999), no. 2, 487--518.
14C17 (14D20 14N10 14N35)
(article,
preprint).
MR1911301 (2003g:14076)
Behrend, K.(3-BC)
Localization and Gromov-Witten invariants. (English. English summary)
Quantum cohomology (Cetraro, 1997), 3--38,
Lecture Notes in Math., 1776,
Springer, Berlin, 2002.
14N35 (14A20 14C17)
MR1664633 (99m:14002)
Quantum cohomology at the Mittag-Leffler Institute.
Edited by P. Aluffi.
Appunti dei Corsi Tenuti da Docenti della Scuola. [Notes of Courses
Given by Teachers at the School]
Scuola Normale Superiore, Pisa, 1997. vi+163 pp.
14-06 (14N10)
(preprint).
MR0268188 (42 #3087)
Artin, M.
Algebraic approximation of structures over complete local rings.
Inst. Hautes Études Sci. Publ. Math. No. 36 1969 23--58.
14.05
(article).
MR0260746 (41 #5369)
Artin, M.
Algebraization of formal moduli. I. 1969
Global Analysis (Papers in Honor of K. Kodaira) pp. 21--71
Univ. Tokyo Press, Tokyo
14.05
MR0260747 (41 #5370)
Artin, M.
Algebraization of formal moduli. II. Existence of modifications.
Ann. of Math. (2) 91 1970 88--135.
14.00
(article).
MR0399094 (53 #2945)
Artin, M.
Versal deformations and algebraic stacks.
Invent. Math. 27 (1974), 165--189.
14D15 (14B05)
MR1034665 (92f:14003)
Keel, Sean(1-UT)
Intersection theory of moduli space of stable $n$-pointed curves of genus zero.
Trans. Amer. Math. Soc. 330 (1992), no. 2, 545--574.
14C15 (14C17 14H10)
(article).
Other useful sources.
A preprint
by Karl Schwede about coproducts in the category of schemes. Artin
proved a general theorem about existence of coproducts in the
category of algebraic spaces, Theorem 6.1 in Algebraization of
Formal Moduli, II.
Some notes on stacks written by Aaron Bertram and Herb Clemens
for the 2001 Park City Math Institute,
(article)
MR1905329 (2003h:14003)
Fantechi, Barbara(I-UDIN-MI)
Stacks for everybody. (English. English summary)
European Congress of Mathematics, Vol. I (Barcelona, 2000), 349--359,
Progr. Math., 201,
Birkhäuser, Basel, 2001.
14A20
(article)
Back to my home page.
Jason Starr
4-108 Math Tower
Department of Mathematics
Stony Brook University
Stony Brook, NY 11794-3651
Phone: 631-632-8270
Fax: 631-632-7631
Jason Starr, jstarr@math.sunysb.edu