18.727 Topics in Algebraic Geometry

Welcome to 18.727! This course aims to introduce techniques for studying intersection theory on moduli spaces. In particular, we will study the geometry of homogeneous varieties, the Deligne-Mumford moduli spaces of stable curves and the Kontsevich moduli spaces of stable maps using intersection theory.

Prerequisites: A first year graduate class in algebraic geometry at the level of the second and third chapters of Hartshorne.

Lecturer: Izzet Coskun, coskun@math.mit.edu

Office: 2-167

Handouts:

  • Course description (pdf)
  • Some very rough notes about Grassmannians. These notes are likely to be full of mistakes. Please use caution when reading them. I will revise these in the future. (pdf)
  • Some rough notes about the construction of the Hilbert scheme. I've added some sketchy notes on G.I.T. and the construction of the moduli spaces. Again they are most likely full of mistakes: use caution when reading them. (pdf)
  • Some rough notes containing a brief survey of the cohomology of the moduli space of curves and the Harer-Zagier formula for the orbifold Euler characterisitic. Again they are most likely full of mistakes: use caution when reading them. I added a few words outlining other work of Harer on the stability of the cohomology of the moduli space of curves. (pdf)
  • Some rough notes describing the Picard group of the moduli space of curves. (pdf). Unfortunately, I did not have enough time to include complete details.
  • Some rough notes about the Kodaira dimension of the moduli space of curves. (pdf). You should also read Chapter 5 and 6F of "Moduli of Curves".
  • Some rough notes on the Kontsevich moduli space of genus zero stable maps. These notes are likely to be full of mistakes. Please use caution when reading them. I will revise these in the future.(pdf).

    Practice problems:

  • A set of exercises in order to gain practice with Schubert calculus: (pdf)
  • A set of exercises in order to gain practice with the Hilbert scheme of conics: (pdf)
  • A set of exercises in order to gain practice with divisor class calculations. Caution: I might have mistyped some of the relations, so correct, then solve (pdf)

    Course description: This course will introduce techniques for studying intersection theory on moduli spaces such as homogeneous varieties, the Deligne-Mumford moduli space of stable curves and the Kontsevich moduli spaces of stable maps. The course will emphasize how one can deduce global geometric properties of moduli spaces and the objects they parameterize using intersection theory.

    The topics will include:

  • The Grassmannian: A great example of a moduli space, the cohomology of the Grassmannian, positivity and Littlewood - Richardson rules.
  • Basic results about the divisor theory and cohomology of the moduli space of curves due to Harer, Zagier, Arbarello and Cornalba
  • A study of Brill-Noether theory and the theory of limit linear series in order to prove that the moduli space of curves is of general type if g > 23.
  • Ample and effective cones of moduli spaces: Recent developments due to Farkas, Gibney, Keel, Khosla and Morrison about the ample and effective cones of the moduli space of curves, the ample and effective cones of Kontsevich moduli spaces
  • The Gromov-Witten invariants of simple homogeneous varieties

    Required Texts: There will be no required texts for this course. I will regularly post lecture notes on the web. Below are a list of suggested papers and books.

    Unit 1. The Grassmannian: An example of a moduli space.

    Suggested reading (For precise references see bibliography):

  • Griffiths and Harris, Principles of Algebraic Geometry, p. 193-211.
  • Fulton, Intersection Theory, chapter 14
  • Harris, Algebraic Geometry: A First Course, Lectures 6 and 16.
  • R. Vakil, A Geometric Littlewood - Richardson Rule (pdf)
  • Kleiman and Laksov have a very pleasant survey article in the American Mathematical Monthly 79 (1972) p. 1061--1082. See (html).

    Unit 2. The moduli space of curves.

    Suggested reading:

  • You can find a sketch of the G.I.T. construction in Harris and Morrison, The moduli space of curves, Chapter 4.
  • For the construction of the Hilbert scheme look at Mumford, Curves on surfaces, Sernesi, Topics on Families of projective schemes or Grothendieck's original lectures.
  • For a very detailed treatment of G.I.T. look at Mumford's Geometric Invariant Theory. (There are newer editions with appendices by Fogarty and Kirwan).
  • For a detailed proof of the Potential Stability Theorem see Gieseker's Lectures on Moduli of Curves.
  • A good account of the G.I.T. construction is contained in Mumford's article Stability of projective varieties.

    Unit 3. The cohomology of the moduli space of curves.

    Suggested reading:

  • A great source is Harer's CIME notes.
  • Check out this paper by Hain and Looijenga, it contains a survey of the stuff we have been talking about (ps )
  • E. Arbarello, M. Cornalba Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves, J. Alg. Geom. 5 (1996), 705-749.
  • Arbarello, Enrico; Cornalba, Maurizio Calculating cohomology groups of moduli spaces of curves via algebraic geometry. Publications Mathématiques de l'IHÉS, 88 (1998), p. 97-127 (pdf )
  • E. Arbarello, M. Cornalba The Picard Groups of the Moduli Spaces of Curves (Topology, Vol. 26, 1987 , pp. 153-171).

    Unit 4. The moduli space of curves is of general type.

    The following papers of Joe (with Mumford and Eisenbud) developped the theory:

  • On the Kodaira dimension of the moduli space of curves. With an appendix by William Fulton. Invent. Math. 67 (1982), no. 1, 23--88.
  • On the Kodaira dimension of the moduli space of curves. II. The even-genus case. Invent. Math. 75 (1984), no. 3, 437--466.
  • The Kodaira dimension of the moduli space of curves of genus $\geq 23$. Invent. Math. 90 (1987), no. 2, 359--387.

    Unit 5. The Kontsevich moduli space of stable maps.

    Fulton and Pandharipande have a great article. This is probably the best introduction to the subject.

  • Fulton and Pandharipande, Notes on stable maps and quantum cohomology (html ) Other papers of interest are:
  • Rahul Pandharipande, Intersections of Q-Divisors on Kontsevich's Moduli Space $\bar{M}_{0,n}(P^r,d)$ and Enumerative Geometry (html )
  • Ravi Vakil, "The enumerative geometry of rational and elliptic curves in projective space", J. Reine Angew. Math. (Crelle's Journal) 529 (2000), 101--153. (html )