Thematic Term Complex Algebraic Geometry

Jason Starr donnera un mini-cours

Title: Geometry of Spaces of Curves on Varieties

• Lectures 1 and 2. Parameter spaces for curves: the Hilbert scheme, the Chow variety and the Kontsevich spaces.
  Introduction to the three most common parameter spaces for curves on varieties including a brief overview of the deformation theory of each parameter space.
  References: Chapters I and II of Koll\'ar's "Rational curves on algebraic varieties", Chapter 2 of Debarre's "Higher-dimensional algebraic geometry". Fulton and Pandharipande's article.

• Lecture 3. Irreducibility of the parameter spaces.
  Discussion of irreducibility of these parameter spaces, particularly when the ambient variety is projective space or a hypersurface in projective space. Application to showing Gromov-Witten invariants are "enumerative".
  References: "The connectedness of the moduli space of maps to homogeneous spaces" by Kim and Pandharipande. "Rational curves on hypersurfaces of low degree" by Harris, Roth and Starr.

• Lecture 4. Ample and effective divisors of Kontsevich moduli spaces.
  Discussion of the ample and effective cones of the Kontsevich moduli spaces. Relation to the F-conjecture. New moduli spaces of curves with an application to "rational simple connectedness".
  References: "The ample cone of the Kontsevich moduli space" and "The effective cone of the Kontsevich moduli space" by Coskun, Harris and Starr. "Higher Fano manifolds and rational surfaces" by de Jong and Starr.

• Lecture 5. Singularities and Kodaira dimensions of Kontsevich moduli spaces.
  Discussion of such birational aspects of the Kontsevich moduli spaces as uniruledness, rational connectedness and Kodaira dimensions. Conjectural relationship with unirationality of the ambient variety.
  References: "Rational surfaces in index-one Fano hypersurfaces" by Beheshti and Starr.
  "Cubic fourfolds and spaces of rational curves" by de Jong and Starr.