MAT 615 Syllabus Curves on Algebraic Varieties Fall 2018

What follows is a tentative syllabus. The pace of these topics (as well as any additional topics to be covered) will be finalized in the first weeks of the semester.

• Week 1. Mori's proof of Hartshorne's Conjecture and Kontsevich's recursion for plane curves of geometric genus 0.
  Tools and techniques. Parameter spaces of curves in varieties. Specialization. Deformation Theory.
• Week 2. The Kollár-Ruan theorem on symplectic uniruledness.
  Tools and techniques. Virtual fundamental classes, Gromov-Witten invariants, free curves.
• Week 3. Rational curves on K3 surfaces.
  Tools and techniques. Hilbert schemes. Additive invariance of cohomology (and motives) under K-equivalence.
• Week 4. The Chevalley-Warning theorem, the Tsen-Lang theorem, and Steinberg's Theorem / Serre's "Conjecture I".
  Tools and techniques. Variants of the Lefschetz fixed point theorem in algebraic geometry.
• Week 5. The Deligne-Mumford theorem on irreducibility and smoothness of the moduli space of stable curves.
  Tools and techniques. Stacks. Semistable reduction theorems.
• Week 6. The Rationally Connected Fibration Theorem.
  Tools and techniques. Deformation theory of reducible curves and of ramified maps.
• Week 7. Irreducibility and smoothness of moduli spaces of stable maps to Fano manifolds.
  Tools and techniques. Inductive structure of the spaces of stable maps. Deformation to the normal cone. Inversion of adjunction.
• Week 8. Esnault's theory for coniveau-one varieties over finite fields. Serre's question for coniveau-one varieties over function fields.
  Tools and techniques. Decomposition of the diagonal. The maximally rationally connected fibration.
• Week 9. The Tsen-Lang Theorem, the Period-Index Theorem, and Serre's "Conjecture II".
  Tools and techniques. The determinant of cohomology. Stacks and discriminant avoidance.
• Week 10. The Rationally Simply Connected Fibration Theorem.
  Tools and techniques. Deformation theory of reducible surfaces.
• Week 11. The Weak Approximation Conjecture: work of Hassett-Tschinkel, de Jong-Starr, Tian-Zong, Chen-Zhu.
  Tools and techniques. Mori theory for moduli spaces.
• Week 12. Symplectic invariance of rational connectedness and rational surfaces.
  Tools and techniques. Gravitational descendants. Mori's explicit theory of threefold flips and flops.
• Week 13. The theorems of Hwang-Mok and Jan Gutt on specializations of projective homogeneous varieties of cominuscule type.
  Tools and techniques. Varieties of minimal rational curves and their tangency morphisms. Crystalline geometry.

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