MAT 614 Syllabus Curves on Algebraic Varieties Fall 2011

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. Castelnuovo's Contractibility Theorem and Uniruledness of Fano Manifolds.
• Week 2. Overview of the Course and Toolkit for the Course.
• Week 3. Several Bend and Break Theorems.
• Week 4. The Cone Theorem and the Contraction Theorem (Smooth Case).
• Week 5. Cursory Overview of the Minimal Model Program.
• Week 6. Rationally Connected Varieties: Definitions and Foundational Results.
• Week 7. Rationally Connected Fibrations over Curves.
• Week 8. The Weak Approximation Problem: the Hassett-Tschinkel Theorem.
• Week 9. Pseudo-sections and Extension of Sections from a Base Curve to a Higher-dimensional Base.
• Week 10. Ax's Conjecture and Rationally Connected Varieties over PAC Fields.
• Week 11. Rationally Simply Connected Varieties: Definitions and Main Examples.
• Week 12. Rational Simple Connectedness and the Weak Approximation Problem: Hassett's Theorem.
• Week 13. Rationally Simply Connected Fibrations over Surfaces.
• Week 14. Applications and Rationally Simply Connected Varieties over Global Function Fields / Function Fields over PAC Fields.

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