NCTS Algebraic Geometry Learning Seminar will focus on the deformation theory in the first half year. This is the 8th talk of Part II.

 

Part II - Deformation of rational curves. The outline of the talks is based of Debarre’s book, [Deb01], with possible references to Kollar's book, [Kol96]. The talks will focus around chapters 3-5 of [Deb01], culminating in proving an effective bound for the volume of a smooth Fano variety based on dimensions which implies the boundedness of family of smooth Fano varieties.

 

Talk 8 (Rational curves on algebraic K3)  We explain the Appendix of Mori and Mukai [MM83]: By applying deformation theory with the moduli of K3 surfaces, it is showed that we can find a rational curve on an algebraic K3 surface.

 

[MM83] Shigefumi Mori and Shigeru Mukai. The uniruledness of the moduli space of curves of genus 11. In Algebraic geometry (Tokyo/Kyoto, 1982), volume 1016 of Lecture Notes in Math., pages 334–353. Springer, Berlin, 1983.

 

Part I - Abstract deformation theory. We mainly follow Hartshorne's Deformation Theory book.

Part II - Deformation of rational curves. We follow Debarre's Higher Dimensional Algebraic Geometry.