Rational curves on algebraic varieties

A working seminar organized by Raymond Cheng and Stefan Schreieder during the fall of 2024

Rational curves, whether through their abundance or absence, provide a powerful method of measuring the complexity of algebraic varieties.

This seminar falls into three parts. In the first third, we work through the basic theory around rational curves in algebraic varieties and rational connectedness. In the second part, we look to understand recent work of Kollár and Zhiyu Tian on lifting algebraic equivalence of \(1\)-cycles to deformation equivalence, and its arithmetic consequences. The final part of the seminar will cover topics of interest proposed by participants of the seminar.

References

Two excellent and comprehensive references for the basic theory of rational curves on algebraic varieties include the following two books:

J. Kollár (1991), Rational curves on algebraic varieties, and
O. Debarre (2001), Higher-Dimensional Algebraic Geometry.

These works have been remixed into brief and introductory notes:

C. Araujo and J. Kollár (2003), Rational curves on varieties.
O. Debarre (2011), Lecture notes on rational curves on algebraic varieties.

Original papers in this subject that we touch upon include:

S. Mori (1979), Projective manifolds with ample tangent bundles.
S. Mori, Y. Miyaoka (1986), A numerical criterion for uniruledness
J. Kollár, Y. Miyaoka, and S. Mori (1992), Rational connectedness and boundedness of Fano manifolds.
T. Graber, J. Harris, and J. Starr (2003), Families of rationally connected varieties.
A. J. de Jong and J. Starr (2003), Every rationally connected variety over the function field of a curve has a rational point.

In the second part of the seminar, we look to understand the recent work of Kollár and Zhiyu Tian:

J. Kollár and Z. Tian (2023), Stable maps of curves and algebraic equivalence of \(1\)-cycles.

Schedule

We meet Thursdays in F107 between 14:15 and 15:45. A program may also be found here.

17.10: Stefan Schreieder; Introduction and organization
24.10: Sophie Friesen; Deforming parameterized curves and bend-and-break
07.11: Simon Pietig; Free and very free curves
14.11: Robin Lahni; Smoothing trees, combs, and chains
28.11: Pascal Fong; Constructing sections
05.12: Fumiaki Suzuki; Lifting algebraic to deformation equivalence, I
12.12: Dominique Mattei; Lifting algebraic to deformation equivalence, II
19.12: Jan Lange; Lifting algebraic to deformation equivalence, III
09.01: Raymond Cheng; Geometric Manin Conjecture
16.01: TBD; TBD
23.01: Tim Gräfnitz; Stable curves and stable maps
30.01: Anneloes Viergever; Counting (real) rational curves