Moduli of curves in varieties

Jun Li

Moduli problem is one of the three main topics in algebraic geometry, besides classification and cycles. Moduli of curves, and moduli of sheaves (of various sorts) are the two most active branches of moduli problem studying.

Take moduli of curves in a variety; the classical question is to enumerate the number of a class of curves subject to incidence relations. The modern question is to enumerate the “topological” number of curves subject to the same condition. This leads to the notion of virtual cycles; a well-known example is GW-invariants of varieties. Inspired by mathematical physics, the drive to understand “topological” enumerations led to a new research field in algebraic geometry, studying invariants of moduli spaces. The progress in this field has huge impacts on algebraic geometry and on other subjects of mathematics.

In this talk, I will touch on several aspects of this new research field, concentrating on enumerating “topological” numbers of curves in algebraic geometry, like constructing GW-invariants, applying degeneration technique, and high genus invariants of quantic Calabi-Yau threefolds. I will conclude with a recent work on applying the technique developed to the function field analogue of Lang’s conjecture on K3 surfaces.

Figure: Rational curves in the sky over Building-383.