Summary

A classical problem in algebraic geometry is to ask how many rational curves of a given degree pass through some generally chosen points on the plane. In the case of a degree d rational curve passing through 3d – 1 points, the answer N_d is known to be finite, and classical counts up to d = 4 date back to the 19th century and earlier. Within the last thirty years, the development of Gromov-Witten theory provided a new approach to these sorts of curve-counting problems, yielding a recursive formula for N_d.

Over the real numbers, the number of curves interpolating these points depends on the placement of the points. However, Welschinger established that a certain signed count of the real curves is invariant of the placement of the points, using methods derived from symplectic geometry. These signs can be interpreted as the local Brouwer degree of an evaluation map out of the moduli of stable curves.

Recent work of Kass, Levine, Solomon, and Wickelgren produces a well-defined A¹-degree of this evaluation map, providing a quadratically enriched count of rational curves interpolating points over an arbitrary field. This leverages the machinery of A¹–homotopy theory, which provides an A¹-Brouwer degree valued in the Grothendieck-Witt ring of quadratic forms over a field k.

Seminar goals

• Learn about virtual fundamental classes and Gromov-Witten invariants, and their relationship to counting curves.
• Understand Welschinger invariants and counting real curves.
• Learn about A¹-enumerative geometry, which provides enumerative answers valued in the Grothendieck-Witt ring GW(k) of a field which specializes to both the complex and real curve counts.
• Work through the details of a recent paper of Kass-Levine-Solomon-Wickelgren which defines a GW(k)-valued count of rational curves on a del Pezzo surface interpolating points as the A¹-degree of an evaluation map.

Here is a tentative weekly outline of topics to be covered.

Participant expectations

Participants will take turns giving presentations. The seminar organizers will make sure that presenters have written source material upon which to base their presentations. Participants are expected to attend regularly.

Participants are encouraged to skim the source material being presented each week. They are also encouraged to participate in the private shared online workspace.

As a general rule, students are expected to obtain 1-2 independent study credits from their home universities for their participation in the seminar. This requirement is waived on an individual basis for students whose home universities do not have formal independent study opportunities.

Organizers

The seminar is organized by Thomas Brazelton (Harvard University) and Sabrina Pauli (Darmstadt). They are assisted by graduate student assistants to be named.

Application

Please submit the application form to apply. Applications received by August 19 will receive full consideration. Contact Dan Isaksen for inquiries about applications.

Schedule

The seminar meets regularly on Mondays at 10:00am eastern time, starting September 9 and ending November 25. Office hours/informal discussion/problem sessions will meet regularly at 10:00am eastern time on Wednesdays.