Degenerating tangent curves

Abstract: It is well-known that a smooth plane cubic $E$ supports $9$ flex lines. In higher degrees we may ask an analogous question: "How many degree $d$ curves intersect $E$ in a single point?" The problem of calculating such numbers has fascinated enumerative geometers for decades. Despite being an extremely classical and concrete problem, it was not until the advent of Gromov-Witten invariants in the 1990s that a general method was discovered. The resulting theory is incredibly rich, and the curve counts satisfy a suite of remarkable properties, some proven and some still conjectural. In this talk I will discuss joint work with Lawrence Barrott, in which we study the behaviour of these tangent curves as the cubic $E$ degenerates to a cycle of lines. Using the machinery of logarithmic Gromov-Witten theory, we obtain detailed information concerning how the tangent curves degenerate along with $E$. The theorems we obtain are purely classical, with no reference to Gromov-Witten theory, but they do not appear to admit a classical proof. No prior knowledge of Gromov-Witten theory will be assumed.

algebraic geometrycombinatorics

Audience: researchers in the topic

( slides | video )

---

Online Nottingham algebraic geometry seminar

Series comments: Online geometry seminar, typically held on Thursday. This seminar takes place online via Microsoft Teams on the Nottingham University "Algebraic Geometry" team.

For recordings of past talks, copies of the speaker's slides, or to be added to the Team, please visit the seminar homepage at: kasprzyk.work/seminars/ag.html

---