Blow-ups and the quantum spectrum of surfaces

TL;DR

This paper studies how the quantum spectrum of smooth projective surfaces changes under blow-ups, showing it asymptotically combines the spectrum of the minimal model with points related to exceptional divisors, confirming a conjecture of Kontsevich.

Contribution

It proves a conjecture of Kontsevich by describing the asymptotic behavior of the quantum spectrum under blow-ups for surfaces.

Findings

Quantum spectrum of a surface decomposes into minimal model spectrum plus points near infinity.

Additional points in the spectrum correspond bijectively to exceptional divisors.

Results hold for small parameter values, confirming conjectural behavior.

Abstract

We investigate the behaviour of the spectrum of the quantum (or Dubrovin) connection of smooth projective surfaces under blow-ups. Our main result is that for small values of the parameters, the quantum spectrum of such a surface is asymptotically the union of the quantum spectrum of a minimal model of the surface and a finite number of additional points located "close to infinity", that correspond bijectively to the exceptional divisors. This proves a conjecture of Kontsevich in the surface case.