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Application of Lagrange inversion to wall-crossing for Quot schemes on surfaces

Arkadij Bojko

Abstract

Motivated by my work on enumerative invariants for Quot schemes, I related two power series obtained by two different means. One of them was computed using geometric arguments via virtual localization methods and the other one came from working with representation theoretic objects called vertex algebras. In this note, I give proof of the equality of the two power series by relying only on techniques related to Lagrange inversion. This makes my work on Quot schemes independent of the previous results in the literature and proves a new combinatorial identity.

Application of Lagrange inversion to wall-crossing for Quot schemes on surfaces

Abstract

Motivated by my work on enumerative invariants for Quot schemes, I related two power series obtained by two different means. One of them was computed using geometric arguments via virtual localization methods and the other one came from working with representation theoretic objects called vertex algebras. In this note, I give proof of the equality of the two power series by relying only on techniques related to Lagrange inversion. This makes my work on Quot schemes independent of the previous results in the literature and proves a new combinatorial identity.
Paper Structure (2 sections, 4 theorems, 27 equations)

This paper contains 2 sections, 4 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Lagrange inversion for Newton--Puiseux generating series

Key Result

Theorem 1

Let $R(t)\in \mathbb{K}\llbracket t\rrbracket$ be a power series over a field $\mathbb{K}$ not involving $q$ and $R(0)\neq 0$. Suppose that $H_k(q)$ are the $e>0$ different Newton--Puiseux solutions to $H_k^e(q) = qR(H_k(q)).$ Set then the following holds:

Theorems & Definitions (8)

  • Theorem 1
  • Remark 2
  • Corollary 3: MOurl
  • Lemma 4: Gessel Gessel
  • Corollary 5
  • proof
  • proof : Formal power series proof of Theorem \ref{['thmCtC']} inspired by esg MOurl
  • proof : Analytic proof inspired by Alex Gavrilov MOurl