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A counterexample to the parity conjecture

Franco Giovenzana, Luca Giovenzana, Michele Graffeo, Paolo Lella

Abstract

Let $[Z]\in\text{Hilb}^d \mathbb A^3$ be a zero-dimensional subscheme of the affine three-dimensional complex space of length $d>0$. Okounkov and Pandharipande have conjectured that the dimension of the tangent space of $\text{Hilb}^d \mathbb A^3$ at $[Z]$ and $d$ have the same parity. The conjecture was proven by Maulik, Nekrasov, Okounkov and Pandharipande for points $[Z]$ defined by monomial ideals and very recently by Ramkumar and Sammartano for homogeneous ideals. In this paper we exhibit a family of zero-dimensional schemes in $\text{Hilb}^{12} \mathbb A^3$, which disproves the conjecture in the general non-homogeneous case.

A counterexample to the parity conjecture

Abstract

Let be a zero-dimensional subscheme of the affine three-dimensional complex space of length . Okounkov and Pandharipande have conjectured that the dimension of the tangent space of at and have the same parity. The conjecture was proven by Maulik, Nekrasov, Okounkov and Pandharipande for points defined by monomial ideals and very recently by Ramkumar and Sammartano for homogeneous ideals. In this paper we exhibit a family of zero-dimensional schemes in , which disproves the conjecture in the general non-homogeneous case.
Paper Structure (8 sections, 9 theorems, 57 equations, 1 figure)

This paper contains 8 sections, 9 theorems, 57 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Comments
  5. Explicit description of the locus of counterexamples
  6. Computational methods: marked bases
  7. Computational details of the proof of Theorem \ref{['thmA']}
  8. Computational details of the proof of Proposition \ref{['prop:smoothability']}

Key Result

Theorem 1

The parity conjecture is false for any $d\geqslant 12$.

Figures (1)

  • Figure 1: A degeneration of four fat points leading to a counterexample to the parity conjecture.

Theorems & Definitions (25)

  • Conjecture 1: Parity conjecture CONGETTURA
  • Theorem : Corollary \ref{['cor:conjecture']}
  • Definition 2.1
  • Theorem 2.2: FGAexplained
  • Definition 2.3: BertoneCioffiRoggero
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • Remark 3.3: cf. SAMMARTANO
  • ...and 15 more