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New irreducible components of $\mathcal{B}(0,c_2)$ and Computation of the Dimension of its tangent space

Aislan Fontes, Maxwell Santos

TL;DR

The paper investigates the moduli spaces $\mathcal{B}(e,c_2)$ of stable rank-2 bundles on $\mathbb{P}^3$, combining monad techniques and Serre correspondence with explicit Ext computations to map tangent spaces and locate new components. Using Macaulay2, it computes $\dim \operatorname{Ext}^1(\mathcal{E},\mathcal{E})$ for bundles arising from minimal Horrocks monads, obtaining a concrete instance with $\dim \mathrm{Ext}^1=79$ and proving $\dim M_4=45$ for $\mathcal{B}(-1,6)$. The authors construct two infinite families, $\mathbf{V}_0$ and $\mathbf{V}_1$, of new irreducible components of $\mathcal{B}(0,c_2)$ by cohomology of minimal Horrocks monads with carefully chosen data, and they prove these components have dimensions exceeding the expected values, thus lying outside the classical Instanton, Ein, and Modified Instanton components. They further propose a general conjecture for monad families $\mathcal{M}(a,b,k)$ that yield new components for large $a$, giving evidence for $k=3,4$ and outlining a framework for higher $k$ with potential gaps in $c_2$ values. Overall, the work advances understanding of the global structure of $\mathcal{B}(e,c_2)$ by locating and characterizing new irreducible components and by providing computational tools to study tangent spaces.

Abstract

We provide a Macaulay2 code for computing the dimension of the tangent space to $\mathcal{B}(e,c_2)$ in certain cases. Using this code, we identify components of $\mathcal{B}(e,c_2)$ containing singular points and compute the dimension of the irreducible component $M_4$ of $\mathcal{B}(-1,6)$, whose existence was proved in \cite{MF2021}. Furthermore, we prove the existence of infinite families of irreducible components of $\mathcal{B}(0,c_2)$.

New irreducible components of $\mathcal{B}(0,c_2)$ and Computation of the Dimension of its tangent space

TL;DR

The paper investigates the moduli spaces of stable rank-2 bundles on , combining monad techniques and Serre correspondence with explicit Ext computations to map tangent spaces and locate new components. Using Macaulay2, it computes for bundles arising from minimal Horrocks monads, obtaining a concrete instance with and proving for . The authors construct two infinite families, and , of new irreducible components of by cohomology of minimal Horrocks monads with carefully chosen data, and they prove these components have dimensions exceeding the expected values, thus lying outside the classical Instanton, Ein, and Modified Instanton components. They further propose a general conjecture for monad families that yield new components for large , giving evidence for and outlining a framework for higher with potential gaps in values. Overall, the work advances understanding of the global structure of by locating and characterizing new irreducible components and by providing computational tools to study tangent spaces.

Abstract

We provide a Macaulay2 code for computing the dimension of the tangent space to in certain cases. Using this code, we identify components of containing singular points and compute the dimension of the irreducible component of , whose existence was proved in \cite{MF2021}. Furthermore, we prove the existence of infinite families of irreducible components of .
Paper Structure (10 sections, 7 theorems, 65 equations, 2 tables)

This paper contains 10 sections, 7 theorems, 65 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Monads
  4. The moduli space $\mathcal{B}(0,c_2)$ and its Known components
  5. Serre Correspondence
  6. Computation of $\mathop{\mathrm{Ext}}\nolimits^1$ Groups
  7. Some families
  8. A family of rank 2 bundles $\mathcal{V}(\boldsymbol{a};\boldsymbol{b})$ with $\boldsymbol{a}=({(a-1)}^2,0^2)$ and $\boldsymbol{b}=(a^2)$
  9. A family of rank 2 bundles $\mathcal{V}(\boldsymbol{a};\boldsymbol{b})$ with $\boldsymbol{a}=({(a-1)}^2,1)$ and $\boldsymbol{b}=(a^2)$
  10. Generalizations

Key Result

Theorem 3.3

Let $\mathrm{M}_4$ be the irreducible component of $\mathcal{B}(-1,6)$ containing stable rank 2 bundles on $\mathbb{P}^3$ given as cohomology of minimal monads of the form Then $\dim \mathrm{M}_4=45$.

Theorems & Definitions (19)

  • Example 2.1
  • Example 3.1
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Remark 4.1
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • proof
  • ...and 9 more