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Exceptional collections on certain Hassett spaces

Ana-Maria Castravet, Jenia Tevelev

Abstract

We construct an $S_2\times S_n$ invariant full exceptional collection on Hassett spaces of weighted stable rational curves with $n+2$ markings and weights $(\frac{1}{2}+η, \frac{1}{2}+η,ε,\ldots,ε)$, for $0<ε, η\ll1$ and can be identified with symmetric GIT quotients of $(\mathbb{P}^1)^n$ by the diagonal action of $\mathbb{G}_m$ when $n$ is odd, and their Kirwan desingularization when $n$ is even. The existence of such an exceptional collection is one of the needed ingredients in order to prove the existence of a full $S_n$-invariant exceptional collection on $\overline{\mathcal{M}}_{0,n}$. To prove exceptionality we use the method of windows in derived categories. To prove fullness we use previous work on the existence of invariant full exceptional collections on Losev-Manin spaces.

Exceptional collections on certain Hassett spaces

Abstract

We construct an invariant full exceptional collection on Hassett spaces of weighted stable rational curves with markings and weights , for and can be identified with symmetric GIT quotients of by the diagonal action of when is odd, and their Kirwan desingularization when is even. The existence of such an exceptional collection is one of the needed ingredients in order to prove the existence of a full -invariant exceptional collection on . To prove exceptionality we use the method of windows in derived categories. To prove fullness we use previous work on the existence of invariant full exceptional collections on Losev-Manin spaces.

Paper Structure

This paper contains 12 sections, 38 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Hassett spaces
  3. Proof of Theorem \ref{['odd']}
  4. Generalities on GIT quotients $(\mathbb{P}^1)^n_{ss}\mathbin{/\mkern-6mu/}\mathbb{G}_m$
  5. The space $Z_N$ as a GIT quotient when $n$ is odd
  6. Exceptionality
  7. Fullness
  8. Proof of Theorem \ref{['even']}
  9. The space $Z_N$ as a GIT quotient, $n$ even
  10. Exceptionality
  11. Fullness
  12. Pushforward of the exceptional collection on the Losev--Manin space $\overline{\text{LM}}_N$ to $Z_N$

Key Result

Theorem 1.2

When $n$ is odd, the space $Z_n$ is isomorphic to the symmetric GIT quotient $Z_n=(\mathbb{P}^1)^n\mathbin{/\mkern-6mu/}_{\mathcal{O}(1,\ldots,1)}\mathbb{G}_m$, with respect to the diagonal action of $\mathbb{G}_m$ on $(\mathbb{P}^1)^n$, coming from $\mathbb{G}_m$ acting on $\mathbb{P}^1$ by $z\cdot

Theorems & Definitions (82)

  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6
  • Definition 1.7
  • Theorem 1.8
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • ...and 72 more