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On the volume of some Fano K-moduli spaces

Salvatore Tambasco

Abstract

We compute the CM volume, that is the degree of the descended CM line bundle, of the Fano K-moduli space of Quartic del Pezzo in any dimension, and of the K-moduli space of the log Fano hyperplane arrangements of dimension one and two. Furthermore, we relate these volumes to the Weil-Petersson volumes by extending the notion of Weil-Petersson metric in the log case.

On the volume of some Fano K-moduli spaces

Abstract

We compute the CM volume, that is the degree of the descended CM line bundle, of the Fano K-moduli space of Quartic del Pezzo in any dimension, and of the K-moduli space of the log Fano hyperplane arrangements of dimension one and two. Furthermore, we relate these volumes to the Weil-Petersson volumes by extending the notion of Weil-Petersson metric in the log case.

Paper Structure

This paper contains 14 sections, 21 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. The CM line bundle and the log Weil-Petersson metric
  3. On the log Weil-Petersson metric.
  4. The CM Volume of quartic del Pezzo moduli space
  5. The moduli space of the intersection of Quadrics
  6. The intersection number of the CM line bundle with a smooth curve in the moduli of quartic del Pezzo's
  7. The CM volume of the moduli space of quartic del Pezzo
  8. The CM Volume of log Fano hyperplane arrangements moduli space
  9. Some generalities on the log Fano hyperplane arrangements and their moduli.
  10. The intersection number of the log CM line bundle
  11. The volume of the moduli space of weighted hyperplane arrangements
  12. The dimension one case
  13. The dimension two case
  14. Final Remarks

Key Result

Theorem 1.1

The CM volume of the $K$-moduli space of quartic del Pezzo quartic of dimension $n,$$M_{\mathrm{dP}^4},$ is given by where $m=n+3$ and $c =8(n+1)(n-1)^n - \sum_{i=1}^n \binom{n+1}{i} (n+1)^{n+1-i}n^i(i-1)2^{i+1}.$

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.1
  • Corollary 2.1
  • Remark 2.1
  • Definition 2.2
  • Theorem 2.1
  • Theorem 2.2
  • Proposition 3.1
  • ...and 19 more