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Surfaces of general type with maximal Picard number near the Noether line

Nguyen Bin, Vicente Lorenzo

Abstract

The first published non-trivial examples of algebraic surfaces of general type with maximal Picard number are due to Persson, who constructed surfaces with maximal Picard number on the Noether line $K^2=2χ-6$ for every admissible pair $(K^2,χ)$ such that $χ\not\equiv 0 \text{ mod } 6$. In this note, given a non-negative integer $k$, algebraic surfaces of general type with maximal Picard number lying on the line $K^2=2χ-6+k$ are constructed for every admissible pair $(K^2,χ)$ such that $χ\geq 2k+10$. These constructions, obtained as bidouble covers of rational surfaces, not only allow to fill in Persson's gap on the Noether line, but they provide infinitely many new examples of algebraic surfaces of general type with maximal Picard number above the Noether line.

Surfaces of general type with maximal Picard number near the Noether line

Abstract

The first published non-trivial examples of algebraic surfaces of general type with maximal Picard number are due to Persson, who constructed surfaces with maximal Picard number on the Noether line for every admissible pair such that . In this note, given a non-negative integer , algebraic surfaces of general type with maximal Picard number lying on the line are constructed for every admissible pair such that . These constructions, obtained as bidouble covers of rational surfaces, not only allow to fill in Persson's gap on the Noether line, but they provide infinitely many new examples of algebraic surfaces of general type with maximal Picard number above the Noether line.
Paper Structure (7 sections, 13 theorems, 41 equations, 1 figure)

This paper contains 7 sections, 13 theorems, 41 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Bidouble covers
  3. Basics on the Picard number
  4. Two families of surfaces with maximal Picard number
  5. A first family of surfaces with maximal Picard number
  6. A second family of surfaces with maximal Picard number
  7. Horikawa surfaces with maximal Picard number

Key Result

Theorem 1

Given an integer $k\geq 0$ let $(K^2,\chi)$ be an admissible pair such that $K^2=2\chi-6+k$. If $\chi\geq 2k+10$, then $\mathfrak{M}_{K^2,\chi}$ contains canonical models whose minimal resolution has maximal Picard number.

Figures (1)

  • Figure 1: Plane $(K^2, \chi)$ where the pairs described in Theorem and the admissible region (i.e. the region delimited by the inequalities $K^2\geq 1, \chi\geq 1, K^2\leq 9\chi, K^2\geq 2\chi-6$) have been highlighted.

Theorems & Definitions (35)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1: MR755236 or MR1103912
  • Remark 1
  • Corollary 1
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • ...and 25 more