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

TL;DR

This paper constructs new algebraic surfaces of general type with maximal Picard number near the Noether line, filling gaps in known examples and expanding the range of such surfaces.

Contribution

It introduces a method to construct surfaces with maximal Picard number on lines close to the Noether line for all admissible pairs, extending previous results.

Findings

Constructed surfaces with maximal Picard number on lines $K^2=2\chi-6+k$

Filled in the gap on the Noether line for $ ext{Noether line}$

Provided infinitely many new examples above 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 $K^2=2\chi-6$ for every admissible pair $(K^2,\chi)$ such that $\chi \not\equiv 0 \text{ mod } 6$. In this note, given a non-negative integer $k$, algebraic surfaces of general type with maximal Picard number lying on the line $K^2=2\chi-6+k$ are constructed for every admissible pair $(K^2,\chi)$ such that $\chi\geq 2k+10$. 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.