On surfaces of general type with q=5

TL;DR

This paper proves that complex surfaces with irregularity 5 and no irrational pencil of genus >1 must have geometric genus greater than 7, aiding classification and addressing a specific conjecture in algebraic geometry.

Contribution

It establishes a lower bound on the geometric genus for certain surfaces with irregularity 5, providing evidence towards a conjecture about their classification.

Findings

Surfaces with q=5 and no irrational pencil of genus >1 have p_g > 7.

Classifies minimal surfaces of general type with q=5 and p_g<8.

Supports the conjecture that the symmetric product of a genus three curve is unique under these conditions.

Abstract

We prove that a complex surface S with irregularity q(S)=5 that has no irrational pencil of genus >1 has geometric genus p_g(S)>7. As a consequence, one is able to classify minimal surfaces S of general type with q(S)=5 and p_g(S)<8. This result is a negative answer, for q=5, to the question asked in arXiv:0811.0390 of the existence of surfaces of general type with irregularity q>3 that have no irrational pencil of genus >1 and with the lowest possible geometric genus p_g=2q-3. This gives some evidence for the conjecture that the only irregular surface with no irrational pencil of genus >1 and p_g=2q-3 is the symmetric product of a genus three curve.