General curves on algebraic surfaces

TL;DR

This paper establishes upper bounds on the genus of curves with general moduli embedded in various types of algebraic surfaces, depending on the surface's Kodaira dimension and other invariants.

Contribution

It provides new genus bounds for curves on algebraic surfaces of intermediate Kodaira dimension and surfaces of general type under specific conditions.

Findings

Bound g ≤ 19 for curves on certain general type surfaces.

Lower bounds for genus on other surface types.

Applicable to surfaces with Castelnuovo inequality constraints.

Abstract

We give upper bounds on the genus of a curve with general moduli assuming that it can be embedded in a projective nonsingular surface $Y$ so that $\dim(|C|) > 0$. We find such bounds for all types of surfaces of intermediate Kodaira dimension and, under mild restrictions, for surfaces of general type whose minimal model $Z$ satisfies the Castelnuovo inequality $K_Z^2 \ge 3\chi(\O_Z) - 10$. In this last case we obtain $g \le 19$. In the other cases considered the bounds are lower.