A few questions about curves on surfaces

TL;DR

This paper investigates conditions under which multiples of divisors on smooth projective surfaces are effective or movable, providing examples, conjectures, and results that suggest such positivity properties are generally limited without strong hypotheses.

Contribution

It offers new insights into the positivity of divisors on surfaces, highlighting limitations and proposing conjectures about their effectivity and movability.

Findings

Examples indicating limitations of positivity properties

Discussion of conjectures related to divisor effectivity

Results suggesting strong hypotheses are necessary for positivity

Abstract

In this note we address the following kind of question: let X be a smooth, irreducible, projective surface and D a divisor on X$satisfying some sort of positivity hypothesis, then is there some multiple of D depending only on X which is effective or movable? We describe some examples, discuss some conjectures and prove some results that suggest that the answer should in general be negative, unless one puts some really strong hypotheses either on D or on X.