Scrolls and hyperbolicity

TL;DR

This paper uses degeneration techniques to scrolls to prove non-existence and boundedness results for low genus curves on general surfaces, constructs hyperbolic surfaces in P3, and establishes new genus bounds for hypersurfaces.

Contribution

It introduces a novel degeneration approach to scrolls for analyzing curves on surfaces and constructs new examples of hyperbolic surfaces in P3.

Findings

Non-existence of low genus curves on general degree d >= 5 surfaces in P3.

Boundedness of families of small genus curves on these surfaces.

Existence of Kobayashi hyperbolic surfaces in P3 of degree 7 and constructions at degree 6.

Abstract

Using degeneration to scrolls, we give an easy proof of non-existence of curves of low genera on general surfaces in P3 of degree d >=5. We show, along the same lines, boundedness of families of curves of small enough genera on general surfaces in P3. We also show that there exist Kobayashi hyperbolic surfaces in P3 of degree d = 7 (a result so far unknown), and give a new construction of such surfaces of degree d = 6. Finally we provide some new lower bounds for geometric genera of surfaces lying on general hypersurfaces of degree 3d > 15 in P4.