On the geometric genus of subvarieties of generic hypersurfaces
Herbert Clemens, Ziv Ran
TL;DR
This paper establishes lower bounds on the canonical bundle twists of subvarieties in generic hypersurfaces, showing that certain low-genus curves do not exist in generic sextic threefolds.
Contribution
It provides new lower bounds on the canonical bundle of subvarieties, leading to nonexistence results for rational, elliptic, and genus 2 curves in generic sextic threefolds.
Findings
No rational or elliptic curves in generic sextic threefolds
No nondegenerate genus 2 curves in generic sextic threefolds
Lower bounds on canonical bundle twists of subvarieties
Abstract
We prove some lower bounds on certain nonegative twists of the canonical bundle of a subvariety of a generic hypersurface in projective space. In particular we prove that the generic sextic threefold contains no rational or elliptic curves and no nondegenerate curves of genus 2.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Geometric Analysis and Curvature Flows · Advanced Algebra and Geometry