Effective cycles on universal hypersurfaces

TL;DR

This paper investigates the structure of effective cones of cycles on universal hypersurfaces, providing explicit descriptions for certain cases such as universal conics and plane curves, advancing understanding in algebraic geometry.

Contribution

It explicitly determines the effective cones of cycles on universal hypersurfaces in specific cases, including universal conics and plane curves, which was previously unknown.

Findings

Effective cone of cycles on universal conic over P^2 determined

Effective cones of cycles of dimension ≤6 or =10 on universal plane curves characterized

Provides new insights into the geometry of universal hypersurfaces

Abstract

We study the effective cones of cycles on universal hypersurfaces on a projective variety $X$, particularly focusing on the case of universal hypersurfaces in $\mathbb{P}^n$. We determine the effective cones of cycles on the universal conic over $\mathbb{P}^2$. We also determine every effective cone of cycles of dimension at most 6 or equal to 10 on any universal plane curve.