Surfaces in $\mathbb{P}^4$ lying on small degree hypersurfaces

TL;DR

This paper investigates smooth surfaces in projective 4-space that lie on low degree hypersurfaces, aiming to understand their properties and finiteness, especially regarding their irregularity and classification.

Contribution

It extends finiteness results and analyzes irregularity for smooth surfaces in $ extbf{P}^4$ on hypersurfaces of degree at most 5, providing new insights into their structure.

Findings

Finiteness results analogous to Hartshorne-Lichtenbaum for surfaces of general type

Constraints on irregularity of such surfaces

Enhanced understanding of surfaces lying on small degree hypersurfaces

Abstract

Since the work of Ellingsrud and Peskine at the end of 1980s, it has been known that, with the exception of a finite number of families, smooth compact complex surfaces in $\mathbb{P}^4$ with prescribed Chern classes must lie on hypersurfaces of degree $m\leq 5$. The study of surfaces lying on a small degree hypersurface in $\mathbb{P}^4$---small meaning $\leq5$---seems to be a way of obtaining empirical data leading to a better conceptual understanding of surfaces in $\mathbb{P}^4$. From this perspective, two main issues are considered in the paper: - an analogue of the Hartshorne-Lichtenbaum finiteness results for smooth surfaces of general type contained in a small degree hypersurface in $\mathbb{P}^4$, - a study of the irregularity of smooth surfaces contained in a small degree hypersurface in $\mathbb{P}^4$.