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

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 $m\le 5$. The study of surfaces lying on a small degree hypersurface in ${\mathbb{P}}^4$—small meaning $\le 5$—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 ${\mathbb{P}}^4$.