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arXiv · 1610.04266

Smooth Surfaces in Smooth Fourfolds with Vanishing First Chern Class

Benjamin Diamond
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Abstract

According to a conjecture attributed to Hartshorne and Lichtenbaum and proven by Ellingsrud and Peskine, the smooth rational surfaces in $\mathbb{P}^4$ belong to only finitely many families. We formulate and study a collection of analogous problems in which $\mathbb{P}^4$ is replaced by a smooth fourfold $X$ with vanishing first integral Chern class. We embed such $X$ into a smooth ambient variety and count families of smooth surfaces which arise in $X$ from the ambient variety. We obtain various finiteness results in such settings. The central technique is the introduction of a new numerical invariant for smooth surfaces in smooth fourfolds with vanishing first Chern class.

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Benjamin Diamond. 2018-01-24. Smooth Surfaces in Smooth Fourfolds with Vanishing First Chern Class. https://doi.org/10.1016/j.jpaa.2017.06.013

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