Smooth Surfaces in Smooth Fourfolds with Vanishing First Chern Class

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Date
2017-06-05
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Publisher
Johns Hopkins University
Abstract
Hartshorne conjectured and Ellingsrud and Peskine proved that 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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Keywords
algebraic geometry, enumerative geometry
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