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Moduli of surfaces in
${{\mathbb {P}}}^{3}$
- Part of
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- Journal:
- Compositio Mathematica / Volume 158 / Issue 6 / June 2022
- Published online by Cambridge University Press:
- 22 August 2022, pp. 1329-1374
- Print publication:
- June 2022
-
- Article
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The main goal of this paper is to construct a compactification of the moduli space of degree
$d \geqslant 5$ surfaces in 
$\mathbb {P}^{3}_{{{\mathbb {C}}}}$, i.e. a parameter space whose interior points correspond to (equivalence classes of) smooth surfaces in 
$\mathbb {P}^{3}$ and whose boundary points correspond to degenerations of such surfaces. We consider a divisor 
$D$ on a Fano variety 
$Z$ as a pair 
$(Z, D)$ satisfying certain properties. We find a modular compactification of such pairs and, in the case of 
$Z = {{\mathbb {P}}}^{3}$ and 
$D$ a surface, use their properties to classify the pairs on the boundary of the moduli space.
