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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Degenerations of $ K3$ surfaces of degree $ 4$


Author: Jayant Shah
Journal: Trans. Amer. Math. Soc. 263 (1981), 271-308
MSC: Primary 14J25; Secondary 14J10, 14J17
MathSciNet review: 594410
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Abstract: A generic $ K3$ surface of degree $ 4$ may be embedded as a nonsingular quartic surface in $ {{\mathbf{P}}_3}$. Let $ f:X \to \operatorname{Spec} \;{\mathbf{C}}[[t]]$ be a family of quartic surfaces such that the generic fiber is regular. Let $ {\Sigma _0}$, $ {\Sigma _2^0}$, $ {\Sigma _4}$ be respectively a nonsingular quadric in $ {{\mathbf{P}}_3}$, a cone in $ {{\mathbf{P}}_3}$ over a nonsingular conic and a rational, ruled surface in $ {{\mathbf{P}}_9}$ which has a section with self intersection $ - 4$. We show that there exists a flat, projective morphism $ f':X' \to {\text{Spec}}\;{\mathbf{C}}[[t]]$ and a map $ \rho :{\text{Spec}}\:{\mathbf{C}}[[t]] \to {\text{Spec}}\:{\mathbf{C}}[[t]]$ such that (i) the generic fiber of $ f'$ and the generic fiber of the pull-back of $ f$ via $ \rho $ are isomorphic, (ii) the fiber $ {X'_0}$ of $ f'$ over the closed point of $ {\text{Spec}}\;{\mathbf{C}}[[t]]$ has only insignificant limit singularities and (iii) $ {X'_0}$ is either a quadric surface or a double cover of $ {\Sigma _0}$, $ {\Sigma _2^0}$ or $ {\Sigma _4}$. The theorem is proved using the geometric invariant theory.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1981-0594410-2
PII: S 0002-9947(1981)0594410-2
Keywords: $ K3$ surfaces, degeneration of surfaces, compactification of moduli, geometric invariant theory, insignificant limit singularities
Article copyright: © Copyright 1981 American Mathematical Society