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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Tetragonal curves, scrolls and $K3$ surfaces

Author: James N. Brawner
Journal: Trans. Amer. Math. Soc. 349 (1997), 3075-3091
MSC (1991): Primary 14J28; Secondary 14H45
MathSciNet review: 1401515
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Abstract: In this paper we establish a theorem which determines the invariants of a general hyperplane section of a rational normal scroll of arbitrary dimension. We then construct a complete intersection surface on a four-dimensional scroll and prove it is regular with a trivial dualizing sheaf. We determine the invariants for which the surface is nonsingular, and hence a $K3$ surface. A general hyperplane section of this surface is a tetragonal curve; we use the first theorem to determine for which tetragonal invariants such a construction is possible. In particular we show that for every genus $g\geq 7$ there is a tetragonal curve of genus $g$ that is a hyperplane section of a $K3$ surface. Conversely, if the tetragonal invariants are not sufficiently balanced, then the complete intersection must be singular. Finally we determine for which additional sets of invariants this construction gives a tetragonal curve as a hyperplane section of a singular canonically trivial surface, and discuss the connection with other recent results on canonically trivial surfaces.

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James N. Brawner

Received by editor(s): November 4, 1994
Article copyright: © Copyright 1997 American Mathematical Society