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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Tetragonal curves, scrolls and $K3$ surfaces
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by James N. Brawner PDF
Trans. Amer. Math. Soc. 349 (1997), 3075-3091 Request permission

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.
References
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Additional Information
  • James N. Brawner
  • Email: brawnerj@sjuvm.stjohns.edu
  • Received by editor(s): November 4, 1994
  • © Copyright 1997 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 349 (1997), 3075-3091
  • MSC (1991): Primary 14J28; Secondary 14H45
  • DOI: https://doi.org/10.1090/S0002-9947-97-01811-4
  • MathSciNet review: 1401515