Tetragonal curves, scrolls and surfaces

Author:
James N. Brawner

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

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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 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 there is a tetragonal curve of genus that is a hyperplane section of a 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.

**[Br]**J.N. Brawner,*The Gaussian map for curves with special linear series*, Ph.D. dissertation (Univ. of N. Carolina, Chapel Hill), 1992.**[DoMo]**R. Donagi and D. Morrison,*Linear systems on**-sections*, J. Differential Geometry**29**(1989) 49-64. MR**90m:14046****[GrHa]**P. Griffiths and J. Harris,*Principles of Algebraic Geometry*, Wiley, New York, 1978. MR**80b:14001****[Ha]**J. Harris,*Algebraic Geometry: a First Course*, Graduate Texts in Mathematics, Springer-Verlag, 1992. MR**93j:14001****[Re]**M. Reid,*Special linear systems on curves lying on a**surface*, J. London Math. Soc. (2)**13**(1976) 454-458. MR**55:8045****[Sc]**F.-O. Schreyer,*Syzygies of canonical curves and special linear series*, Math. Ann.**275**(1986), 105-137. MR**87j:14052****[Wa]**J.M. Wahl,*Curves on canonically trivial surfaces*, to appear.

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Additional Information

**James N. Brawner**

Email:
brawnerj@sjuvm.stjohns.edu

DOI:
https://doi.org/10.1090/S0002-9947-97-01811-4

Received by editor(s):
November 4, 1994

Article copyright:
© Copyright 1997
American Mathematical Society