Tetragonal curves, scrolls and 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 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]**Giorgio Bolondi and Rosa M. Miró-Roig,*Two-codimensional Buchsbaum subschemes of 𝑃ⁿ via their hyperplane sections*, Comm. Algebra**17**(1989), no. 8, 1989–2016. MR**1013479**, 10.1080/00927878908823832**[GrHa]**Phillip Griffiths and Joseph Harris,*Principles of algebraic geometry*, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. MR**507725****[Ha]**Joe Harris,*Algebraic geometry*, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. A first course. MR**1182558****[Re]**Miles Reid,*Special linear systems on curves lying on a K3 surface*, J. London Math. Soc. (2)**13**(1976), no. 3, 454–458. MR**0435083****[Sc]**Frank-Olaf Schreyer,*Syzygies of canonical curves and special linear series*, Math. Ann.**275**(1986), no. 1, 105–137. MR**849058**, 10.1007/BF01458587**[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:
http://dx.doi.org/10.1090/S0002-9947-97-01811-4

Received by editor(s):
November 4, 1994

Article copyright:
© Copyright 1997
American Mathematical Society