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Theorem of Enriques-Petri type for a very ample invertible sheaf on a curve of genus three. (English) Zbl 0509.14030
A well known theorem of Noether-Enriques-Petri says that a canonical curve \(C\) (i.e.: \(C\subseteq {\mathbb{P}}^{g-1}; g(C)=g; \omega_ C\simeq {\mathcal O}_ C(1))\) is projectively normal and is intersection of quadrics unless \(C\) is trigonal. Furthermore if C is trigonal the intersection of the quadrics through \(C\) is a rational ruled surface or the Veronese surface [P. Griffiths and J. Harris, ”Principles of algebraic geometry” (1978; Zbl 0408.14001), p. 535]. In the paper under review the author studies a similar problem for curves of genus three embedded by complete linear series. Let L be a very ample line bundle of degree d on the curve X of genus three. Denote by \(\phi_ L(X)\) the image of X in the embedding: \(\phi_ L:X\hookrightarrow {\mathbb{P}}(H^ 0(L)^{\nu})=:{\mathbb{P}}^ n.\) Note that if \(d\geq 7\) then \(\phi_ L(X)\) is projectively normal [D. Mumford, CIME \(3^ o\) Ciclo Varenna 1969, Quest. algebr. Varieties, 29-100 (1970; Zbl 0198.258)]. The cases \(d\leq 6\) have been previously studied by the author [Tsukuba J. Math. 4, 269-279 (1980; Zbl 0473.14015)]. If \(d\geq 8\) it follows from a theorem of B. Saint-Donat [C. R. Acad. Sci., Paris, Sér. A 274, 324-327 (1972; Zbl 0234.14012)] that \(\phi_ L(X)\) is intersection of quadrics. So the only case left is \(d=7\). In the first part of the paper, the author describes the scheme \(Q(\phi_ L(X))\) defined by the ideal generated by the quadrics through \(\phi_ L(X)\). The last section contains a comment on the general case. From the result of Saint-Donat quoted above it follows that if \(g(X)\geq 2\) and \(\deg(L)=2g+1,\) then the homogeneous ideal of \(\phi_ L(X)\) can be generated by its elements of degree 2 and 3. The author makes the following conjecture: ”Assume \(X\) non hyperelliptic then: (1) if \(h^ 0(L\otimes \omega_ X^{-1})=2, Q(\phi_ L(X))\) is a rational ruled surface; (2) if \(h^ 0(L\otimes \omega_ X^{-1})=1, Q(\phi_ L(X))\) is the union of \(\phi_ L(X)\) and a line: (3) if \(h^ 0(L\otimes \omega_ X^{-1})=0, Q(\phi_ L(X))=\phi_ L(X).\)” - Section 1 proves the conjecture for \(g=3\) and the paper ends with the proof of (1) for every \(g\).
Reviewer: J.Brun

14H10 Families, moduli of curves (algebraic)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14C20 Divisors, linear systems, invertible sheaves
14H45 Special algebraic curves and curves of low genus
Full Text: DOI EuDML
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