Theorem of Enriques-Petri type for a very ample invertible sheaf on a curve of genus three
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 [\textit{P. Griffiths} and \textit{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 [\textit{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 \textit{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 \(g\).
scientific article; zbMATH DE number 3803630