Theorem of Enriques-Petri type for a very ample invertible sheaf on a curve of genus three.

*(English)*Zbl 0509.14030A 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

##### MSC:

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 |

##### Keywords:

curve of genus three; complete linear system; intersection of quadrics; invertible sheaf; hyperelliptic curve##### References:

[1] | Arbarello, E., Sernesi, E.: Petri’s approach to the study of the ideal associated to a special divisor. Invent. Math.49, 99-119 (1978) · Zbl 0399.14019 · doi:10.1007/BF01403081 |

[2] | Hartshorne, R.: Algebraic geometry. Graduate Text in Math.52, Berlin-Heidelberg-New York: Springer 1977 · Zbl 0367.14001 |

[3] | Homma, M.: On projective normality and defining equations of a projective curve of genus three embedded by a complete linear system. Tsukuba J. Math.4, 269-279 (1980). · Zbl 0473.14015 |

[4] | Homma, Y.: Projective normality and the defining equations of ample invertible sheaves on elliptic ruled surfaces withe?0. Natur. Sci. Rep. Ochanomizu Univ.31, 61-73 (1980) · Zbl 0486.14001 |

[5] | Mumford, D.: Varieties defined by quadratic equations. In: Questions on Algebraic Varieties. Corso C.I.M.E. (Varenna 1969). Roma: Cremonese 1970 · Zbl 0169.23301 |

[6] | Saint-Donat, B.: Sur les équations definissant une courbe algébrique. C. R. Acad. Sci. Paris Sér. A274, 324-327 (1972) · Zbl 0234.14012 |

[7] | Saint-Donat, B.: On Petri’s analysis of the linear system of quadrics through a canonical curve. Math. Ann.206, 157-175 (1973) · Zbl 0315.14010 · doi:10.1007/BF01430982 |

[8] | Saint-Donat, B.: Projective model of K-3 surfaces. Amer. J. Math.96, 602-639 (1974) · Zbl 0301.14011 · doi:10.2307/2373709 |

[9] | Sokurov, V. V.: The Noether-Enriques theorem on canonical curves. Math. USSR-Sb.15, 361-403 (1971) · Zbl 0249.14009 · doi:10.1070/SM1971v015n03ABEH001552 |

[10] | X.X.X.: Correspondence. Amer. J. Math.79, 951-952 (1957) · Zbl 0079.15001 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.