On \(\gamma \)-hyperelliptic Weierstrass semigroups of genus \(6\gamma +1\) and \(6\gamma \).

*(English)*Zbl 1388.14099Let \(\gamma\) be a non-negative integer. A pointed curve \((C,P)\) is called \(\gamma\)-hyperelliptic if the Weierstrass semigroup \(H(P)\) at \(P\) has exactly \(\gamma\) even gaps; here by a curve we mean a projective, non-singular, irreducible algebraic curve defined over an algebraically closed field of characteristic zero. Then the very semigroup property of \(H(P)\), see e.g. [F. Torres, Semigroup Forum 55, No. 3, 364–379 (1997; Zbl 0931.14017)], implies \(H(P)=2\tilde H\cup\{u_\gamma<\ldots<u_1\}\cup\{2g+i: i\in{\mathbb N}_0\}\), where \(g=g(C)\) is the genus of \(C\), \(\tilde H\) is a numerical semigroup of genus \(\gamma\), and the \(u_i's\) are odd integers with \(u_1<2g\); in addition \(u_\gamma\geq 2g-4\gamma+1\, (*)\).

This paper deals with the question \(D(C,P,\gamma)\): If \((C,P)\) is \(\gamma\)-hyperelliptic, do exist a double covering of curves \(F:C\to \tilde C\) which is ramified at \(P\)? If the answer is positive, the Weierstrass semigroup at \(F(P)\) equals \(\tilde H\) above so that \(g(\tilde C)=\gamma\); in particular, \(g\geq 2\gamma\) by the Riemann-Hurwitz formula. If \(\gamma\leq 3\), \(D(C,P,g,\gamma)\) is indeed true; see [J. Komeda, Semigroup Forum 83, No. 3, 479–488 (2011; Zbl 1244.14025)] and the references therein. From now we let \(\gamma\geq 4\).

If \(g(C)\geq 6\gamma+4\), \(D(C,P,\gamma)\) is true [F. Torres, Manuscr. Math. 83, No. 1, 39–58 (1994; Zbl 0838.14025)]. To see this we consider the linear system \(D_{\gamma+1}:=|(6\gamma+2)P|\) which has dimension \(2\gamma+1\) by \((*)\) above (indeed, this follows provided that \(g(C)\geq 5\gamma+1\)). Then the degree \(t\) of the morphism \(F_1: C\to {\mathbb P}^{2\gamma+1}\) associated to \(D_{\gamma+1}\) is at most \(2\). If \(t=2\), the claimed answer follows. On the contrary, Castelnuovo’s genus bound gives \(g(C)\leq \pi_0(6\gamma+2,2\gamma+1)=6\gamma+3\), a contradiction.

The present paper proves that \(D(C,P,\gamma)\) is even true whenever \(g(C)= 6\gamma+1, 6\gamma\). As a matter of fact, \(D(C,P,\gamma\) is also true for \(g(C)=6\gamma+3, 6\gamma+2\) which follow from the techniques used by the authors here.

Let \(g(C)=6\gamma+1\) and notation as above. We claim that \(t=2\). Let \(C_0:=F_1(C)\) and assume \(t=1\). Then \(g(C)=g(C_0)\leq g_a(C_0)\leq c_0(6\gamma+2, 2\gamma+1)=6\gamma+3\), where \(g_a\) is the arithmetic genus of \(C_0\). If \(g(C)=g_a(C_0)\), \(C\) is isomorphic to \(C_0\) and hence \((6\gamma+2)P\sim P+D\) with \(D\) a divisor on \(C\) such that \(P\not\in \text{supp}(D)\). Hence \(6\gamma+1\in H(P)\), a contradiction according to \((*)\) above. Now the number \(\pi_1(6\gamma+2,2\gamma+1)\) in Theorem 3.15 [J. Harris, Curves in projective space. Montreal, Quebec, Canada: Les Presses de l’Universite de Montreal (1982; Zbl 0511.14014)] equals \(6\gamma+1\); hence \(C_0\subseteq S\subseteq {\mathbb P}^{2\gamma+1}\), being \(S\) a surface of degree \(2\gamma\) [loc. cit.]. Then by considering the minimal resolution of \(S\) and the adjunction formula, \(g_a(C_0)\) can be computed. Finally the proof that \(t=1\) is a contradiction proceeds via a carefully study of the condition \(1\leq g_a(C_0)-g(C_0)\leq 2\).

The case \(g(C)=6\gamma\) is treated in a similar way; however, here the linear system \(|(6\gamma-2)P|\), which is of dimension \(2\gamma-1\), is used.

This paper deals with the question \(D(C,P,\gamma)\): If \((C,P)\) is \(\gamma\)-hyperelliptic, do exist a double covering of curves \(F:C\to \tilde C\) which is ramified at \(P\)? If the answer is positive, the Weierstrass semigroup at \(F(P)\) equals \(\tilde H\) above so that \(g(\tilde C)=\gamma\); in particular, \(g\geq 2\gamma\) by the Riemann-Hurwitz formula. If \(\gamma\leq 3\), \(D(C,P,g,\gamma)\) is indeed true; see [J. Komeda, Semigroup Forum 83, No. 3, 479–488 (2011; Zbl 1244.14025)] and the references therein. From now we let \(\gamma\geq 4\).

If \(g(C)\geq 6\gamma+4\), \(D(C,P,\gamma)\) is true [F. Torres, Manuscr. Math. 83, No. 1, 39–58 (1994; Zbl 0838.14025)]. To see this we consider the linear system \(D_{\gamma+1}:=|(6\gamma+2)P|\) which has dimension \(2\gamma+1\) by \((*)\) above (indeed, this follows provided that \(g(C)\geq 5\gamma+1\)). Then the degree \(t\) of the morphism \(F_1: C\to {\mathbb P}^{2\gamma+1}\) associated to \(D_{\gamma+1}\) is at most \(2\). If \(t=2\), the claimed answer follows. On the contrary, Castelnuovo’s genus bound gives \(g(C)\leq \pi_0(6\gamma+2,2\gamma+1)=6\gamma+3\), a contradiction.

The present paper proves that \(D(C,P,\gamma)\) is even true whenever \(g(C)= 6\gamma+1, 6\gamma\). As a matter of fact, \(D(C,P,\gamma\) is also true for \(g(C)=6\gamma+3, 6\gamma+2\) which follow from the techniques used by the authors here.

Let \(g(C)=6\gamma+1\) and notation as above. We claim that \(t=2\). Let \(C_0:=F_1(C)\) and assume \(t=1\). Then \(g(C)=g(C_0)\leq g_a(C_0)\leq c_0(6\gamma+2, 2\gamma+1)=6\gamma+3\), where \(g_a\) is the arithmetic genus of \(C_0\). If \(g(C)=g_a(C_0)\), \(C\) is isomorphic to \(C_0\) and hence \((6\gamma+2)P\sim P+D\) with \(D\) a divisor on \(C\) such that \(P\not\in \text{supp}(D)\). Hence \(6\gamma+1\in H(P)\), a contradiction according to \((*)\) above. Now the number \(\pi_1(6\gamma+2,2\gamma+1)\) in Theorem 3.15 [J. Harris, Curves in projective space. Montreal, Quebec, Canada: Les Presses de l’Universite de Montreal (1982; Zbl 0511.14014)] equals \(6\gamma+1\); hence \(C_0\subseteq S\subseteq {\mathbb P}^{2\gamma+1}\), being \(S\) a surface of degree \(2\gamma\) [loc. cit.]. Then by considering the minimal resolution of \(S\) and the adjunction formula, \(g_a(C_0)\) can be computed. Finally the proof that \(t=1\) is a contradiction proceeds via a carefully study of the condition \(1\leq g_a(C_0)-g(C_0)\leq 2\).

The case \(g(C)=6\gamma\) is treated in a similar way; however, here the linear system \(|(6\gamma-2)P|\), which is of dimension \(2\gamma-1\), is used.

Reviewer: Fernando Torres (Campinas)

##### MSC:

14H55 | Riemann surfaces; Weierstrass points; gap sequences |

14H30 | Coverings of curves, fundamental group |

14J26 | Rational and ruled surfaces |

##### Keywords:

Weierstrass semigroup; double cover of a curve; rational ruled surface Plane curves of degree 4
PDF
BibTeX
XML
Cite

\textit{J. Komeda} and \textit{A. Ohbuch}, Bull. Braz. Math. Soc. (N.S.) 48, No. 2, 209--218 (2017; Zbl 1388.14099)

Full Text:
DOI

##### References:

[1] | Arbarello, E., Cornalba, M., Griffiths, P.A., Harris, J.: Geometry of Algebraic Curves, vol. I. Springer, New York (1985) · Zbl 0559.14017 |

[2] | Coppens, M, The Weierstrass gap sequences of the total ramification points of trigonal coverings of \(\mathbb{P}^1\), Indag. Math., 47, 245-270, (1985) · Zbl 0592.14025 |

[3] | Coppens, M, The Weierstrass gap sequences of the ordinary ramification points of trigonal coverings of \(\mathbb{P}^1\): existence of a kind of Weierstrass gap sequence, J. Pure Appl. Algebra, 43, 11-25, (1986) · Zbl 0616.14012 |

[4] | Coppens, M, Weierstrass points with two prescribed nongaps, Pac. J. Math., 131, 71-104, (1988) · Zbl 0592.14018 |

[5] | Harris, J.: Curves in projective space/Joe Harris with the collaboration of David Eisenbud. Presses de L’Université de Montréal (1982) · Zbl 0511.14014 |

[6] | Kato, T; Horiuchi, R, Weierstrass gap sequences at the ramification points of trigonal Riemann surfaces, J. Pure Appl. Algebra, 50, 271-285, (1988) · Zbl 0649.14009 |

[7] | Kim, SJ, On the existence of Weierstrass gap sequences on trigonal curves, J. Pure Appl. Algebra, 63, 171-180, (1990) · Zbl 0712.14019 |

[8] | Komeda, J, Double coverings of curves and non-Weierstrass semigroups, Commun. Algebra, 41, 312-324, (2013) · Zbl 1270.14014 |

[9] | Mumford, D.: The red book of varieties and schemes. In: Lecture Notes in Mathematics, vol. 1358 (1999) · Zbl 0945.14001 |

[10] | Torres, F, Weierstrass points and double coverings of curves with application: symmetric numerical semigroups which cannot be realized as Weierstrass semigroups, Manuscripta Math., 83, 39-58, (1994) · Zbl 0838.14025 |

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.