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On \(\gamma \)-hyperelliptic Weierstrass semigroups of genus \(6\gamma +1\) and \(6\gamma \). (English) Zbl 1388.14099
Let \(\gamma\) 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.
MSC:
14H55 Riemann surfaces; Weierstrass points; gap sequences
14H30 Coverings of curves, fundamental group
14J26 Rational and ruled surfaces
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