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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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##### References:
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