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Double coverings of curves and non-Weierstrass semigroups. (English) Zbl 1270.14014
Let \(X\) be a (non-singular, projective, irreducible, algebraic) curve defined over an algebraically closed field of characteristic zero, and let \(P\in X\). The Weierstrass semigroup \(H(P)\) of \(X\) at \(P\) is the set of poles of regular function on \(X\setminus\{P\}\). Thus \(H(P)\) is indeed a subsemigroup of the additive semigroup \(\mathbb N_0\) such that \(\#(\mathbb N_0\setminus H(P)\) equals the genus of \(X\) (The Weierstrass gap theorem).
Let \(H\) be a subsemigroup of \((\mathbb N_0,+)\) which is named numerical provided that \(G(H):=\mathbb N_0\setminus H)\) is finite; the genus of \(H\) is \(\#G(H)\). The subject matter addressed in the paper under review is related to the following question posed by Hurwitz around 1892 [A. Hurwitz, Math. Ann. XLI, 403–442 (1893; JFM 24.0380.02)]: Is any numerical semigroup \(H\) equal to the Weierstrass semigroup at some point of a curve? If this is so, \(H\) is called Weierstrass. The answer to this question is in general negative as R.-O. Buchweitz pointed out around 1980 [Lect. Notes Math. 777, 205–220 (1980; Zbl 0428.32016)]. He observed the following by considering elements of \(G(H)\). For an integer \(m\geq 2\), let \(G_m(H)\) be the set of all sums of \(m\) elements of \(G(H)\). Thus if \(H\) of genus \(g\) were Weierstrass, then \(\#G_m(H)\leq (2m-1)(g-1)\, (*)\), the dimension of a \(m\)-pluricanonical divisor of a curve of genus \(g\). In fact, Buchweitz constructed non-Weierstrass semigroups by contradicting condition \((*)\); see also [J. Komeda, Semigroup Forum 57, No. 2, 157–185 (1998; Zbl 0922.14022)]. The least genus of Buchweitz’s examples is \(g=16\). Numerical semigroups of genus at most eight are always Weierstrass; see [J. Komeda and A. Ohbuchi, Bull. Braz. Math. Soc. (N.S.) 39, No. 1, 109–121 (2008; Zbl 1133.14307)] and the references therein. Let \(\ell_g=\ell_g(H)\) be the biggest element of \(G(H)\). By the semigroup property, \(\ell_g\leq 2g-1\). If \(\ell_g\geq 2g-4\), then condition \((*)\) is always true for any numerical semigroup; cf. [G. Oliveira, Semigroup Forum 69, No. 3, 423–430 (2004; Zbl 1076.20052)] and the references therein. However, non-Weierstrass numerical semigroups with \(\ell_g\geq 2g-4\) do exist: G. Oliveira and K.-O. Stöhr [Geom. Dedicata 67, No. 1, 45–63 (1997; Zbl 0904.14018)], F. Torres [Commun. Algebra 23, No. 11, 4211–4228 (1995; Zbl 0842.14023)]; see also N. Medeiros [J. Pure Appl. Algebra 170, No. 2–3, 267–285 (2002; Zbl 1039.14015)]. The basic tool in constructing such non-Weierstrass semigroups is the use of certain covering of curves and Buchweitz’s examples as a building block (Stöhr).
On the other hand, let \(m(H)\) be the first positive element of a numerical semigroup \(H\). If \(m(H)\leq 5\), \(H\) is always Weierstrass; see [J. Komeda, Manuscripta. Math. 76, No. 2, 193–211 (1992; Zbl 0770.30038)] and the references therein. There exist non-Weierstrass semigroups \(H\) whenever \(m(H)\geq 13\) (e.g. Buchweitz, loc. cit.). In the article under review, the author constructs examples of non-Weierstrass semigroups \(H\) with \(m(H)=8\) and with \(m(H)=12\). To explain his method, for a numerical semigroup \(\tilde H\) let us consider the associated numerical semigroup \(d_2(\tilde H):=\{h/2: \text{}h\) is even

MSC:
14H55 Riemann surfaces; Weierstrass points; gap sequences
14H30 Coverings of curves, fundamental group
20M14 Commutative semigroups
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