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Singular rational curves with points of nearly-maximal weight. (English) Zbl 1394.14019
In this review, for the sake of simplicity, a curve \(C\) is a rational integral and projective one-dimensional \(\mathbb C\)-scheme of arithmetic genus \(g\), with a unique singular point \(P\), which is unibranch; the authors consider sometimes a larger class of curves. The integral closure of \(\mathcal O_{C,P}\) is a discrete valuation ring \(V\). Let \(v\) be the valuation defined by \(V\); \(S(P):=v(\mathcal O_{C,P})\) is the value semigroup of \(P\). Instead of using the usual notion of weight \(w(P)\) of \(P\), namely the weight of the numerical semigroup \(S(P)\), the authors define a weight \(w(P)\) of \(P\), using pole orders of differentials, similarly as S. Kato [Math. Ann. 239, 141–147 (1979; Zbl 0401.30037)]. For a numerical semigroup, F. Torres [Manuscr. Math. 83, 39–58 (1994; Zbl 0838.14025); Semigroup forum 55, 364–379 (1997; Zbl 0931.14017] introduced the notion of being \(\kappa\)-hyperelliptic; in this paper, the authors say that \(P\) is hyperelliptic if \(S(P)\) is \(0\)-hyperelliptic, i.e., \(2\in S(P)\), and that \(P\) is bielliptic if \(P\) is 1-hyperelliptic, i.e., \(4\) and \(6\) are the smallest integers in \(S(P)\). In Theorem 2, resp. Theorem 4, the case that \(P\) is hyperelliptic resp. bielliptic is characterized by a condition on the weight \(w(P)\). Section 2 starts with a review of linear series on \(C\). \(g^r_d\) denotes a linear series of degree \(d\) and dimension \(r\); \(\text{gon}(C)\) is the gonality of \(C\), i.e., the smallest \(k\) for which \(C\) carries a \(g^1_k\). In Theorem 2.2 it is shown that \(\text{gon}(C)\leq g+1\). For a particular class of curves \(C\), it is shown that \(C\) carries a \(g^1_k\) with a non-removable base-point and that \(C\) lies on a scroll. The curve \(C\) is called hyperelliptic whenever a degree-\(2 \) morphisms \(C\to\mathbb P^1\) exists, and it is called bielliptic whenever a degree-\(2\) morphism \(C\to E\) exists where \(E\) is an elliptic curve. In section 3 the authors study hyperelliptic and bielliptic curves. They obtain a partial classification of these curves.

14H20 Singularities of curves, local rings
14H45 Special algebraic curves and curves of low genus
14H51 Special divisors on curves (gonality, Brill-Noether theory)
20M20 Semigroups of transformations, relations, partitions, etc.
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