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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.

##### MSC:
 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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