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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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##### References:
 [1] Arbarello, E.; Cornalba, M.; Griffiths, P. A.; Harris, J., Geometry of algebraic curves, (1985), Springer · Zbl 0559.14017 [2] Barucci, V.; D’Anna, M.; Fröberg, R., Analytically unramified one-dimensional semi local rings and their value semigroups, J. Pure Appl. Algebra, 147, 215-254, (2000) · Zbl 0963.13021 [3] Barucci, V.; Fröberg, R., One-dimensional almost Gorenstein rings, J. Algebra, 188, 2, 418-442, (1997) · Zbl 0874.13018 [4] Bras-Amorós, M.; de Mier, A., Representation of numerical semigroups by Dyck paths, Semigroup Forum, 75, 3, 676-681, (2007) · Zbl 1128.20046 [5] Bras-Amorós, M.; Bulygin, S., Towards a better understanding of the semigroup tree, Semigroup Forum, 79, 3, 561-574, (2009) · Zbl 1230.05018 [6] Contiero, A.; Feital, L.; Martins, R. V., MAX Noether’s theorem for integral curves · Zbl 1386.14112 [7] Coppens, M., Free linear systems on integral Gorenstein curves, J. Algebra, 145, 209-218, (1992) · Zbl 0770.14002 [8] Cox, D.; Kustin, A.; Polini, C.; Ulrich, B., A study of singularities on rational curves via syzygies, Mem. Am. Math. Soc., 222, 1045, (2013) · Zbl 1305.14014 [9] Eisenbud, D.; Koh, J.; Stillman, M., Determinantal equations for curves of high degree, Am. J. Math., 110, 3, 513-539, (1988) · Zbl 0681.14027 [10] Fernández de Bobadilla, J.; Luengo, I.; Melle-Hernández, A.; Némethi, A., On rational cuspidal plane curves, open surfaces, and local singularities, (Singularity Theory, (2007), World Sci. Publ.), 411-442 · Zbl 1124.14026 [11] Feital, L., Gonalidade e o teorema de MAX Noether para curvas não-Gorenstein, (2013), Univ. Federal de Minas Gerais, viewable at [12] Feital, L.; Martins, R. V., Gonality of non-Gorenstein curves of genus five, Bull. Braz. Math. Soc., 45, 4, 649-670, (2014) · Zbl 1308.14029 [13] Flenner, H.; Zaidenberg, M., Rational cuspidal plane curves of type $$(d, d - 3)$$, Math. Nachr., 210, 93-110, (2000) · Zbl 0948.14020 [14] Kato, T., Non-hyperelliptic Weierstrass points of maximal weight, Math. Ann., 239, 141-147, (1979) · Zbl 0401.30037 [15] Koras, M.; Palka, K., The coolidge-Nagata conjecture · Zbl 1393.14029 [16] Martins, R. V., On trigonal non-Gorenstein curves with zero maroni invariant, J. Algebra, 275, 453-470, (2004) · Zbl 1060.14036 [17] Kleiman, S.; Martins, R. V., The canonical model of a singular curve, Geom. Dedic., 139, 139-166, (2009) · Zbl 1172.14019 [18] Moe, T. Karoline, Rational cuspidal curves, (2008), Univ. Oslo, master’s thesis [19] Orevkov, S. Y., On rational cuspidal curves I. sharp estimate for degree via multiplicities, Math. Ann., 324, 4, 657-673, (2002) · Zbl 1014.14010 [20] Oliveira, G.; Torres, F.; Villanueva, J., On the weight of numerical semigroups, J. Pure Appl. Algebra, 214, 1955-1961, (2010) · Zbl 1194.14048 [21] Piontkowski, J., On the number of cusps of rational cuspidal plane curves, Exp. Math., 16, 2, 251-255, (2007) · Zbl 1147.14011 [22] Rosa, R.; Stöhr, K.-O., Trigonal Gorenstein curves, J. Pure Appl. Algebra, 174, 187-205, (2002) · Zbl 1059.14038 [23] Rosenlicht, M., Equivalence relations on algebraic curves, Ann. Math., 56, 169-191, (1952) · Zbl 0047.14503 [24] Schreyer, F.-O., Syzygies of canonical curves and special linear systems, Math. Ann., 275, 105-137, (1986) · Zbl 0578.14002 [25] Stöhr, K.-O., On the poles of regular differentials of singular curves, Bull. Braz. Math. Soc., 24, 105-135, (1993) · Zbl 0788.14020 [26] Tono, K., On the number of cusps of cuspidal plane curves, Math. Nachr., 278, 1-2, 216-221, (2005) · Zbl 1069.14029 [27] Torres, F., Weierstrass points and double coverings of curves with application: symmetric numerical semigroups which cannot be realized as Weierstrass semigroups, Manuscr. Math., 83, 39-58, (1994) · Zbl 0838.14025 [28] Torres, F., On γ-hyperelliptic numerical semigroups, Semigroup Forum, 55, 364-379, (1997) · Zbl 0931.14017
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