Curves of genus \(g\) whose canonical model lies on a surface of degree \(g+1\). (English) Zbl 1260.14067
Let \(C\subset \mathbb {P}^{g-1}\) be a canonically embedded smooth curve of genus \(g\). Let \(\rho\) be the minimal degree of a surface \(S\subset \mathbb {P}^{g-1}\) containing \(C\). A classical problem is the classification of all \(C\) with low \(\rho\). The case \(\rho \leq g\) was known ([E. Ballico, G. Casnati and R. Notari, J. Algebra 332, No. 1, 229–243 (2011; Zbl 1242.14030); G. Casnati, Proc. Am. Math. Soc. 140, No. 4, 1185–1197 (2012; Zbl 1258.14065)]).
The paper under review gives the complete classification of all \(C\) with \(\rho =g+1\): either \(7 \leq g \leq 15\) and \(C\) is birationally isomorphic to a plane septic with \(15-g\) ordinary (possibly infinitely near) singular point (here \(S\) comes from \(\mathbb {P}^2\)) or \(g\geq 9\) and \(C\) has a unique \(g^1_4\) (here \(S\) is a conic bundle; if \(g\geq 16\) the \(g^1_4\) is composed with an involution onto a curve of genus \(2\)). At the end of the introduction the author announces generalizations due to I. Coşkun, which have now been published [“Surfaces of low degree containing a canonical curve”, in: Computational algebraic and analytic geometry. Contemp. Math. 572, 57–70 (2012; doi:10.1099/conm572)].
The paper under review gives the complete classification of all \(C\) with \(\rho =g+1\): either \(7 \leq g \leq 15\) and \(C\) is birationally isomorphic to a plane septic with \(15-g\) ordinary (possibly infinitely near) singular point (here \(S\) comes from \(\mathbb {P}^2\)) or \(g\geq 9\) and \(C\) has a unique \(g^1_4\) (here \(S\) is a conic bundle; if \(g\geq 16\) the \(g^1_4\) is composed with an involution onto a curve of genus \(2\)). At the end of the introduction the author announces generalizations due to I. Coşkun, which have now been published [“Surfaces of low degree containing a canonical curve”, in: Computational algebraic and analytic geometry. Contemp. Math. 572, 57–70 (2012; doi:10.1099/conm572)].
Reviewer: Edoardo Ballico (Povo)
MSC:
| 14N25 | Varieties of low degree |
| 14H51 | Special divisors on curves (gonality, Brill-Noether theory) |
| 14H30 | Coverings of curves, fundamental group |
| 14N05 | Projective techniques in algebraic geometry |
References:
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