On the gonality sequence of smooth curves
A smooth curve \(X\) is said to be \(d\)-gonal if \(d\) is the minimum degree of a rational map \(X \longrightarrow \mathbb P^1\), i.e. there exists a \(g^1_d\) on \(X\) but not a \(g^1_{d-1}\). The notion of gonality has been generalized in recent years. Essentially, the invariant \(d_r\) of \(X\) is defined to be the minimum degree of a rational map \(X \longrightarrow \mathbb P^r\) (here ``degree'' stands for the product of the usual degree of a map and the degree of the image of \(X\) in \(\mathbb P^r\)). Clearly, \(d_1 = d\) is the usual gonality and \((d_1, d_2, d_3, \dots)\) is called the \textit{gonality sequence} of \(X\). Let us observe that its interesting elements are the initial ones, up to \(d_{g-1}\), and recall that the sequence is known for general curves of genus \(g\). More, it plays an important role in Brill--Noether Theory. The present paper deals with the so called ``slope inequality'' (introduced by \textit{H. Lange} and \textit{G. Martens} [Manuscr. Math. 137, No. 3--4, 457--473 (2012; Zbl 1238.14022)] in a wide study of this subject): \[ {{d_r}\over{r}} \; \geq \; {{d_{r+1}}\over{r+1}} \] which is satisfied in ``most'' cases. In [loc. cit.], several violations of the inequality are described, mainly concerning extremal curves, and a question is posed (see 5.4): Is there a number \(g_0 \geq 9\) such that for every integer \(g \geq g_0\) there is a curve of genus \(g\) not satisfying all slope inequalities? The present paper answers to this proving that (see Corollary 1.1) for each integer \(g \geq 31\) there is a smooth curve \(C\) of genus \(g\) with \[ {{d_2}\over{2}} \; < \; {{d_{3}}\over{3}} . \] The techniques here used regard nodal plane curves. More precisely, the invariants \(d_1, d_2, d_3\) of the normalization of such a curve are computed (or bounded) and, with a skilful computation, the result is proved. This study ends with an interesting question, which specializes the one originating this paper and whose answer in the case \(r=2\) is the cited main result (1.1): Fix an integer \(r \geq 3\). Is there an integer \(g_r\) such that for every \(g\geq g_r\) there is a smooth curve \(C\) with \[ {{d_r}\over{r}} \; < \; {{d_{r+1}}\over{r+1}} \quad ? \]
scientific article; zbMATH DE number 6065341