Remarks on Brill-Noether divisors and Hilbert schemes
Let \(\mathcal {M}_g\) be the moduli space of smooth curves of genus \(g\). Set \(\rho (g,r,d):= g-(r+1)(g-d+r)\) (the Brill-Noether number). Set \(\mathcal {M}^{r}_{g,d}:= \{C\in \mathcal {M}_g:\) \(C\) has a \(g^r_k\}\). If \(\rho (g,r,d)=-1\), then \(\mathcal {M}^{r}_{g,d}\) is a divisor of \(\mathcal {M}_g\) [\textit{D. Eisenbud} and \textit{J. Harris}, Ann. Sci. Éc. Norm. Supér. (4) 22, No. 1, 33--53 (1989; Zbl 0691.14006); \textit{F. Steffen}, Invent. Math. 132, No. 1, 73--89 (1998; Zbl 0935.14018)]. Here the authors prove that if \(\rho (g,r,d) = \rho (g,s,e)=-1\) and \(r\neq s\), then \(\mathcal {M}^{r}_{g,d} \neq \mathcal {M}^{s}_{g,e}\), unless \(e=2g-2-d\) (Serre duality gives that the two divisors are the same if \(e =2g-2-d\)). This result is very useful: previously just proving that \(\mathcal {M}^{1}_{23,12}\), \(\mathcal {M}^{2}_{23,17}\) and \(\mathcal {M}^{3}_{23,20}\) are mutually distinct was a key step to prove that \(\mathcal {M}_{23}\) has Kodaira dimension \(\geq 2\) [\textit{G. Farkas}, J. Reine Angew. Math. 539, 185--200 (2001; Zbl 0994.14022)]. In the paper under review the authors also give other applications (the existence of a unique irreducible component of the Hilbert scheme of smooth curves in \(\mathbb {P}^r\) with fixed degree and genus and the non-esistence of \((d-e)\)-secant \((r-s-1)\)-plane on embeddings by the general \(g^r_d\) on a general \(C\in \mathcal {M}^{r}_{g,d}\) if \(e\neq 2g-2-d\)).
moduli space of curves, divisors on the moduli space of curves
scientific article; zbMATH DE number 5999972