Remarks on Brill–Noether divisors and Hilbert schemes

AbstractLet Mg,dr be the sublocus of Mg, whose points correspond to smooth curves possessing gdr. If the Brill–Noether number ρ(g,r,d)=−1, it is known that Mg,dr is irreducible. In this paper, we prove that if g is odd, and r,s,d,e (r≠s) are positive integers satisfying ρ(g,r,d)=ρ(g,s,e)=−1 and e≠2g−2−d, then the supports of Mg,dr and Mg,es are distinct. As an application, we show that in the case d>g there is a unique irreducible component Dd,g,r of Hd,g,r dominating Mg,dr and that a general member C∈Dd,g,r has no (d−e)-secant (r−s−1)-plane for ρ(g,s,e)=−1,e≠2g−2−d