Remarks on Brill–Noether divisors and Hilbert schemes

Let be the sublocus of Mg, whose points correspond to smooth curves possessing . If the Brill–Noether number ρ(g,r,d)=−1, it is known that 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 and 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 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.