Reducible Hilbert scheme of smooth curves with positive Brill-Noether number

Abstract:

In this paper we demonstrate various reducible examples of the scheme $\mathcal {I}{’ _{d,g,r}}$ of smooth curves of degee d and genus g in ${\mathbb {P}^r}$ with positive Brill-Noether number. An example of a reducible $\mathcal {I}{’ _{d,g,r}}$ with positive $\rho (d,g,r)$, namely, the example $\mathcal {I}{’ _{2g - 8,g,g - 8}},$, has been known to some people and seems to have first appeared in the literature in Eisenbud and Harris, Irreducibility of some families of linear series with Brill-Noether number $-1$, Ann. Sci. École Norm. Sup. (4) 22 (1989), 33-53. The purpose of this paper is to add a wider class of examples to the list of such reducible examples by using general k-gonal curves. We also show that $\mathcal {I}{’ _{d,g,r}}$ is irreducible for the range of $d \geq 2g - 7$ and $g - d + r \leq 0$.