Remarks on Brill–Noether divisors and Hilbert schemes

Linear series on a curve of compact type bridged by a chain of elliptic curves

“…This cannot occur. Hence we have g 1 ≥ g 2 + h. Therefore we get the result (1). If ρ = −2 and r ≥ 2, the results ( 1) and ( 2) tell that the inequality t ≤ g − d + 2r − 2 becomes a necessary and sufficient condition on t for a TCBE(⌈ g−2 2 ⌉, ⌊ g−2 2 ⌋; 2, t) curve with t ≥ r + 4 to carry a smoothable limit g r d .…”

“…Our result combined with Theorem 1.1 [10] gives rise to some relations among Brill-Noether loci corresponding to ρ = −1, −2 in the moduli space M g . In particular, it will be shown that Brill-Noether loci of codimension two have mutually distinct supports in M g as in the case of Brill-Noether divisors [1,2].…”

“…Relationships among Brill-Noether divisors inside M g are useful for dealing with the birational geometry of M g [5,7,8]. It was known that any two Brill-Noether divisors have mutually distinct supports in M g [1,2]. On the one hand, every component of M r g,d has codimension at most −ρ(g, r, d) in M g ; further, in case −3 ≤ ρ(g, r, d) ≤ −1, each component of M r g,d is of codimension −ρ(g, r, d) [3,6,11].…”