Linear series on a stable curve of compact type and relations among Brill-Noether loci

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

“…Along the line, the focus of this work lies on finding conditions on t for a TCBE(g 1 , g 2 ; 2, t) curve not to admit a g r d . This work combined with the existence conditions given in [10] will enable us to see some relations among Brill-Noether loci in M g or M g . More precisely, we describe differences and intersections among Brill-Noether loci by using the family of TCBE(g 1 , g 2 ; 2, t) curves in M g , whose dimension equals 3g − 8.…”

“…Here δ g 1 ,g 2 denotes the Kronecker delta. This result is a kind of counter part of Theorem 1.1 in [10] for the case n = 2: under the hypotheses ρ(g, r, d) = −2 + h and g 1 − h ≥ g 2 ≥ 2 with h = 0, 1, we get…”

“…In this section we investigate necessary conditions on t for a TCBE(g 1 , g 2 ; 2, t) curve to admit a limit linear series g r d with ρ(g, r, d) < 0, whereas sufficient conditions for the existence of a smoothable limit g r d with ρ(g, r, d) = −1, −2 were given in Theorem 1.1 in [10]. 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].…”

“…More generally, a TCBE(g 1 , g 2 ; n, t) curve is defined in [10] by the same way as a TCBE(g 1 , g 2 ; 2, t) curve except that the chain of elliptic curves has length n.…”

“…The second author [10] gave conditions on t in terms of g, r, d under which a TCBE(g 1 , g 2 ; n, t) curve in M g carries a smoothable limit linear series g r d . Along the line, the focus of this work lies on finding conditions on t for a TCBE(g 1 , g 2 ; 2, t) curve not to admit a g r d .…”