Du Val Curves and the Pointed Brill-noether Theorem

We show that a general curve in an explicit class of what we call Du Val pointed curves satisfies the Brill-Noether Theorem for pointed curves. Furthermore, we prove that a generic pencil of Du Val pointed curves is disjoint from all Brill-Noether divisors on the universal curve. This provides explicit examples of smooth pointed curves of arbitrary genus defined over Q which are Brill-Noether general. A similar result is proved for 2-pointed curves as well using explicit curves on elliptic ruled surfaces. The pointed Brill-Noether Theorem concerns the study of linear series on a general pointed algebraic curve [C, p] with prescribed ramification at the marked point p. Recall that for a point p ∈ C and a linear series l = (L, V ) ∈ Gd(C), one denotes by α(p) : 0 ≤ α0(p) ≤ . . . ≤ α l r(p) ≤ d− r the ramification sequence of l at p. One says that p ∈ C is a ramification point of l if αr(p) > 0. For instance, the ramification points of the canonical linear series are precisely the Weierstrass points of C. The total number of ramification points of l, counted with appropriate multiplicities, is given by the Plücker formula, see for instance [EH1] Proposition 1.1. Fixing a Schubert index α : 0 ≤ α0 ≤ . . . ≤ αr ≤ d− r, one can ask when a general pointed curve [C, p] of genus g carries a linear series l ∈ Gd(C) with ramification sequence αl(p) ≥ α. The locus Gd(C, p, α) of linear series on C satisfying this condition is a generalized determinantal variety of expected dimension ρ(g, r, d, α) := ρ(g, r, d) − w(α), where ρ(g, r, d) := g − (r + 1)(g − d + r) and w(α) := α0 + · · · + αr is the weight of α. It is proved in [EH2] Theorem 1.1 that for a general pointed curve [C, p] ∈ Mg,1, each component of Gd(C, p, α), if nonempty, has dimension precisely ρ(g, r, d, α). Moreover, [EH2] Proposition 1.2 establishes that Gd(C, p, α) 6= ∅ if and only if r ∑ i=0 max{αi + g − d+ r, 0} ≤ g. The proofs in [EH2] rely on limit linear series and degeneration to the boundary of the universal curve Cg := Mg,1. Up to now, no examples whatsoever of smooth pointed curves [C, p] ∈ Cg verifying the pointed Brill-Noether Theorem have been known. This situation contrasts the classical Brill-Noether Theorem; even though the original proof in [GH] used degeneration to nodal curves, soon afterwards, in his well-known paper [Laz], Lazarsfeld showed that sections of general polarized K3 surfaces are Brill-Noether-Petri general. 2010 Mathematics Subject Classification. 14H99 (primary), 14J26 (secondary).