$k$-canonical divisors through Brill-Noether special points

Abstract

Inside the projectivized k-th Hodge bundle, we construct a collection of divisors obtained by imposing vanishing at a Brill-Noether special point. We compute the classes of the closures of such divisors in two ways, using incidence geometry and restrictions to various families, including pencils of curves on K3 surfaces and pencils of Du Val curves. We also show the extremality and rigidity of the closure of the incidence divisor consisting of smooth pointed curves together with a canonical or 2-canonical divisor passing through the marked point. MSC2020. 14H10, 14H51, 14C25 (primary), 30F30 (secondary).

Effective divisors on projectivized Hodge bundles and modular Forms

“…This effective divisor class can also be given by the cycle as given by Gheorghita in [12]. Here, we get a section of Sym 24 (∧ 2 𝔼) ⊗ det(𝔼) 44 (−6 𝛿 0 − 12 𝛿 1 ).…”

“…Via the projection 𝑝 8,0 ∶ Sym 4 (Sym 2 (𝑊)) → 𝑊 8,0 , a section of Sym 4 (𝔼) ⊗ det(𝔼) 8 defines a covariant of bidegree (8, 24∕2) = (8,12) for the action of GL(𝑊). Proposition 6.1.…”

“…Proof. By restricting and projecting we obtain a covariant of bidegree (8,12). This covariant is divisible by the discriminant and does not vanish on the locus of smooth hyperelliptic curves.…”

“…Lemma 14.3. The projections to the three summands in (12) of the pull back of 𝜒 4,0,8 to  3,2 define modular forms on  3,2 of weights (4,0,8), (2, 0, 4), and (0,0,14) and these are up to a scalar given by the covariants 𝑓 8,−2 𝔡, 𝑓 4,−1 𝔡 and the discriminant 𝔡.…”

“…Corollary 15.1. The dual of the canonical curve defines a Siegel modular cusp form of degree 3 of weight (0, 12,12) vanishing with multiplicity 2 at infinity. The Weierstrass divisor defines a cusp form of weight (0,24,44) vanishing with multiplicity 6 at infinity.…”