Jets of line bundles on curves and Wronskians

Abstract

a b s t r a c tWe collect a few results about jets of line bundles on curves and Wronskians, with a special emphasis to those arising from the canonical involution of a hyperelliptic curve.

On generalized Wrońskians

“…In [8] one shows as in Example 7.3 that a hyperelliptic curve of genus g ≥ 2 satisfies a Weierstrass type equation expressing the relation between the Wronskian and the product of the sections vanishing twice at its Weierstrass points. It is also intrinsically shown, again using the notion of Wronskian, that the hyperelliptic curve of genus g ≥ 2 can be realized, for each a ≥ 0, as a curve of degree 2(g + 1 + a) lying on a rational normal scroll S(a, g + 1 + a) of P g+2+a .…”

“…It may thought of as a global derivative of order h of the section v. In fact it is a local representation of v together with its first h derivatives. The truncation morphism occurring in (14), [8] for further details.…”

“…We notice now that each g r d (L), i.e. V ∈ G(r + 1, H 0 (L)), can be seen in fact as an element of Γ triv (̺ r,d ), because [8], and then D d v = 0 implies v = 0.…”

“…A. Nigro proposes to extend the notion of ramification locus of a linear system on a curve to that of ramification locus of a holomorphic section of a Grassmann bundle [39]. The construction was motivated by the following observation (see also [8]): Let Γ triv (ρ r,d ) be the set of all the sections γ : C → G(r + 1, J d L) such that the pull back of the tautological bundle S r over G(r+1, J d L) is trivial. Then each…”

“…, it coincides with the usual Wro ńskian of V -see Section 5. In particular, if M is a line bundle defining the unique g 1 2 over a hyperelliptic curve of genus g ≥ 2, the extended Wro ński map Γ triv (ρ 1,2 ) → PH 0 (M ⊗2 ⊗ K) is dominant (see [8]), its behavior is closer to the surjectivity of the Wro ński map defined on the space of g r d s on P 1 . The latter, in this case, coincides with Γ triv (ρ r,d ) modulo identification of V with γ V .…”