Degree-2 Abel Maps for Nodal Curves

Abstract

We present numerical conditions for the existence of natural degree-2 Abel maps for any given nodal curve. CoCoA scripst were written and have so far verified the validity of the conditions for numerous curves.curves of compact type, developed by Eisenbud and Harris; see [21]. Other results have been obtained through the theory; see [17].The theory of Eisenbud and Harris points out to a partial compactification of the variety of linear series over the moduli space of curves of compact type. It is natural to ask whether one can extend this compactification over the whole M g , as Eisenbud and Harris themselves asked in [18]. The supposedly intermediate step, that of constructing a compactification of the relative Picard scheme over M g , has been carried out by Caporaso [3]. But the final step has proved to be very difficult.Sketches on how to deal with limit linear series for stable curves not of compact type are sparse in the literature. Recently, Medeiros and the second author [14] were able to describe limit canonical series on curves with two components, using the theory presented in [11], the ingredients of which having already appeared in [29]. But their description is rather complicated, and relies on the strong assumption that the components intersect each other at points in general position on each component. The same assumption is present in [16], where the case of curves with more components is partially considered.In view of the difficulties, one might ask whether we stand a better chance of compactifying the variety of linear series over M g by looking at degenerations of Abel maps. Indeed, Abel maps of all degrees have been constructed and studied for integral curves in [1]. Degree-1 Abel maps for stable curves were constructed in [6] and studied in [7]. Higher-degree Abel maps for curves of compact type appeared soon afterwards in [10].At this point one might say that the theory of degenerations of Abel maps is on even terms with that of limit linear series. A comparison between the two theories has in fact appeared in [15], based on the more refined notion of limit linear series introduced by Osserman [26], and thus limited to two-component curves.The present article advances the theory of degenerations of Abel maps, by presenting purely numerical conditions for the existence of degree-2 Abel maps for any given nodal curve. Whether the curve is stable or not is immaterial. The specific moduli, even the genera of its irreducible components is immaterial. Where the points of intersection of components lie on each component is immaterial, in stark contrast with [14] and [16]. In fact, for nodal curves with the same dual graph, either all of them or none of them have degree-2 Abel maps. An algorithm has been implemented by means of CoCoA scripts, available at http://w3.impa.br/∼esteves/CoCoAScripts/Abelmaps to check the validity of these conditions for any given curve and has, so far, always returned a positive answer. Our approach is simply to extend the construction done in [9] for two-component two...

The resolution of the degree‐2 Abel‐Jacobi map for nodal curves‐I

“…Consider We recall some facts about the local and global geometry of and ; we refer the reader to [, Sections and ] for more details and for the proofs of the statements.…”

“…If A is a point of , then and identifies respectively with and C , and identifies with the contraction map (see [, Proposition 4.5]). For each γ in , we will denote by the strict transform to of the component of via .…”

“…A slice of through A is a section of sending the special point of B to A and such that the pullbacks of , and via λ are all prime. Form the Cartesian diagram By [, Proposition 4.6] we have that ρ is a smoothing of , which we call the λ‐ smoothing of .…”

“…Since Abel maps exist for any nodal curve only in degree 1 (at least for certain polarizations), the natural next step is to consider the degree‐2 case. In , the question whether or not it is possible to obtain a resolution of the rational map through a sequence of blowups along certain divisors of , is reduced to a series of combinatorial issues. In this paper and in we solve the posed combinatorial problems when is the canonical polarization on and for , where is the ideal sheaf of , i.e.…”

Brill–Noether Locus of Rank 1 and Degreeg− 1 on a Nodal Curve

“…The first is towards the resolution of the rational Abel map. Few maps have been fully resolved, namely Abel maps of degree 1 for stable curves [7] and Gorenstein curves [8], and of degree 2 for nodal curves in [9], [15], and [16]. Also, Abel maps of any degree were constructed for stable curves of compact type in [10], and for nodal curves with two components in [1].…”