Limit Weierstrass points on nodal reducible curves

Esteves, Eduardo; Salehyan, Parham

doi:10.1090/s0002-9947-07-04193-1

“…The reasoning behind these conclusions is laid out right before the statement of Lemma 3.2. Now, it is just a matter of using the entries of the second table of Lemma 3.2 to conclude that ξ * ∆ 1 is of the stated form, with the coefficients of E R,i and E R,j as prescribed by (16).…”

Section: Formentioning

confidence: 99%

“…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.…”

mentioning

confidence: 99%

“…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.…”

mentioning

confidence: 99%

“…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].…”

mentioning

confidence: 99%

See 2 more Smart Citations

Degree-2 Abel Maps for Nodal Curves

Coelho

¹

,

Esteves

²

,

Pacini

³

2015

Int Math Res Notices

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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...

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“…On the other hand, an approach similar to that by Eisenbud and Harris, choosing for each component of the limit curve a "best" limit line bundle, was tentatively carried out by many: Ziv Ran had an early draft on this already in the 80's, whereas the second author, in collaboration with Medeiros and Salehyan [EM02,ES07], used this approach to study limits of Weierstrass points for a wide class of stable curves in the nineties. However, one could not carry the approach further.…”

Section: Introductionmentioning

confidence: 99%

Voronoi tilings, toric arrangements and degenerations of line bundles III

Amini,

Esteves

2021

Preprint

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We describe limits of line bundles on nodal curves in terms of toric arrangements associated to Voronoi tilings of Euclidean spaces. These tilings encode information on the relationship between the possibly infinitely many limits, and ultimately give rise to a new definition of limit linear series. This article and the first two that preceded it are the first in a series aimed to explore this new approach.In Part I, we set up the combinatorial framework and showed how graphs weighted with integer lengths associated to the edges provide tilings of Euclidean spaces by certain polytopes associated to the graph itself and to its subgraphs.In Part II, we described the arrangements of toric varieties associated to the tilings of Part I in several ways: using normal fans, as unions of orbits, by equations and as degenerations of tori.In the present Part III, we show how these combinatorial and toric frameworks allow us to describe all stable limits of a family of line bundles along a degenerating family of curves. Our main result asserts that the collection of all these limits is parametrized by a connected 0-dimensional closed substack of the Artin stack of all torsion-free rank-one sheaves on the limit curve. Moreover, we thoroughly describe this closed substack and all the closed substacks that arise in this way as certain torus quotients of the arrangements of toric varieties of Part II determined by the Voronoi tilings of Euclidean spaces studied in Part I.

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Inflectionary Invariants for Isolated Complete Intersection Curve Singularities

Patel¹,

Swaminathan²

2023

Memoirs of the AMS

0

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We investigate the role played by curve singularity germs in the enumeration of inflection points in families of curves acquiring singular members. Let N ≥ 2 N \geq 2 , and consider an isolated complete intersection curve singularity germ f : ( C N , 0 ) → ( C N − 1 , 0 ) f \colon (\mathbb {C}^N,0) \to (\mathbb {C}^{N-1},0) . We define a numerical function m ↦ AD ( 2 ) m ⁡ ( f ) m \mapsto \operatorname {AD}_{(2)}^m(f) that naturally arises when counting m t h m^{\mathrm {th}} -order weight- 2 2 inflection points with ramification sequence ( 0 , … , 0 , 2 ) (0, \dots , 0, 2) in a 1 1 -parameter family of curves acquiring the singularity f = 0 f = 0 , and we compute AD ( 2 ) m ⁡ ( f ) \operatorname {AD}_{(2)}^m(f) for several interesting families of pairs ( f , m ) (f,m) . In particular, for a node defined by f : ( x , y ) ↦ x y f \colon (x,y) \mapsto xy , we prove that AD ( 2 ) m ⁡ ( x y ) = ( m + 1 4 ) , \operatorname {AD}_{(2)}^m(xy) = {{m+1} \choose 4}, and we deduce as a corollary that AD ( 2 ) m ⁡ ( f ) ≥ ( mult 0 ⁡ Δ f ) ⋅ ( m + 1 4 ) \operatorname {AD}_{(2)}^m(f) \geq (\operatorname {mult}_0 \Delta _f) \cdot {{m+1} \choose 4} for any f f , where mult 0 ⁡ Δ f \operatorname {mult}_0 \Delta _f is the multiplicity of the discriminant Δ f \Delta _f at the origin in the deformation space. Significantly, we prove that the function m ↦ AD ( 2 ) m ⁡ ( f ) − ( mult 0 ⁡ Δ f ) ⋅ ( m + 1 4 ) m \mapsto \operatorname {AD}_{(2)}^m(f) -(\operatorname {mult}_0 \Delta _f) \cdot {{m+1} \choose 4} is an analytic invariant measuring how much the singularity “counts as” an inflection point. We prove similar results for weight- 2 2 inflection points with ramification sequence ( 0 , … , 0 , 1 , 1 ) (0, \dots , 0, 1,1) and for weight- 1 1 inflection points, and we apply our results to solve a number of related enumerative problems.

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Limit Weierstrass points on nodal reducible curves

Cited by 6 publications

References 7 publications

Degree-2 Abel Maps for Nodal Curves

Degree-2 Abel Maps for Nodal Curves

Voronoi tilings, toric arrangements and degenerations of line bundles III

Inflectionary Invariants for Isolated Complete Intersection Curve Singularities

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