Limit linear series for vector bundles

We study concepts of stability associated to a smooth complex curve together with a linear series on it. In particular we investigate the relation between stability of the associated dual span bundle and linear stability. Our results imply that stability of the dual span holds under a hypothesis related to the Clifford index of the curve. Furthermore, in some of the cases, we prove that a stronger stability holds: cohomological stability. Finally, using our results we obtain stable vector bundles of slope 3, and prove that they admit theta-divisors.

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Smoothing of Limit Linear Series of Rank One on Saturated Metrized Complexes of Algebraic Curves

Canadian Journal of Mathematics ◽

10.4153/cjm-2017-027-2 ◽

2018 ◽

Vol 70 (3) ◽

pp. 628-682 ◽

Cited By ~ 1

Author(s):

Ye Luo ◽

Madhusudan Manjunath

Keyword(s):

Algebraic Curves ◽

Linear Series ◽

Compact Type ◽

Nodal Curves ◽

Dual Graphs ◽

Rank One ◽

Limit Linear Series

AbstractWe investigate the smoothing problem of limit linear series of rank one on an enrichment of the notions of nodal curves and metrized complexes called saturated metrized complexes. We give a finitely verifiable full criterion for smoothability of a limit linear series of rank one on saturated metrized complexes, characterize the space of all such smoothings, and extend the criterion to metrized complexes. As applications, we prove that all limit linear series of rank one are smoothable on saturated metrized complexes corresponding to curves of compact-type, and we prove an analogue for saturated metrized complexes of a theorem of Harris and Mumford on the characterization of nodal curves contained in a given gonality stratum. In addition, we give a full combinatorial criterion for smoothable limit linear series of rank one on saturated metrized complexes corresponding to nodal curves whose dual graphs are made of separate loops.

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Connectedness of Brill–Noether Loci via Degenerations

International Mathematics Research Notices ◽

10.1093/imrn/rnx325 ◽

2018 ◽

Vol 2019 (19) ◽

pp. 6162-6178 ◽

Cited By ~ 1

Author(s):

Brian Osserman

Keyword(s):

Linear Series ◽

Positive Dimension ◽

Recent Advances ◽

Limit Linear Series

Abstract We show that limit linear series spaces for chains of curves are reduced. Using recent advances in the foundations of limit linear series, we then use degenerations to study the question of connectedness for spaces of linear series with imposed ramification at up to two points. We find that in general, these spaces may not be connected even when they have positive dimension, but we prove a criterion for connectedness which generalizes the theorem previously proved by Fulton and Lazarsfeld in the case without imposed ramification.

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