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

2018 ◽  
Vol 70 (3) ◽  
pp. 628-682 ◽  
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.

scholarly journals Limit linear series for curves not of compact type

2019 ◽  
Vol 2019 (753) ◽  
pp. 57-88 ◽  
Author(s):  
Brian Osserman
Keyword(s):  
Moduli Space ◽  
Linear Series ◽  
Restricted Class ◽  
Compact Type ◽  
Nodal Curves ◽  
Equivalent Definition ◽  
Two Component ◽  
Limit Linear Series ◽  
Definition Of

AbstractWe introduce a notion of limit linear series for nodal curves which are not of compact type. We give a construction of a moduli space of limit linear series, which works also in smoothing families, and we prove a corresponding specialization result. For a more restricted class of curves which simultaneously generalizes two-component curves and curves of compact type, we give an equivalent definition of limit linear series, which is visibly a generalization of the Eisenbud–Harris definition. Finally, for the same class of curves, we prove a smoothing theorem which constitutes an improvement over known results even in the compact-type case.

scholarly journals Smoothing of Limit Linear Series on Curves and Metrized Complexes of Pseudocompact Type

2018 ◽  
Vol 71 (03) ◽  
pp. 629-658 ◽  
Author(s):  
Xiang He
Keyword(s):  
Linear Series ◽  
Metric Graph ◽  
Rank One ◽  
Limit Linear Series

AbstractWe investigate the connection between Osserman limit series (on curves of pseudocompact type) and Amini–Baker limit linear series (on metrized complexes with corresponding underlying curve) via a notion of pre-limit linear series on curves of the same type. Then, applying the smoothing theorems of Osserman limit linear series, we deduce that, fixing certain metrized complexes, or for certain types of Amini–Baker limit linear series, the smoothability is equivalent to a certain “weak glueing condition”. Also for arbitrary metrized complexes of pseudocompact type the weak glueing condition (when it applies) is necessary for smoothability. As an application we confirm the lifting property of specific divisors on the metric graph associated with a certain regular smoothing family, and give a new proof of a result of Cartright, Jensen, and Payne for vertex-avoiding divisors, and generalize it for divisors of rank one in the sense that, for the metric graph, there could be at most three edges (instead of two) between any pair of adjacent vertices.

scholarly journals Brill-Noether generality of binary curves

2020 ◽  
pp. 1-21
Author(s):  
Xiang He
Keyword(s):  
Potential Application ◽  
Algebraic Curves ◽  
Linear Series ◽  
Arbitrary Rank ◽  
Limit Linear Series ◽  
Effective Divisors

Abstract We show that the space $G^r_{\underline d}(X)$ of linear series of certain multi-degree $\underline d=(d_1,d_2)$ (including the balanced ones) and rank r on a general genus-g binary curve X has dimension $\rho _{g,r,d}=g-(r+1)(g-d+r)$ if nonempty, where $d=d_1+d_2$ . This generalizes Caporaso’s result from the case $r\leq 2$ to arbitrary rank, and shows that the space of Osserman-limit linear series on a general binary curve has the expected dimension, which was known for $r\leq 2$ . In addition, we show that the space $G^r_{\underline d}(X)$ is still of expected dimension after imposing certain ramification conditions with respect to a sequence of increasing effective divisors supported on two general points $P_i\in Z_i$ , where $i=1,2$ and $Z_1,Z_2$ are the two components of X. Our result also has potential application to the lifting problem of divisors on graphs to divisors on algebraic curves.

A Geometric Characterization of Nonnegative Bands

2004 ◽  
Vol 47 (2) ◽  
pp. 257-263
Author(s):  
Alka Marwaha
Keyword(s):  
Invariant Subspace ◽  
Geometric Structure ◽  
Finite Rank ◽  
Special Kind ◽  
Maximal Rank ◽  
Geometric Characterization ◽  
Idempotent Operators ◽  
Rank One ◽  
Standard Subspace

AbstractA band is a semigroup of idempotent operators. A nonnegative band S in having at least one element of finite rank and with rank (S) > 1 for all S in S is known to have a special kind of common invariant subspace which is termed a standard subspace (defined below).Such bands are called decomposable. Decomposability has helped to understand the structure of nonnegative bands with constant finite rank. In this paper, a geometric characterization of maximal, rank-one, indecomposable nonnegative bands is obtained which facilitates the understanding of their geometric structure.

scholarly journals Injective linear series of algebraic curves on quadrics

2020 ◽  
Vol 66 (2) ◽  
pp. 231-254
Author(s):  
Edoardo Ballico ◽  
Emanuele Ventura
Keyword(s):  
Algebraic Geometry ◽  
Projective Space ◽  
Algebraic Curves ◽  
Linear Series ◽  
Central Importance ◽  
Arbitrary Characteristic ◽  
Space Curves

Abstract We study linear series on curves inducing injective morphisms to projective space, using zero-dimensional schemes and cohomological vanishings. Albeit projections of curves and their singularities are of central importance in algebraic geometry, basic problems still remain unsolved. In this note, we study cuspidal projections of space curves lying on irreducible quadrics (in arbitrary characteristic).

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