scholarly journals A metric graph satisfying w41=1$w_4^1 = 1$ that cannot be lifted to a curve satisfying dim⁡ (W41)=1$\dim \;(W_4^1 ) = 1$

2016 ◽  
Vol 14 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Marc Coppens
Keyword(s):  
Linear Systems ◽  
Harmonic Morphism ◽  
Metric Graphs ◽  
Clifford Index ◽  
Special Linear ◽  
Metric Graph ◽  
Index 2

AbstractFor all integers g ≥ 6 we prove the existence of a metric graph G with $w_4^1 = 1$ such that G has Clifford index 2 and there is no tropical modification G′ of G such that there exists a finite harmonic morphism of degree 2 from G′ to a metric graph of genus 1. Those examples show that not all dimension theorems on the space classifying special linear systems for curves have immediate translation to the theory of divisors on metric graphs.

scholarly journals EMBEDDING OF METRIC GRAPHS ON HYPERBOLIC SURFACES

2019 ◽  
Vol 99 (03) ◽  
pp. 508-520
Author(s):  
BIDYUT SANKI
Keyword(s):  
Euler Characteristic ◽  
Isometric Embedding ◽  
Hyperbolic Surface ◽  
Hyperbolic Surfaces ◽  
Metric Graphs ◽  
Metric Graph ◽  
Complementary Region

An embedding of a metric graph $(G,d)$ on a closed hyperbolic surface is essential if each complementary region has a negative Euler characteristic. We show, by construction, that given any metric graph, its metric can be rescaled so that it admits an essential and isometric embedding on a closed hyperbolic surface. The essential genus $g_{e}(G)$ of $(G,d)$ is the lowest genus of a surface on which such an embedding is possible. We establish a formula to compute $g_{e}(G)$ and show that, for every integer $g\geq g_{e}(G)$ , there is an embedding of $(G,d)$ (possibly after a rescaling of $d$ ) on a surface of genus $g$ . Next, we study minimal embeddings where each complementary region has Euler characteristic $-1$ . The maximum essential genus $g_{e}^{\max }(G)$ of $(G,d)$ is the largest genus of a surface on which the graph is minimally embedded. We describe a method for an essential embedding of $(G,d)$ , where $g_{e}(G)$ and $g_{e}^{\max }(G)$ are realised.

scholarly journals Special linear systems and syzygies

2008 ◽  
Vol 59 (3) ◽  
pp. 239-254 ◽  
Author(s):  
Alberto Alzati
Keyword(s):  
Linear Systems ◽  
Special Linear

scholarly journals On a notion of toric special linear systems

2019 ◽  
Vol 223 (8) ◽  
pp. 3225-3237
Author(s):  
Joaquín Moraga
Keyword(s):  
Linear Systems ◽  
Special Linear

scholarly journals The Gonality Sequence of Complete Graphs

10.37236/6876 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Filip Cools ◽  
Marta Panizzut
Keyword(s):  
Linear Systems ◽  
Complete Graph ◽  
Algebraic Curve ◽  
Plane Curve ◽  
Finite Graph ◽  
Complete Graphs ◽  
Metric Graphs ◽  
Smooth Plane ◽  
Graph Metric ◽  
Gonality Sequence

The gonality sequence $(\gamma_r)_{r\geq1}$ of a finite graph/metric graph/algebraic curve comprises the minimal degrees $\gamma_r$ of linear systems of rank $r$. For the complete graph $K_d$, we show that $\gamma_r =  kd - h$ if $r<g=\frac{(d-1)(d-2)}{2}$, where $k$ and $h$ are the uniquely determined integers such that $r = \frac{k(k+3)}{2} - h$ with $1\leq k\leq d-3$ and $0 \leq h \leq k $. This shows that the graph $K_d$ has the gonality sequence of a smooth plane curve of degree $d$. The same result holds for the corresponding metric graphs.

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