Geometric combinatorics in the study of compact toric surfaces

2006 ◽  
pp. 71-123 ◽  
Author(s):  
Dimitrios I. Dais
Keyword(s):  
Toric Surfaces ◽  
Geometric Combinatorics

Weierstrass n-semigroups with even n and curves on toric surfaces

2021 ◽  
Vol 225 (12) ◽  
pp. 106759
Author(s):  
Ryo Kawaguchi ◽  
Jiryo Komeda
Keyword(s):  
Toric Surfaces

scholarly journals Enumeration of Hybrid Domino-Lozenge Tilings II: Quasi-Octagonal Regions

10.37236/4669 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Tri Lai
Keyword(s):  
Product Formula ◽  
Cambridge University ◽  
Replacement Method ◽  
Lozenge Tilings ◽  
Simple Product ◽  
Geometric Combinatorics

We use the subgraph replacement method to prove a simple product formula for the tilings of an  8-vertex counterpart of Propp's quasi-hexagons (Problem 16 in New Perspectives in Geometric Combinatorics, Cambridge University Press, 1999), called quasi-octagon.

Toric surfaces

2011 ◽  
pp. 459-512
Author(s):  
David Cox ◽  
John Little ◽  
Henry Schenck
Keyword(s):  
Toric Surfaces

scholarly journals Refined curve counting with tropical geometry

2015 ◽  
Vol 152 (1) ◽  
pp. 115-151 ◽  
Author(s):  
Florian Block ◽  
Lothar Göttsche
Keyword(s):  
Tropical Geometry ◽  
Ruled Surfaces ◽  
Plane Curves ◽  
Toric Surfaces ◽  
Formula One ◽  
Tropical Curves ◽  
Tropical Curve ◽  
Severi Variety ◽  
Floor Diagrams

The Severi degree is the degree of the Severi variety parametrizing plane curves of degree $d$ with ${\it\delta}$ nodes. Recently, Göttsche and Shende gave two refinements of Severi degrees, polynomials in a variable $y$, which are conjecturally equal, for large $d$. At $y=1$, one of the refinements, the relative Severi degree, specializes to the (non-relative) Severi degree. We give a tropical description of the refined Severi degrees, in terms of a refined tropical curve count for all toric surfaces. We also refine the equivalent count of floor diagrams for Hirzebruch and rational ruled surfaces. Our description implies that, for fixed ${\it\delta}$, the refined Severi degrees are polynomials in $d$ and $y$, for large $d$. As a consequence, we show that, for ${\it\delta}\leqslant 10$ and all $d\geqslant {\it\delta}/2+1$, both refinements of Göttsche and Shende agree and equal our refined counts of tropical curves and floor diagrams.

Optic systems with spherical, cylindrical, and toric surfaces

1995 ◽  
Vol 34 (22) ◽  
pp. 4900 ◽  
Author(s):  
J. Barcala ◽  
M. C. Vazquez ◽  
A. Garcia
Keyword(s):  
Toric Surfaces ◽  
Optic Systems

scholarly journals Exceptional sequences of invertible sheaves on rational surfaces

2011 ◽  
Vol 147 (4) ◽  
pp. 1230-1280 ◽  
Author(s):  
Lutz Hille ◽  
Markus Perling
Keyword(s):  
Large Class ◽  
Complete Classification ◽  
Toric Surface ◽  
Structural Result ◽  
Toric Surfaces ◽  
Rational Surfaces ◽  
Exceptional Sequences

AbstractIn this article we consider exceptional sequences of invertible sheaves on smooth complete rational surfaces. We show that to every such sequence one can associate a smooth complete toric surface in a canonical way. We use this structural result to prove various theorems on exceptional and strongly exceptional sequences of invertible sheaves on rational surfaces. We construct full strongly exceptional sequences for a large class of rational surfaces. For the case of toric surfaces we give a complete classification of full strongly exceptional sequences of invertible sheaves.

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