Pub Date : 2019-06-01DOI: 10.1142/9789811200489_0009
Giulia Codenotti, Lena Walter
{"title":"Finding a fully mixed cell in a mixed subdivision of polytopes","authors":"Giulia Codenotti, Lena Walter","doi":"10.1142/9789811200489_0009","DOIUrl":"https://doi.org/10.1142/9789811200489_0009","url":null,"abstract":"","PeriodicalId":322478,"journal":{"name":"Algebraic and Geometric Combinatorics on Lattice Polytopes","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127967766","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-06-01DOI: 10.1142/9789811200489_0019
L. Ng
{"title":"Technically, squares are polytopes","authors":"L. Ng","doi":"10.1142/9789811200489_0019","DOIUrl":"https://doi.org/10.1142/9789811200489_0019","url":null,"abstract":"","PeriodicalId":322478,"journal":{"name":"Algebraic and Geometric Combinatorics on Lattice Polytopes","volume":"63 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126377997","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-06-01DOI: 10.1142/9789811200489_0029
Gabriele Balletti, F. Castillo, Liam Solus, Bach Tran, Akiyoshi Tsuchiya
{"title":"Open problems from the 2018 Summer Workshop on Lattice Polytopes at Osaka University","authors":"Gabriele Balletti, F. Castillo, Liam Solus, Bach Tran, Akiyoshi Tsuchiya","doi":"10.1142/9789811200489_0029","DOIUrl":"https://doi.org/10.1142/9789811200489_0029","url":null,"abstract":"","PeriodicalId":322478,"journal":{"name":"Algebraic and Geometric Combinatorics on Lattice Polytopes","volume":"72 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131680985","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-06-01DOI: 10.1142/9789811200489_0012
Laura Escobar
{"title":"A brief survey about moment polytopes of subvarieties of products of Grassmanians","authors":"Laura Escobar","doi":"10.1142/9789811200489_0012","DOIUrl":"https://doi.org/10.1142/9789811200489_0012","url":null,"abstract":"","PeriodicalId":322478,"journal":{"name":"Algebraic and Geometric Combinatorics on Lattice Polytopes","volume":"87 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114520371","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-06-01DOI: 10.1142/9789811200489_0011
Florian Kohl, Alexander Engström
{"title":"Lattice polytopes in mathematical physics","authors":"Florian Kohl, Alexander Engström","doi":"10.1142/9789811200489_0011","DOIUrl":"https://doi.org/10.1142/9789811200489_0011","url":null,"abstract":"","PeriodicalId":322478,"journal":{"name":"Algebraic and Geometric Combinatorics on Lattice Polytopes","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129702189","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-23DOI: 10.1142/9789811200489_0027
Bach Tran
Let $L$ be an ample line bundle over a smooth projective toric surface $X$. Then $L$ corresponds to a very ample lattice polytope $P$ that encodes many geometric properties of $L$. In this article, by studying $P$, we will give some necessary and sufficient numerical criteria for the adjoint series $|K_X+L|$ to be either nef or (very) ample.
{"title":"A Reider-type result for smooth projective toric surfaces","authors":"Bach Tran","doi":"10.1142/9789811200489_0027","DOIUrl":"https://doi.org/10.1142/9789811200489_0027","url":null,"abstract":"Let $L$ be an ample line bundle over a smooth projective toric surface $X$. Then $L$ corresponds to a very ample lattice polytope $P$ that encodes many geometric properties of $L$. In this article, by studying $P$, we will give some necessary and sufficient numerical criteria for the adjoint series $|K_X+L|$ to be either nef or (very) ample.","PeriodicalId":322478,"journal":{"name":"Algebraic and Geometric Combinatorics on Lattice Polytopes","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127068576","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-16DOI: 10.1142/9789811200489_0018
Makoto Miura
In this article, we summarize combinatorial description of complete intersection Calabi-Yau threefolds in Hibi toric varieties. Such Calabi-Yau threefolds have at worst conifold singularities, and are often smoothable to non-singular Calabi-Yau threefolds. We focus on such non-singular Calabi-Yau threefolds of Picard number one, and illustrate the calculation of topological invariants, using new motivating examples.
{"title":"Complete intersection Calabi–Yau threefolds in Hibi toric varieties and their smoothing","authors":"Makoto Miura","doi":"10.1142/9789811200489_0018","DOIUrl":"https://doi.org/10.1142/9789811200489_0018","url":null,"abstract":"In this article, we summarize combinatorial description of complete intersection Calabi-Yau threefolds in Hibi toric varieties. Such Calabi-Yau threefolds have at worst conifold singularities, and are often smoothable to non-singular Calabi-Yau threefolds. We focus on such non-singular Calabi-Yau threefolds of Picard number one, and illustrate the calculation of topological invariants, using new motivating examples.","PeriodicalId":322478,"journal":{"name":"Algebraic and Geometric Combinatorics on Lattice Polytopes","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129328713","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-03DOI: 10.1142/9789811200489_0017
Sebastian Manecke
Eberhard-type theorems are statements about the realizability of a polytope (or more general polyhedral maps) given the valency of its vertices and sizes of its polygonal faces up to a linear linear degree of freedom. We present new theorems of Eberhard-type where we allow adding two kinds of polygons and one type of vertices. We also hint towards a full classification of these types of results.
{"title":"Eberhard-type theorems with two kinds of polygons","authors":"Sebastian Manecke","doi":"10.1142/9789811200489_0017","DOIUrl":"https://doi.org/10.1142/9789811200489_0017","url":null,"abstract":"Eberhard-type theorems are statements about the realizability of a polytope (or more general polyhedral maps) given the valency of its vertices and sizes of its polygonal faces up to a linear linear degree of freedom. We present new theorems of Eberhard-type where we allow adding two kinds of polygons and one type of vertices. We also hint towards a full classification of these types of results.","PeriodicalId":322478,"journal":{"name":"Algebraic and Geometric Combinatorics on Lattice Polytopes","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125392263","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-12-17DOI: 10.1142/9789811200489_0023
Maren H. Ring
McMullen's formulas or local formulas for Ehrhart coefficients are functions on rational cones that determine the $i$-th coefficient of the Ehrhart polynomial as a weighted sum of the volumes of the i-dimensional faces of a polytope. This work focuses on the RS-$mu$-construction as given in a previous paper by Achill Sch"urmann and the author. We give an explicit description of the construction from the dual point of view, i.e. given the cone of feasible directions instead of the normal cone as input value. We further show some properties of the construction in special cases, namely in case of symmetry and for the codimension one case.
{"title":"Special cases and a dual view on the local formulas for Ehrhart coefficients from lattice tiles","authors":"Maren H. Ring","doi":"10.1142/9789811200489_0023","DOIUrl":"https://doi.org/10.1142/9789811200489_0023","url":null,"abstract":"McMullen's formulas or local formulas for Ehrhart coefficients are functions on rational cones that determine the $i$-th coefficient of the Ehrhart polynomial as a weighted sum of the volumes of the i-dimensional faces of a polytope. This work focuses on the RS-$mu$-construction as given in a previous paper by Achill Sch\"urmann and the author. We give an explicit description of the construction from the dual point of view, i.e. given the cone of feasible directions instead of the normal cone as input value. We further show some properties of the construction in special cases, namely in case of symmetry and for the codimension one case.","PeriodicalId":322478,"journal":{"name":"Algebraic and Geometric Combinatorics on Lattice Polytopes","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130946894","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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