Snapshots from the History of Toric Geometry David A. Cox Geometry - PowerPoint PPT Presentation

Snapshots from the History of Toric Snapshots from the History of Toric Geometry David A. Cox Geometry 1970–1988 Toric Geometry and its Applications Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments David A. Cox Some Quotes Since 1988 Secondary Fan Department of Mathematics Homogeneous Amherst College Coordinates Mirror Symmetry dac@math.amherst.edu Survey Papers The 21st Century Leuven May 2011 Conclusion 6 June 2011

Outline 1970–1988 1 Snapshots from the Demazure History of Toric KKMS Geometry Other Early Papers David A. Cox The Russian School 1970–1988 Polytopes Demazure KKMS Other Developments Other Early Papers The Russian School Some Quotes Polytopes Other Developments Since 1988 2 Some Quotes Since 1988 Secondary Fan Secondary Fan Homogeneous Coordinates Homogeneous Coordinates Mirror Symmetry Mirror Symmetry Survey Papers Survey Papers The 21st Century May 2011 Conclusion The 21st Century May 2011 Conclusion

Outline 1970–1988 1 Snapshots from the Demazure History of Toric KKMS Geometry Other Early Papers David A. Cox The Russian School 1970–1988 Polytopes Demazure KKMS Other Developments Other Early Papers The Russian School Some Quotes Polytopes Other Developments Since 1988 2 Some Quotes Since 1988 Secondary Fan Secondary Fan Homogeneous Coordinates Homogeneous Coordinates Mirror Symmetry Mirror Symmetry Survey Papers Survey Papers The 21st Century May 2011 Conclusion The 21st Century May 2011 Conclusion

Demazure 1970 Toric varieties such as C n , ( C ∗ ) n , P n , P n × P m , have been Snapshots from the around for a long time. The general definition came in 1970: History of Toric Geometry SOUS-GROUPES ALGÉBRIQUES David A. Cox DE RANG MAXIMUM DU GROUPE DE CREMONA 1970–1988 P AR M ICHEL DEMAZURE Demazure KKMS Other Early Papers He studied groups of birational automorphisms of P n : The Russian School Polytopes Other Developments Some Quotes . . . ces schémas en groupes se réalisent commes groupes Since 1988 Secondary Fan d’automorphismes de certains Z -schémas à décomposition Homogeneous Coordinates cellulaire obtenus en “ajoutant à un tore déployé certains Mirror Symmetry Survey Papers points á l’infini”. The 21st Century May 2011 Un rôle important est joué par les schémas précédents; la Conclusion manière “ajouter des points á l’infini à un tore” est décrite par un “éventail” . . .

Demazure 1970 Toric varieties such as C n , ( C ∗ ) n , P n , P n × P m , have been Snapshots from the around for a long time. The general definition came in 1970: History of Toric Geometry SOUS-GROUPES ALGÉBRIQUES David A. Cox DE RANG MAXIMUM DU GROUPE DE CREMONA 1970–1988 P AR M ICHEL DEMAZURE Demazure KKMS Other Early Papers He studied groups of birational automorphisms of P n : The Russian School Polytopes Other Developments Some Quotes . . . ces schémas en groupes se réalisent commes groupes Since 1988 Secondary Fan d’automorphismes de certains Z -schémas à décomposition Homogeneous Coordinates cellulaire obtenus en “ajoutant à un tore déployé certains Mirror Symmetry Survey Papers points á l’infini”. The 21st Century May 2011 Un rôle important est joué par les schémas précédents; la Conclusion manière “ajouter des points á l’infini à un tore” est décrite par un “éventail” . . .

Definition and Name Snapshots from the D ÉFINITION 1. History of Toric Soit M ∗ un groupe abélien libre de type fini. On appelle Geometry David A. Cox éventail dans M ∗ un ensemble fini Σ de parties de M ∗ tel que: 1970–1988 Demazure a. chaque élément de Σ est une partie d’une base de M ∗ ; KKMS Other Early Papers b. toute partie d’un élément de Σ appartient á Σ ; The Russian School Polytopes c. si K , L ∈ Σ , on a N . K ∩ N . L = N . ( K ∩ L ) . Other Developments Some Quotes Since 1988 D ÉFINITION 2. Secondary Fan Homogeneous Coordinates On appelle schéma défini par l’éventail Σ le schéma X Mirror Symmetry Survey Papers obtenu part recollement des V K , K parcourant Σ , á l’aide The 21st Century May 2011 des immersions ouvertes V K ∩ L → V K , V K ∩ L → V L , pour Conclusion K , L ∈ Σ .

Definition and Name Snapshots from the D ÉFINITION 1. History of Toric Soit M ∗ un groupe abélien libre de type fini. On appelle Geometry David A. Cox éventail dans M ∗ un ensemble fini Σ de parties de M ∗ tel que: 1970–1988 Demazure a. chaque élément de Σ est une partie d’une base de M ∗ ; KKMS Other Early Papers b. toute partie d’un élément de Σ appartient á Σ ; The Russian School Polytopes c. si K , L ∈ Σ , on a N . K ∩ N . L = N . ( K ∩ L ) . Other Developments Some Quotes Since 1988 D ÉFINITION 2. Secondary Fan Homogeneous Coordinates On appelle schéma défini par l’éventail Σ le schéma X Mirror Symmetry Survey Papers obtenu part recollement des V K , K parcourant Σ , á l’aide The 21st Century May 2011 des immersions ouvertes V K ∩ L → V K , V K ∩ L → V L , pour Conclusion K , L ∈ Σ .

Some Results Snapshots P ROPOSITION 4. from the History of Soit k un corps. Les conditions suivantes sont équivalentes: Toric Geometry (i) le Z -schéma X est propre; David A. Cox (ii) le k-schéma X k est propre; (iii) l’éventail Σ est complet. 1970–1988 Demazure KKMS Other Early Papers C OROLLAIRE 1. The Russian School Polytopes Suppose Σ complet et soit n ∈ Z | Σ | . Les conditions Other Developments Some Quotes suivantes sont équivalentes: Since 1988 (i) L n est très ample; Secondary Fan Homogeneous Coordinates (ii) L n est ample (i.e. L ⊗ m est très ample pour m assez n Mirror Symmetry Survey Papers grand); The 21st Century May 2011 (iii) pour tout élément maximal K de Σ , l’unique m K de M Conclusion tel que � ρ , m K � = − n ρ pour ρ ∈ K est tel que � ρ , m K � > − n ρ pour ρ ∈ | Σ | , ρ / ∈ K.

Some Results Snapshots P ROPOSITION 4. from the History of Soit k un corps. Les conditions suivantes sont équivalentes: Toric Geometry (i) le Z -schéma X est propre; David A. Cox (ii) le k-schéma X k est propre; (iii) l’éventail Σ est complet. 1970–1988 Demazure KKMS Other Early Papers C OROLLAIRE 1. The Russian School Polytopes Suppose Σ complet et soit n ∈ Z | Σ | . Les conditions Other Developments Some Quotes suivantes sont équivalentes: Since 1988 (i) L n est très ample; Secondary Fan Homogeneous Coordinates (ii) L n est ample (i.e. L ⊗ m est très ample pour m assez n Mirror Symmetry Survey Papers grand); The 21st Century May 2011 (iii) pour tout élément maximal K de Σ , l’unique m K de M Conclusion tel que � ρ , m K � = − n ρ pour ρ ∈ K est tel que � ρ , m K � > − n ρ pour ρ ∈ | Σ | , ρ / ∈ K.

Kempf, Knudsen, Mumford, Saint-Donat 1975 Snapshots Introduction from the History of The goal of these notes is to formalize and Toric Geometry illustrate the power of a technique which has David A. Cox cropped up independently in the work of at least a dozen people, ... When teaching algebraic geometry 1970–1988 Demazure and illustrating simple singularities, varieties, KKMS Other Early Papers and morphisms, one almost invariably tends to The Russian School Polytopes choose examples of a “monomial” type: i.e., Other Developments Some Quotes varieties defined by equations Since 1988 Secondary Fan X a 1 r = X a r + 1 Homogeneous 1 ··· X a r r + 1 ··· X a n Coordinates n Mirror Symmetry Survey Papers The 21st Century *) After this was written, I received a paper by K. May 2011 Conclusion Miyake and T. Oda entitled Almost homogeneous algebraic varieties under algebraic torus action also on this topic.

Kempf, Knudsen, Mumford, Saint-Donat 1975 Snapshots Introduction from the History of The goal of these notes is to formalize and Toric Geometry illustrate the power of a technique which has David A. Cox cropped up independently in the work of at least a dozen people, ... When teaching algebraic geometry 1970–1988 Demazure and illustrating simple singularities, varieties, KKMS Other Early Papers and morphisms, one almost invariably tends to The Russian School Polytopes choose examples of a “monomial” type: i.e., Other Developments Some Quotes varieties defined by equations Since 1988 Secondary Fan X a 1 r = X a r + 1 Homogeneous 1 ··· X a r r + 1 ··· X a n Coordinates n Mirror Symmetry Survey Papers The 21st Century *) After this was written, I received a paper by K. May 2011 Conclusion Miyake and T. Oda entitled Almost homogeneous algebraic varieties under algebraic torus action also on this topic.

A Definition and Some Names Snapshots from the Definition 3: History of Toric A finite rational partial polyhedral decomposition Geometry (we abbreviate this to f.r.p.p. decomposition) of David A. Cox N R is a finite set { σ α } of convex rational 1970–1988 polyhedral cones in N R such that: Demazure KKMS (i) if σ is a face of σ α , then σ = σ β for some β Other Early Papers The Russian School (ii) ∀ α , β , σ α ∩ σ β is a face of σ α and σ β . Polytopes Other Developments Some Quotes Since 1988 Some Names Secondary Fan Homogeneous Coordinates • T-equivariant embedding of a torus T Mirror Symmetry Survey Papers • T-space The 21st Century May 2011 • torus embedding Conclusion The last name became standard terminology several years.