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Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Snapshots from the History of Toric Geometry

Toric Geometry and its Applications David A. Cox

Department of Mathematics Amherst College dac@math.amherst.edu

Leuven 6 June 2011

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Outline

1

1970–1988 Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

2

Since 1988 Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Outline

1

1970–1988 Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

2

Since 1988 Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Demazure 1970

Toric varieties such as Cn, (C∗)n, Pn, Pn ×Pm, have been around for a long time. The general definition came in 1970: SOUS-GROUPES ALGÉBRIQUES DE RANG MAXIMUM DU GROUPE DE CREMONA PAR MICHEL DEMAZURE He studied groups of birational automorphisms of Pn: . . . ces schémas en groupes se réalisent commes groupes d’automorphismes de certains Z-schémas à décomposition cellulaire obtenus en “ajoutant à un tore déployé certains points á l’infini”. Un rôle important est joué par les schémas précédents; la manière “ajouter des points á l’infini à un tore” est décrite par un “éventail” . . .

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Demazure 1970

Toric varieties such as Cn, (C∗)n, Pn, Pn ×Pm, have been around for a long time. The general definition came in 1970: SOUS-GROUPES ALGÉBRIQUES DE RANG MAXIMUM DU GROUPE DE CREMONA PAR MICHEL DEMAZURE He studied groups of birational automorphisms of Pn: . . . ces schémas en groupes se réalisent commes groupes d’automorphismes de certains Z-schémas à décomposition cellulaire obtenus en “ajoutant à un tore déployé certains points á l’infini”. Un rôle important est joué par les schémas précédents; la manière “ajouter des points á l’infini à un tore” est décrite par un “éventail” . . .

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Definition and Name

DÉFINITION 1. Soit M∗ un groupe abélien libre de type fini. On appelle éventail dans M∗ un ensemble fini Σ de parties de M∗ tel que:

• a. chaque élément de Σ est une partie d’une base de M∗;
• b. toute partie d’un élément de Σ appartient á Σ;
• c. si K,L ∈ Σ, on a N.K ∩N.L = N.(K ∩L).

DÉFINITION 2. On appelle schéma défini par l’éventail Σ le schéma X

• btenu part recollement des VK, K parcourant Σ, á l’aide

des immersions ouvertes VK∩L → VK, VK∩L → VL, pour K,L ∈ Σ.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Definition and Name

DÉFINITION 1. Soit M∗ un groupe abélien libre de type fini. On appelle éventail dans M∗ un ensemble fini Σ de parties de M∗ tel que:

• a. chaque élément de Σ est une partie d’une base de M∗;
• b. toute partie d’un élément de Σ appartient á Σ;
• c. si K,L ∈ Σ, on a N.K ∩N.L = N.(K ∩L).

DÉFINITION 2. On appelle schéma défini par l’éventail Σ le schéma X

• btenu part recollement des VK, K parcourant Σ, á l’aide

des immersions ouvertes VK∩L → VK, VK∩L → VL, pour K,L ∈ Σ.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Some Results

PROPOSITION 4. Soit k un corps. Les conditions suivantes sont équivalentes: (i) le Z-schéma X est propre; (ii) le k-schéma Xk est propre; (iii) l’éventail Σ est complet. COROLLAIRE 1. Suppose Σ complet et soit n ∈ Z|Σ|. Les conditions suivantes sont équivalentes: (i) Ln est très ample; (ii) Ln est ample (i.e. L ⊗m

n

est très ample pour m assez grand); (iii) pour tout élément maximal K de Σ, l’unique mK de M tel que ρ,mK = −nρ pour ρ ∈ K est tel que ρ,mK > −nρ pour ρ ∈ |Σ|, ρ / ∈ K.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Some Results

PROPOSITION 4. Soit k un corps. Les conditions suivantes sont équivalentes: (i) le Z-schéma X est propre; (ii) le k-schéma Xk est propre; (iii) l’éventail Σ est complet. COROLLAIRE 1. Suppose Σ complet et soit n ∈ Z|Σ|. Les conditions suivantes sont équivalentes: (i) Ln est très ample; (ii) Ln est ample (i.e. L ⊗m

n

est très ample pour m assez grand); (iii) pour tout élément maximal K de Σ, l’unique mK de M tel que ρ,mK = −nρ pour ρ ∈ K est tel que ρ,mK > −nρ pour ρ ∈ |Σ|, ρ / ∈ K.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Kempf, Knudsen, Mumford, Saint-Donat 1975

Introduction The goal of these notes is to formalize and illustrate the power of a technique which has cropped up independently in the work of at least a dozen people, ... When teaching algebraic geometry and illustrating simple singularities, varieties, and morphisms, one almost invariably tends to choose examples of a “monomial” type: i.e., varieties defined by equations Xa1

1 ···Xar r = Xar+1 r+1 ···Xan n

*) After this was written, I received a paper by K. Miyake and T. Oda entitled Almost homogeneous algebraic varieties under algebraic torus action also on this topic.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Kempf, Knudsen, Mumford, Saint-Donat 1975

Introduction The goal of these notes is to formalize and illustrate the power of a technique which has cropped up independently in the work of at least a dozen people, ... When teaching algebraic geometry and illustrating simple singularities, varieties, and morphisms, one almost invariably tends to choose examples of a “monomial” type: i.e., varieties defined by equations Xa1

1 ···Xar r = Xar+1 r+1 ···Xan n

*) After this was written, I received a paper by K. Miyake and T. Oda entitled Almost homogeneous algebraic varieties under algebraic torus action also on this topic.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

A Definition and Some Names

Definition 3: A finite rational partial polyhedral decomposition (we abbreviate this to f.r.p.p. decomposition) of NR is a finite set {σα} of convex rational polyhedral cones in NR such that: (i) if σ is a face of σα, then σ = σβ for some β (ii) ∀ α,β, σα ∩σβ is a face of σα and σβ. Some Names

• T-equivariant embedding of a torus T
• T-space
• torus embedding

The last name became standard terminology several years.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

A Definition and Some Names

Definition 3: A finite rational partial polyhedral decomposition (we abbreviate this to f.r.p.p. decomposition) of NR is a finite set {σα} of convex rational polyhedral cones in NR such that: (i) if σ is a face of σα, then σ = σβ for some β (ii) ∀ α,β, σα ∩σβ is a face of σα and σβ. Some Names

• T-equivariant embedding of a torus T
• T-space
• torus embedding

The last name became standard terminology several years.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

The First Picture

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Oda and Miyake 1975

From the Introduction: An almost homogeneous variety under the action of T is an algebraic variety X over k endowed with an action of T and which has a dense orbit. The dense orbit is open. Demazure [1] studied non-singular ones associated to a “fan” in connection with algebraic subgroups of the Cremona group. His “fan” is our complex of cones in the non-singular case. Our result says, in particular, that, conversely, a non-singular almost homogeneous variety is always associated to a Demazure fan. We learned recently that our Theorem 6 was also obtained by Mumford [8].

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

A Great Picture

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Hochster and Ehlers

Hochster 1972 Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes THEOREM 1. Let M be a normal semigroup of monomials in variables x1,...,xn. Then R[M] is Cohen-Macaulay for every Cohen-Macaulay ring R. Ehlers 1975 Eine Klasse komplexer Mannigfaltigkeiten und die Auflösung einiger isolierter Singularitäten SATZ 1. Sei Σ ein Komplex in (E,M). Dann ist XΣ kompakt genau dann, wenn |Σ|(=

σ∈Σ σ) = E ist.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Hochster and Ehlers

Hochster 1972 Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes THEOREM 1. Let M be a normal semigroup of monomials in variables x1,...,xn. Then R[M] is Cohen-Macaulay for every Cohen-Macaulay ring R. Ehlers 1975 Eine Klasse komplexer Mannigfaltigkeiten und die Auflösung einiger isolierter Singularitäten SATZ 1. Sei Σ ein Komplex in (E,M). Dann ist XΣ kompakt genau dann, wenn |Σ|(=

σ∈Σ σ) = E ist.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

The Russian School

In the mid 1970s, Bernstein, Khovanskii and Kusnirenko studied subvarieties of (C∗)n defined by the vanishing of Laurent polynomials fi. For example, the number of solutions of a generic system f1 = ··· = fn = 0 is given by the mixed volume MV(P1,P2,,...,Pn), where Pi is the Newton polytope of fi. Khovanskii studied toric varieties in 1977. His paper is notable for several reasons: It introduced support functions in the toric context. It proved the Demazure vanishing theorem Hp(X,OX(D)) = 0 for p > 0 when D is basepoint free. It gave the first toric proof of the properties of the Ehrhart polynomial.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

The Russian School

In the mid 1970s, Bernstein, Khovanskii and Kusnirenko studied subvarieties of (C∗)n defined by the vanishing of Laurent polynomials fi. For example, the number of solutions of a generic system f1 = ··· = fn = 0 is given by the mixed volume MV(P1,P2,,...,Pn), where Pi is the Newton polytope of fi. Khovanskii studied toric varieties in 1977. His paper is notable for several reasons: It introduced support functions in the toric context. It proved the Demazure vanishing theorem Hp(X,OX(D)) = 0 for p > 0 when D is basepoint free. It gave the first toric proof of the properties of the Ehrhart polynomial.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

The Russian School

In the mid 1970s, Bernstein, Khovanskii and Kusnirenko studied subvarieties of (C∗)n defined by the vanishing of Laurent polynomials fi. For example, the number of solutions of a generic system f1 = ··· = fn = 0 is given by the mixed volume MV(P1,P2,,...,Pn), where Pi is the Newton polytope of fi. Khovanskii studied toric varieties in 1977. His paper is notable for several reasons: It introduced support functions in the toric context. It proved the Demazure vanishing theorem Hp(X,OX(D)) = 0 for p > 0 when D is basepoint free. It gave the first toric proof of the properties of the Ehrhart polynomial.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

The Name

Khovanskii 1977

Mnogogranniki Nьtona i toriqeskie mnogoobrazi.

This was translated as Newton polyhedra and toroidal varieties, but “toroidal” is not the right word for

toriqeskie (toricheskie), because the toroidal varieties

are slightly different from toric varieties. Danilov 1979

Geometri toriqeskih mnogoobrazi i.

One translation was Geometry of toral varieties. Yuck! Miles Reid translated the paper into English for the Russian Math Surveys as the Geometry of toric varieties. This is the

• rigin of the name “toric variety”.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

The Name

Khovanskii 1977

Mnogogranniki Nьtona i toriqeskie mnogoobrazi.

This was translated as Newton polyhedra and toroidal varieties, but “toroidal” is not the right word for

toriqeskie (toricheskie), because the toroidal varieties

are slightly different from toric varieties. Danilov 1979

Geometri toriqeskih mnogoobrazi i.

One translation was Geometry of toral varieties. Yuck! Miles Reid translated the paper into English for the Russian Math Surveys as the Geometry of toric varieties. This is the

• rigin of the name “toric variety”.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

The Name

Khovanskii 1977

Mnogogranniki Nьtona i toriqeskie mnogoobrazi.

This was translated as Newton polyhedra and toroidal varieties, but “toroidal” is not the right word for

toriqeskie (toricheskie), because the toroidal varieties

are slightly different from toric varieties. Danilov 1979

Geometri toriqeskih mnogoobrazi i.

One translation was Geometry of toral varieties. Yuck! Miles Reid translated the paper into English for the Russian Math Surveys as the Geometry of toric varieties. This is the

• rigin of the name “toric variety”.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

The Name

Khovanskii 1977

Mnogogranniki Nьtona i toriqeskie mnogoobrazi.

This was translated as Newton polyhedra and toroidal varieties, but “toroidal” is not the right word for

toriqeskie (toricheskie), because the toroidal varieties

are slightly different from toric varieties. Danilov 1979

Geometri toriqeskih mnogoobrazi i.

One translation was Geometry of toral varieties. Yuck! Miles Reid translated the paper into English for the Russian Math Surveys as the Geometry of toric varieties. This is the

• rigin of the name “toric variety”.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Teissier and Khovanskii 1979

Recall that MV(P1,...,Pn) is the mixed volume of polytopes P1,...,Pn. Alexandrov-Fenchel Inequality MV(P1,P2,P3,...,Pn)2 ≥ MV(P1,P1,P3 ...,Pn)MV(P2,P2,P3,...,Pn) Proof

• Construct a toric variety X such that each Pi gives a

divisor Di on X.

• Interpret the mixed volume MV(P1,...,Pn) as an

intersection product of the Di.

• Apply the Hodge Index Theorem.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Teissier and Khovanskii 1979

Recall that MV(P1,...,Pn) is the mixed volume of polytopes P1,...,Pn. Alexandrov-Fenchel Inequality MV(P1,P2,P3,...,Pn)2 ≥ MV(P1,P1,P3 ...,Pn)MV(P2,P2,P3,...,Pn) Proof

• Construct a toric variety X such that each Pi gives a

divisor Di on X.

• Interpret the mixed volume MV(P1,...,Pn) as an

intersection product of the Di.

• Apply the Hodge Index Theorem.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

McMullen Conjecture

Let fi be the number of i-dimensional faces of an n-dimensional simplicial polytope P. Define hi = ∑i

j=0(−1)i−jn−j n−i

• fj−1.

The hi satisfy the Dehn-Sommerville equations hi = hn−i, 0 ≤ i ≤ n, and in 1971 McMullen conjectured that hi −hi−1 ≥ 0, 1 ≤ i ≤ ⌊n

2⌋,

and that if hi −hi−1 = ni

i

• +

ni−1

i−1

• +···+

nr

r

• with 1 ≤ i ≤ ⌊n

2⌋−1 and ni > ni−1 > ··· > nr ≥ r ≥ 1, then

hi+1 −hi ≤ ni+1

i+1

• +

ni−1+1

i

• +···+

nr +1

r+1

• .

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Stanley 1980

Let X be the simplicial toric variety whose fan consists of cones over the faces of P. Since X is a projective orbifold, we have: hi = dimH2i(X,Q) = dimIH2i(X,Q). Poincaré duality for IH ⇒ Dehn-Sommerville. Hard Lefschetz1 for IH ⇒ hi −hi−1 ≥ 0 for 1 ≤ i ≤ ⌊n

2⌋.

Stanley’s 1980 paper in Advances is three pages long:

• The first page recalls the conjecture.
• The third page is mostly references.
• A one-page proof!

1 Hard Lefschetz for intersection cohomology was not fully

proved until 1990.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Stanley 1980

Let X be the simplicial toric variety whose fan consists of cones over the faces of P. Since X is a projective orbifold, we have: hi = dimH2i(X,Q) = dimIH2i(X,Q). Poincaré duality for IH ⇒ Dehn-Sommerville. Hard Lefschetz1 for IH ⇒ hi −hi−1 ≥ 0 for 1 ≤ i ≤ ⌊n

2⌋.

Stanley’s 1980 paper in Advances is three pages long:

• The first page recalls the conjecture.
• The third page is mostly references.
• A one-page proof!

1 Hard Lefschetz for intersection cohomology was not fully

proved until 1990.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Stanley 1980

Let X be the simplicial toric variety whose fan consists of cones over the faces of P. Since X is a projective orbifold, we have: hi = dimH2i(X,Q) = dimIH2i(X,Q). Poincaré duality for IH ⇒ Dehn-Sommerville. Hard Lefschetz1 for IH ⇒ hi −hi−1 ≥ 0 for 1 ≤ i ≤ ⌊n

2⌋.

Stanley’s 1980 paper in Advances is three pages long:

• The first page recalls the conjecture.
• The third page is mostly references.
• A one-page proof!

1 Hard Lefschetz for intersection cohomology was not fully

proved until 1990.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Stanley 1980

Let X be the simplicial toric variety whose fan consists of cones over the faces of P. Since X is a projective orbifold, we have: hi = dimH2i(X,Q) = dimIH2i(X,Q). Poincaré duality for IH ⇒ Dehn-Sommerville. Hard Lefschetz1 for IH ⇒ hi −hi−1 ≥ 0 for 1 ≤ i ≤ ⌊n

2⌋.

Stanley’s 1980 paper in Advances is three pages long:

• The first page recalls the conjecture.
• The third page is mostly references.
• A one-page proof!

1 Hard Lefschetz for intersection cohomology was not fully

proved until 1990.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Stanley 1980

Let X be the simplicial toric variety whose fan consists of cones over the faces of P. Since X is a projective orbifold, we have: hi = dimH2i(X,Q) = dimIH2i(X,Q). Poincaré duality for IH ⇒ Dehn-Sommerville. Hard Lefschetz1 for IH ⇒ hi −hi−1 ≥ 0 for 1 ≤ i ≤ ⌊n

2⌋.

Stanley’s 1980 paper in Advances is three pages long:

• The first page recalls the conjecture.
• The third page is mostly references.
• A one-page proof!

1 Hard Lefschetz for intersection cohomology was not fully

proved until 1990.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Some Other Developments

Reid 1983 Decomposition of toric morphisms Danilov and Khovanskii 1986 Newton polyhedra and an algorithm for calculating Hodge-Deligne numbers Kleinschmidt 1988 A classification of toric varieties with few generators Oda 1988 Convex Bodies and Algebraic Geometry Fulton lectures in St. Louis 1989, published 1993 Introduction to Toric Varieties

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Some Other Developments

Reid 1983 Decomposition of toric morphisms Danilov and Khovanskii 1986 Newton polyhedra and an algorithm for calculating Hodge-Deligne numbers Kleinschmidt 1988 A classification of toric varieties with few generators Oda 1988 Convex Bodies and Algebraic Geometry Fulton lectures in St. Louis 1989, published 1993 Introduction to Toric Varieties

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Some Other Developments

Reid 1983 Decomposition of toric morphisms Danilov and Khovanskii 1986 Newton polyhedra and an algorithm for calculating Hodge-Deligne numbers Kleinschmidt 1988 A classification of toric varieties with few generators Oda 1988 Convex Bodies and Algebraic Geometry Fulton lectures in St. Louis 1989, published 1993 Introduction to Toric Varieties

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Some Other Developments

Reid 1983 Decomposition of toric morphisms Danilov and Khovanskii 1986 Newton polyhedra and an algorithm for calculating Hodge-Deligne numbers Kleinschmidt 1988 A classification of toric varieties with few generators Oda 1988 Convex Bodies and Algebraic Geometry Fulton lectures in St. Louis 1989, published 1993 Introduction to Toric Varieties

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Some Other Developments

Reid 1983 Decomposition of toric morphisms Danilov and Khovanskii 1986 Newton polyhedra and an algorithm for calculating Hodge-Deligne numbers Kleinschmidt 1988 A classification of toric varieties with few generators Oda 1988 Convex Bodies and Algebraic Geometry Fulton lectures in St. Louis 1989, published 1993 Introduction to Toric Varieties

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

My Favorite Quotes

Reid 1983 This construction has been of considerable use within algebraic geometry in the last 10 years . . . and has also been amazingly successful as a tool of algebro-geometric imperialism, infiltrating areas of combinatorics. Oda 1988 The theory of toric varieties . . . relates algebraic geometry to the geometry of convex figures in real affine spaces. Ever since the the foundations of the theory were laid down in the 1970’s, tremendous progress has been made and various applications have been found. Fulton 1993 . . . toric varieties have provided a remarkably fertile testing ground for general theories

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

My Favorite Quotes

Reid 1983 This construction has been of considerable use within algebraic geometry in the last 10 years . . . and has also been amazingly successful as a tool of algebro-geometric imperialism, infiltrating areas of combinatorics. Oda 1988 The theory of toric varieties . . . relates algebraic geometry to the geometry of convex figures in real affine spaces. Ever since the the foundations of the theory were laid down in the 1970’s, tremendous progress has been made and various applications have been found. Fulton 1993 . . . toric varieties have provided a remarkably fertile testing ground for general theories

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

My Favorite Quotes

Reid 1983 This construction has been of considerable use within algebraic geometry in the last 10 years . . . and has also been amazingly successful as a tool of algebro-geometric imperialism, infiltrating areas of combinatorics. Oda 1988 The theory of toric varieties . . . relates algebraic geometry to the geometry of convex figures in real affine spaces. Ever since the the foundations of the theory were laid down in the 1970’s, tremendous progress has been made and various applications have been found. Fulton 1993 . . . toric varieties have provided a remarkably fertile testing ground for general theories

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

An Explosion

Many new ideas and applications entered toric geometry in the last decade of the 20th century, including: Nonnormal toric varieties Secondary fans Discriminants and resultants A-Hypergeometric functions Homogeneous coordinates Mirror Symmetry

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

An Explosion

Many new ideas and applications entered toric geometry in the last decade of the 20th century, including: Nonnormal toric varieties Secondary fans Discriminants and resultants A-Hypergeometric functions Homogeneous coordinates Mirror Symmetry

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

An Explosion

Many new ideas and applications entered toric geometry in the last decade of the 20th century, including: Nonnormal toric varieties Secondary fans Discriminants and resultants A-Hypergeometric functions Homogeneous coordinates Mirror Symmetry

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

An Explosion

Many new ideas and applications entered toric geometry in the last decade of the 20th century, including: Nonnormal toric varieties Secondary fans Discriminants and resultants A-Hypergeometric functions Homogeneous coordinates Mirror Symmetry

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

An Explosion

Many new ideas and applications entered toric geometry in the last decade of the 20th century, including: Nonnormal toric varieties Secondary fans Discriminants and resultants A-Hypergeometric functions Homogeneous coordinates Mirror Symmetry

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

An Explosion

Many new ideas and applications entered toric geometry in the last decade of the 20th century, including: Nonnormal toric varieties Secondary fans Discriminants and resultants A-Hypergeometric functions Homogeneous coordinates Mirror Symmetry

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

The Secondary Fan

Gelfand, Kapranov and Zelevinsky 1989 Newton polyhedra of principal A-determinants Billera, Filliman and Sturmfels 1990 Constructions and complexity of secondary polytopes Oda and Park 1991 Linear Gale transforms and Gelfand-Kapranov-Zelevinskij decompositions Gelfand, Kapranov and Zelevinsky 1994 Discriminants, Resultants and Multidimensional Determinants Sturmfels 1996 Gröbner Bases and Convex Polytopes

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

The Secondary Fan

Gelfand, Kapranov and Zelevinsky 1989 Newton polyhedra of principal A-determinants Billera, Filliman and Sturmfels 1990 Constructions and complexity of secondary polytopes Oda and Park 1991 Linear Gale transforms and Gelfand-Kapranov-Zelevinskij decompositions Gelfand, Kapranov and Zelevinsky 1994 Discriminants, Resultants and Multidimensional Determinants Sturmfels 1996 Gröbner Bases and Convex Polytopes

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

The Secondary Fan

Gelfand, Kapranov and Zelevinsky 1989 Newton polyhedra of principal A-determinants Billera, Filliman and Sturmfels 1990 Constructions and complexity of secondary polytopes Oda and Park 1991 Linear Gale transforms and Gelfand-Kapranov-Zelevinskij decompositions Gelfand, Kapranov and Zelevinsky 1994 Discriminants, Resultants and Multidimensional Determinants Sturmfels 1996 Gröbner Bases and Convex Polytopes

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

The Secondary Fan

Gelfand, Kapranov and Zelevinsky 1989 Newton polyhedra of principal A-determinants Billera, Filliman and Sturmfels 1990 Constructions and complexity of secondary polytopes Oda and Park 1991 Linear Gale transforms and Gelfand-Kapranov-Zelevinskij decompositions Gelfand, Kapranov and Zelevinsky 1994 Discriminants, Resultants and Multidimensional Determinants Sturmfels 1996 Gröbner Bases and Convex Polytopes

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

The Secondary Fan

Gelfand, Kapranov and Zelevinsky 1989 Newton polyhedra of principal A-determinants Billera, Filliman and Sturmfels 1990 Constructions and complexity of secondary polytopes Oda and Park 1991 Linear Gale transforms and Gelfand-Kapranov-Zelevinskij decompositions Gelfand, Kapranov and Zelevinsky 1994 Discriminants, Resultants and Multidimensional Determinants Sturmfels 1996 Gröbner Bases and Convex Polytopes

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

An Example

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Homogeneous Coordinates

The quotient construction Pn = (Cn+1 \{0})/C∗ applies to any toric variety. Discovers

• Audin/Delzant/Kirwan (Symplectic geometry)
• Batyrev (Quantum cohomology)
• Cox (Primitive cohomology of hypersurfaces)
• Krasauskas (Geometric modeling)
• Musson (Differential operators)

The quotient construction involves a multigraded polynomial

• ring. Hence toric geometry has three types of objects:
• Geometric: Toric varieties
• Combinatorial: Fan and polytopes
• Algebraic: Toric ideals, total coordinate rings (Cox rings)

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Homogeneous Coordinates

The quotient construction Pn = (Cn+1 \{0})/C∗ applies to any toric variety. Discovers

• Audin/Delzant/Kirwan (Symplectic geometry)
• Batyrev (Quantum cohomology)
• Cox (Primitive cohomology of hypersurfaces)
• Krasauskas (Geometric modeling)
• Musson (Differential operators)

The quotient construction involves a multigraded polynomial

• ring. Hence toric geometry has three types of objects:
• Geometric: Toric varieties
• Combinatorial: Fan and polytopes
• Algebraic: Toric ideals, total coordinate rings (Cox rings)

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Mirror Symmetry

Candelas, de la Ossa, Green and Parkes 1991 A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory Witten 1993 Phases of N = 2 theories in two dimensions Batyrev 1994 Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties Aspinwall, Greene and Morrison 1994 Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Mirror Symmetry

Candelas, de la Ossa, Green and Parkes 1991 A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory Witten 1993 Phases of N = 2 theories in two dimensions Batyrev 1994 Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties Aspinwall, Greene and Morrison 1994 Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Mirror Symmetry

Candelas, de la Ossa, Green and Parkes 1991 A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory Witten 1993 Phases of N = 2 theories in two dimensions Batyrev 1994 Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties Aspinwall, Greene and Morrison 1994 Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Mirror Symmetry

Candelas, de la Ossa, Green and Parkes 1991 A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory Witten 1993 Phases of N = 2 theories in two dimensions Batyrev 1994 Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties Aspinwall, Greene and Morrison 1994 Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Four Survey Papers

Oda 1989 Geometry of toric varieties, 114 references Oda 1994 Recent topics on toric varieties, 64 references Cox 1997 Recent developments in toric geometry, 157 references Cox 2001 Update on toric geometry, 240 references

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Four Survey Papers

Oda 1989 Geometry of toric varieties, 114 references Oda 1994 Recent topics on toric varieties, 64 references Cox 1997 Recent developments in toric geometry, 157 references Cox 2001 Update on toric geometry, 240 references

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Four Survey Papers

Oda 1989 Geometry of toric varieties, 114 references Oda 1994 Recent topics on toric varieties, 64 references Cox 1997 Recent developments in toric geometry, 157 references Cox 2001 Update on toric geometry, 240 references

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Four Survey Papers

Oda 1989 Geometry of toric varieties, 114 references Oda 1994 Recent topics on toric varieties, 64 references Cox 1997 Recent developments in toric geometry, 157 references Cox 2001 Update on toric geometry, 240 references

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

The 21st Century

Another explosion! Toric stacks T-varieties Tropical geometry Algebraic statistics Phylogenetic models Geometric modeling Toric codes

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

The 21st Century

Another explosion! Toric stacks T-varieties Tropical geometry Algebraic statistics Phylogenetic models Geometric modeling Toric codes

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

The 21st Century

Another explosion! Toric stacks T-varieties Tropical geometry Algebraic statistics Phylogenetic models Geometric modeling Toric codes

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

The 21st Century

Another explosion! Toric stacks T-varieties Tropical geometry Algebraic statistics Phylogenetic models Geometric modeling Toric codes

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

The 21st Century

Another explosion! Toric stacks T-varieties Tropical geometry Algebraic statistics Phylogenetic models Geometric modeling Toric codes

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

The 21st Century

Another explosion! Toric stacks T-varieties Tropical geometry Algebraic statistics Phylogenetic models Geometric modeling Toric codes

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

The 21st Century

Another explosion! Toric stacks T-varieties Tropical geometry Algebraic statistics Phylogenetic models Geometric modeling Toric codes

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Papers from May 2011

To give you a better idea of

• the level of activity in toric geometry, and
• the range of topics in toric geometry,

I will show some (not all!) papers from May 2011. I learned about these papers from:

• the Journal of Symbolic Computation (one paper)
• the arXiv (the rest).

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Journal of Symbolic Computation 46 (2011)

Castryck and Vercauteren Toric forms of elliptic curves and their arithmetic Abstract: We scan a large class of one-parameter families of elliptic curves for efficient arithmetic. The construction of the class is inspired by toric geometry, which provides a natural framework for the study of various forms of elliptic curves. The class both encompasses many prominent known forms and includes thousands of new forms. A powerful algorithm is described that automatically computes the most compact group operation formulas for any parameterized family of elliptic curves. . . .

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

3 May 2011 math.AG (math.CO)

Cueto Implicitization of surfaces via geometric tropicalization Abstract: In this paper we describe tropical methods for implicitization

• f surfaces. We construct the corresponding tropical

surfaces via the theory of geometric tropicalization due to Hacking, Keel and Tevelev, which we enrich with a formula for computing tropical multiplicities of regular points in any

• dimension. We extend previous results for tropical

implicitization of generic surfaces due to Sturmfels, Tevelev and Yu and provide methods for the non-generic case.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

5 May 2011 math.AG (physics.hep-th)

Gasparim, Köppe, Majumdar and Ray BPS state counting on singular varieties Abstract: We define new partition functions for theories with targets

• n toric singularities via products of old partition functions
• n crepant resolutions. We compute explicit examples and

show that the new partition functions turn out to be homogeneous on MacMahon factors.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

12 May 2011 math.CO (math.AG)

Di Rocco, Haase, Nill and Paffenholz Polyhedral adjunction theory Abstract: In this paper we give a combinatorial view on the adjunction theory of toric varieties. Inspired by classical adjunction theory of polarized algebraic varieties we define two convex-geometric notions: the Q-codegree and the nef value of a rational polytope P. We define the adjoint polytope P(s) as the set of those points in P, whose lattice distance to every facet of P is at least s. We prove a structure theorem for lattice polytopes with high Q-codegree. . . . Moreover, we illustrate how classification results in adjunction theory can be translated into new classification results for lattice polytopes.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

18 May 2011 math.AC (math.CO)

Olteanu Monomial cut ideals Abstract:

• B. Sturmfels and S. Sullivant associated to any graph a toric

ideal, called the cut ideal. We consider monomial cut ideals and we show that their algebraic properties such as the minimal primary decomposition, the property of having a linear resolution or being Cohen–Macaulay may be derived from the combinatorial structure of the graph.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

22 May 2011 math.AC

Ohsugi and Hibi Centrally symmetric configurations of integer matrices Abstract: The concept of centrally symmetric configurations of integer matrices is introduced. We study the problem when the toric ring of a centrally symmetric configuration is normal as well as is Gorenstein. In addition, Gröbner bases of toric ideals

• f centrally symmetric configurations will be discussed.

Special attentions will be given to centrally symmetric configurations of unimodular matrices and those of incidence matrices of finite graphs.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

25 May 2011 math.CO (math.AG)

Joswig and Paffenholz Defect polytopes and counter-examples with polymake Abstract: It is demonstrated how the software system polymake can be used for computations in toric geometry. More precisely, counter-examples to conjectures related to A-determinants and defect polytopes are constructed.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

27 May 2011 math.AG (math.CO, math.NT)

Burgos Gil, Philippon and Sombra Arithmetic geometry of toric varieties: Metrics, measures and heights Abstract: We show that the height of a toric variety with respect to a toric metrized line bundle can be expressed as the integral

• ver a polytope of a certain adelic family of concave
• functions. . . . We also present a closed formula for the

integral over a polytope of a function of one variable composed with a linear form. This allows us to compute the height of toric varieties with respect to some interesting metrics arising from polytopes. We also compute the height

• f toric projective curves with respect to the Fubini-Study

metric, and of some toric bundles.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

31 May 2011 math.AG (math.SG)

Hochenegger and Witt On complex and symplectic toric stacks Abstract: Toric varieties play an important role both in symplectic and complex geometry. In symplectic geometry, the construction

• f a symplectic toric manifold from a smooth polytope is due

to Delzant. . . . For rational but not necessarily smooth polytopes the Delzant construction was refined by Lerman and Tolman, leading to symplectic toric orbifolds or more generally, symplectic toric DM stacks (Lerman and Malkin). . . . we hope that this text serves as an example driven introduction to symplectic toric geometry for the algebraically minded reader.

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Conclusion

Toric geometry has

• a rich history,
• a diverse community, and
• a bright future.

This week, we will get a glimpse of what lies ahead.

Let’s have fun with Toric Geometry and Applications!

Snapshots from the History of Toric Geometry David A. Cox 1970–1988

Demazure KKMS Other Early Papers The Russian School Polytopes Other Developments Some Quotes

Since 1988

Secondary Fan Homogeneous Coordinates Mirror Symmetry Survey Papers The 21st Century May 2011 Conclusion

Conclusion

Toric geometry has

• a rich history,
• a diverse community, and
• a bright future.

This week, we will get a glimpse of what lies ahead.

Let’s have fun with Toric Geometry and Applications!