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Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
Snapshots from the History of ToricGeometry
Toric Geometry and its Applications
David A. Cox
Department of MathematicsAmherst College
dac@math.amherst.edu
Leuven6 June 2011
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
Outline
1 1970–1988DemazureKKMSOther Early PapersThe Russian SchoolPolytopesOther DevelopmentsSome Quotes
2 Since 1988Secondary FanHomogeneous CoordinatesMirror SymmetrySurvey PapersThe 21st CenturyMay 2011Conclusion
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
Outline
1 1970–1988DemazureKKMSOther Early PapersThe Russian SchoolPolytopesOther DevelopmentsSome Quotes
2 Since 1988Secondary FanHomogeneous CoordinatesMirror SymmetrySurvey PapersThe 21st CenturyMay 2011Conclusion
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
Demazure 1970
Toric varieties such as Cn, (C∗)n, Pn, Pn ×Pm, have beenaround for a long time. The general definition came in 1970:
SOUS-GROUPES ALGÉBRIQUESDE RANG MAXIMUM DU GROUPE DE CREMONA
PAR MICHEL DEMAZURE
He studied groups of birational automorphisms of Pn:
. . . ces schémas en groupes se réalisent commes groupesd’automorphismes de certains Z-schémas à décompositioncellulaire obtenus en “ajoutant à un tore déployé certainspoints á l’infini”.
Un rôle important est joué par les schémas précédents; lamanière “ajouter des points á l’infini à un tore” est décritepar un “éventail” . . .
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
Demazure 1970
Toric varieties such as Cn, (C∗)n, Pn, Pn ×Pm, have beenaround for a long time. The general definition came in 1970:
SOUS-GROUPES ALGÉBRIQUESDE RANG MAXIMUM DU GROUPE DE CREMONA
PAR MICHEL DEMAZURE
He studied groups of birational automorphisms of Pn:
. . . ces schémas en groupes se réalisent commes groupesd’automorphismes de certains Z-schémas à décompositioncellulaire obtenus en “ajoutant à un tore déployé certainspoints á l’infini”.
Un rôle important est joué par les schémas précédents; lamanière “ajouter des points á l’infini à un tore” est décritepar un “éventail” . . .
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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 appelleéventail dans M∗ un ensemble fini Σ de parties de M∗ telque: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 Xobtenu part recollement des VK , K parcourant Σ, á l’aidedes immersions ouvertes VK∩L → VK , VK∩L → VL, pourK ,L ∈ Σ.
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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 appelleéventail dans M∗ un ensemble fini Σ de parties de M∗ telque: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 Xobtenu part recollement des VK , K parcourant Σ, á l’aidedes immersions ouvertes VK∩L → VK , VK∩L → VL, pourK ,L ∈ Σ.
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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 conditionssuivantes sont équivalentes:
(i) Ln est très ample;(ii) Ln est ample (i.e. L
⊗mn 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 .
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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 conditionssuivantes sont équivalentes:
(i) Ln est très ample;(ii) Ln est ample (i.e. L
⊗mn 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 .
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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= X
ar+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.
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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= X
ar+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.
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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.
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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.
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
The First Picture
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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 isan algebraic variety X over k endowed with an action of Tand 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 theCremona group. His “fan” is our complex of cones in thenon-singular case. Our result says, in particular, that,conversely, a non-singular almost homogeneous variety isalways associated to a Demazure fan. We learned recentlythat our Theorem 6 was also obtained by Mumford [8].
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
A Great Picture
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
Hochster and Ehlers
Hochster 1972
Rings of invariants of tori, Cohen-Macaulay rings generatedby monomials, and polytopes
THEOREM 1. Let M be a normal semigroup of monomialsin variables x1, . . . ,xn. Then R[M] is Cohen-Macaulay forevery Cohen-Macaulay ring R.
Ehlers 1975
Eine Klasse komplexer Mannigfaltigkeiten und dieAuflösung 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.
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
Hochster and Ehlers
Hochster 1972
Rings of invariants of tori, Cohen-Macaulay rings generatedby monomials, and polytopes
THEOREM 1. Let M be a normal semigroup of monomialsin variables x1, . . . ,xn. Then R[M] is Cohen-Macaulay forevery Cohen-Macaulay ring R.
Ehlers 1975
Eine Klasse komplexer Mannigfaltigkeiten und dieAuflösung 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.
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
The Russian School
In the mid 1970s, Bernstein, Khovanskii and Kusnirenkostudied subvarieties of (C∗)n defined by the vanishing ofLaurent polynomials fi . For example, the number ofsolutions of a generic system f1 = · · · = fn = 0 is given by themixed volume
MV (P1,P2, , . . . ,Pn),
where Pi is the Newton polytope of fi .
Khovanskii studied toric varieties in 1977. His paper isnotable for several reasons:
It introduced support functions in the toric context.
It proved the Demazure vanishing theoremHp(X ,OX (D)) = 0 for p > 0 when D is basepoint free.
It gave the first toric proof of the properties of theEhrhart polynomial.
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
The Russian School
In the mid 1970s, Bernstein, Khovanskii and Kusnirenkostudied subvarieties of (C∗)n defined by the vanishing ofLaurent polynomials fi . For example, the number ofsolutions of a generic system f1 = · · · = fn = 0 is given by themixed volume
MV (P1,P2, , . . . ,Pn),
where Pi is the Newton polytope of fi .
Khovanskii studied toric varieties in 1977. His paper isnotable for several reasons:
It introduced support functions in the toric context.
It proved the Demazure vanishing theoremHp(X ,OX (D)) = 0 for p > 0 when D is basepoint free.
It gave the first toric proof of the properties of theEhrhart polynomial.
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
The Russian School
In the mid 1970s, Bernstein, Khovanskii and Kusnirenkostudied subvarieties of (C∗)n defined by the vanishing ofLaurent polynomials fi . For example, the number ofsolutions of a generic system f1 = · · · = fn = 0 is given by themixed volume
MV (P1,P2, , . . . ,Pn),
where Pi is the Newton polytope of fi .
Khovanskii studied toric varieties in 1977. His paper isnotable for several reasons:
It introduced support functions in the toric context.
It proved the Demazure vanishing theoremHp(X ,OX (D)) = 0 for p > 0 when D is basepoint free.
It gave the first toric proof of the properties of theEhrhart polynomial.
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
The Name
Khovanskii 1977
Mnogogranniki Nь�tona i toriqeskiemnogoobrazi�.
This was translated as Newton polyhedra and toroidalvarieties, but “toroidal” is not the right word fortoriqeskie (toricheskie), because the toroidal varietiesare slightly different from toric varieties.
Danilov 1979
Geometri� toriqeskih mnogoobrazii.
One translation was Geometry of toral varieties. Yuck!Miles Reid translated the paper into English for the RussianMath Surveys as the Geometry of toric varieties. This is theorigin of the name “toric variety”.
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
The Name
Khovanskii 1977
Mnogogranniki Nь�tona i toriqeskiemnogoobrazi�.
This was translated as Newton polyhedra and toroidalvarieties, but “toroidal” is not the right word fortoriqeskie (toricheskie), because the toroidal varietiesare slightly different from toric varieties.
Danilov 1979
Geometri� toriqeskih mnogoobrazii.
One translation was Geometry of toral varieties. Yuck!Miles Reid translated the paper into English for the RussianMath Surveys as the Geometry of toric varieties. This is theorigin of the name “toric variety”.
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
The Name
Khovanskii 1977
Mnogogranniki Nь�tona i toriqeskiemnogoobrazi�.
This was translated as Newton polyhedra and toroidalvarieties, but “toroidal” is not the right word fortoriqeskie (toricheskie), because the toroidal varietiesare slightly different from toric varieties.
Danilov 1979
Geometri� toriqeskih mnogoobrazii.
One translation was Geometry of toral varieties. Yuck!Miles Reid translated the paper into English for the RussianMath Surveys as the Geometry of toric varieties. This is theorigin of the name “toric variety”.
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
The Name
Khovanskii 1977
Mnogogranniki Nь�tona i toriqeskiemnogoobrazi�.
This was translated as Newton polyhedra and toroidalvarieties, but “toroidal” is not the right word fortoriqeskie (toricheskie), because the toroidal varietiesare slightly different from toric varieties.
Danilov 1979
Geometri� toriqeskih mnogoobrazii.
One translation was Geometry of toral varieties. Yuck!Miles Reid translated the paper into English for the RussianMath Surveys as the Geometry of toric varieties. This is theorigin of the name “toric variety”.
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
Teissier and Khovanskii 1979
Recall that MV (P1, . . . ,Pn) is the mixed volume of polytopesP1, . . . ,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 adivisor Di on X .
• Interpret the mixed volume MV (P1, . . . ,Pn) as anintersection product of the Di .
• Apply the Hodge Index Theorem.
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
Teissier and Khovanskii 1979
Recall that MV (P1, . . . ,Pn) is the mixed volume of polytopesP1, . . . ,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 adivisor Di on X .
• Interpret the mixed volume MV (P1, . . . ,Pn) as anintersection product of the Di .
• Apply the Hodge Index Theorem.
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
McMullen Conjecture
Let fi be the number of i-dimensional faces of ann-dimensional simplicial polytope P. Define
hi = ∑ij=0(−1)i−j(n−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 ≤ ⌊n2⌋,
and that if
hi −hi−1 =(ni
i
)
+(ni−1
i−1
)
+ · · ·+(nr
r
)
with 1 ≤ i ≤ ⌊n2⌋−1 and ni > ni−1 > · · · > nr ≥ r ≥ 1, then
hi+1−hi ≤(ni+1
i+1
)
+(ni−1+1
i
)
+ · · ·+(nr +1
r+1
)
.
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
Stanley 1980
Let X be the simplicial toric variety whose fan consists ofcones over the faces of P. Since X is a projective orbifold,we have:
hi = dimH2i(X ,Q) = dim IH2i(X ,Q).
Poincaré duality for IH ⇒ Dehn-Sommerville.
Hard Lefschetz1 for IH ⇒ hi −hi−1 ≥ 0 for 1 ≤ i ≤ ⌊n2⌋.
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.
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
Stanley 1980
Let X be the simplicial toric variety whose fan consists ofcones over the faces of P. Since X is a projective orbifold,we have:
hi = dimH2i(X ,Q) = dim IH2i(X ,Q).
Poincaré duality for IH ⇒ Dehn-Sommerville.
Hard Lefschetz1 for IH ⇒ hi −hi−1 ≥ 0 for 1 ≤ i ≤ ⌊n2⌋.
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.
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
Stanley 1980
Let X be the simplicial toric variety whose fan consists ofcones over the faces of P. Since X is a projective orbifold,we have:
hi = dimH2i(X ,Q) = dim IH2i(X ,Q).
Poincaré duality for IH ⇒ Dehn-Sommerville.
Hard Lefschetz1 for IH ⇒ hi −hi−1 ≥ 0 for 1 ≤ i ≤ ⌊n2⌋.
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.
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
Stanley 1980
Let X be the simplicial toric variety whose fan consists ofcones over the faces of P. Since X is a projective orbifold,we have:
hi = dimH2i(X ,Q) = dim IH2i(X ,Q).
Poincaré duality for IH ⇒ Dehn-Sommerville.
Hard Lefschetz1 for IH ⇒ hi −hi−1 ≥ 0 for 1 ≤ i ≤ ⌊n2⌋.
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.
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
Stanley 1980
Let X be the simplicial toric variety whose fan consists ofcones over the faces of P. Since X is a projective orbifold,we have:
hi = dimH2i(X ,Q) = dim IH2i(X ,Q).
Poincaré duality for IH ⇒ Dehn-Sommerville.
Hard Lefschetz1 for IH ⇒ hi −hi−1 ≥ 0 for 1 ≤ i ≤ ⌊n2⌋.
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.
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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 calculatingHodge-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
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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 calculatingHodge-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
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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 calculatingHodge-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
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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 calculatingHodge-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
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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 calculatingHodge-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
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
My Favorite Quotes
Reid 1983
This construction has been of considerable use withinalgebraic geometry in the last 10 years . . . and has alsobeen amazingly successful as a tool of algebro-geometricimperialism, infiltrating areas of combinatorics.
Oda 1988
The theory of toric varieties . . . relates algebraic geometryto the geometry of convex figures in real affine spaces. Eversince the the foundations of the theory were laid down in the1970’s, tremendous progress has been made and variousapplications have been found.
Fulton 1993
. . . toric varieties have provided a remarkably fertile testingground for general theories
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
My Favorite Quotes
Reid 1983
This construction has been of considerable use withinalgebraic geometry in the last 10 years . . . and has alsobeen amazingly successful as a tool of algebro-geometricimperialism, infiltrating areas of combinatorics.
Oda 1988
The theory of toric varieties . . . relates algebraic geometryto the geometry of convex figures in real affine spaces. Eversince the the foundations of the theory were laid down in the1970’s, tremendous progress has been made and variousapplications have been found.
Fulton 1993
. . . toric varieties have provided a remarkably fertile testingground for general theories
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
My Favorite Quotes
Reid 1983
This construction has been of considerable use withinalgebraic geometry in the last 10 years . . . and has alsobeen amazingly successful as a tool of algebro-geometricimperialism, infiltrating areas of combinatorics.
Oda 1988
The theory of toric varieties . . . relates algebraic geometryto the geometry of convex figures in real affine spaces. Eversince the the foundations of the theory were laid down in the1970’s, tremendous progress has been made and variousapplications have been found.
Fulton 1993
. . . toric varieties have provided a remarkably fertile testingground for general theories
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
An Explosion
Many new ideas and applications entered toric geometry inthe last decade of the 20th century, including:
Nonnormal toric varieties
Secondary fans
Discriminants and resultants
A-Hypergeometric functions
Homogeneous coordinates
Mirror Symmetry
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
An Explosion
Many new ideas and applications entered toric geometry inthe last decade of the 20th century, including:
Nonnormal toric varieties
Secondary fans
Discriminants and resultants
A-Hypergeometric functions
Homogeneous coordinates
Mirror Symmetry
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
An Explosion
Many new ideas and applications entered toric geometry inthe last decade of the 20th century, including:
Nonnormal toric varieties
Secondary fans
Discriminants and resultants
A-Hypergeometric functions
Homogeneous coordinates
Mirror Symmetry
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
An Explosion
Many new ideas and applications entered toric geometry inthe last decade of the 20th century, including:
Nonnormal toric varieties
Secondary fans
Discriminants and resultants
A-Hypergeometric functions
Homogeneous coordinates
Mirror Symmetry
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
An Explosion
Many new ideas and applications entered toric geometry inthe last decade of the 20th century, including:
Nonnormal toric varieties
Secondary fans
Discriminants and resultants
A-Hypergeometric functions
Homogeneous coordinates
Mirror Symmetry
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
An Explosion
Many new ideas and applications entered toric geometry inthe last decade of the 20th century, including:
Nonnormal toric varieties
Secondary fans
Discriminants and resultants
A-Hypergeometric functions
Homogeneous coordinates
Mirror Symmetry
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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-Zelevinskijdecompositions
Gelfand, Kapranov and Zelevinsky 1994
Discriminants, Resultants and MultidimensionalDeterminants
Sturmfels 1996
Gröbner Bases and Convex Polytopes
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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-Zelevinskijdecompositions
Gelfand, Kapranov and Zelevinsky 1994
Discriminants, Resultants and MultidimensionalDeterminants
Sturmfels 1996
Gröbner Bases and Convex Polytopes
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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-Zelevinskijdecompositions
Gelfand, Kapranov and Zelevinsky 1994
Discriminants, Resultants and MultidimensionalDeterminants
Sturmfels 1996
Gröbner Bases and Convex Polytopes
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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-Zelevinskijdecompositions
Gelfand, Kapranov and Zelevinsky 1994
Discriminants, Resultants and MultidimensionalDeterminants
Sturmfels 1996
Gröbner Bases and Convex Polytopes
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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-Zelevinskijdecompositions
Gelfand, Kapranov and Zelevinsky 1994
Discriminants, Resultants and MultidimensionalDeterminants
Sturmfels 1996
Gröbner Bases and Convex Polytopes
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
An Example
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
Homogeneous Coordinates
The quotient construction Pn = (Cn+1 \{0})/C∗ applies toany 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 polynomialring. Hence toric geometry has three types of objects:
• Geometric: Toric varieties• Combinatorial: Fan and polytopes• Algebraic: Toric ideals, total coordinate rings (Cox rings)
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
Homogeneous Coordinates
The quotient construction Pn = (Cn+1 \{0})/C∗ applies toany 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 polynomialring. Hence toric geometry has three types of objects:
• Geometric: Toric varieties• Combinatorial: Fan and polytopes• Algebraic: Toric ideals, total coordinate rings (Cox rings)
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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 solublesuperconformal theory
Witten 1993
Phases of N = 2 theories in two dimensions
Batyrev 1994
Dual polyhedra and mirror symmetry for Calabi-Yauhypersurfaces in toric varieties
Aspinwall, Greene and Morrison 1994
Calabi-Yau moduli space, mirror manifolds and spacetimetopology change in string theory
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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 solublesuperconformal theory
Witten 1993
Phases of N = 2 theories in two dimensions
Batyrev 1994
Dual polyhedra and mirror symmetry for Calabi-Yauhypersurfaces in toric varieties
Aspinwall, Greene and Morrison 1994
Calabi-Yau moduli space, mirror manifolds and spacetimetopology change in string theory
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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 solublesuperconformal theory
Witten 1993
Phases of N = 2 theories in two dimensions
Batyrev 1994
Dual polyhedra and mirror symmetry for Calabi-Yauhypersurfaces in toric varieties
Aspinwall, Greene and Morrison 1994
Calabi-Yau moduli space, mirror manifolds and spacetimetopology change in string theory
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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 solublesuperconformal theory
Witten 1993
Phases of N = 2 theories in two dimensions
Batyrev 1994
Dual polyhedra and mirror symmetry for Calabi-Yauhypersurfaces in toric varieties
Aspinwall, Greene and Morrison 1994
Calabi-Yau moduli space, mirror manifolds and spacetimetopology change in string theory
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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).
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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 ellipticcurves for efficient arithmetic. The construction of the classis inspired by toric geometry, which provides a naturalframework for the study of various forms of elliptic curves.The class both encompasses many prominent known formsand includes thousands of new forms. A powerful algorithmis described that automatically computes the most compactgroup operation formulas for any parameterized family ofelliptic curves. . . .
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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 implicitizationof surfaces. We construct the corresponding tropicalsurfaces via the theory of geometric tropicalization due toHacking, Keel and Tevelev, which we enrich with a formulafor computing tropical multiplicities of regular points in anydimension. We extend previous results for tropicalimplicitization of generic surfaces due to Sturmfels, Tevelevand Yu and provide methods for the non-generic case.
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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 targetson toric singularities via products of old partition functionson crepant resolutions. We compute explicit examples andshow that the new partition functions turn out to behomogeneous on MacMahon factors.
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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 adjunctiontheory of toric varieties. Inspired by classical adjunctiontheory of polarized algebraic varieties we define twoconvex-geometric notions: the Q-codegree and the nefvalue of a rational polytope P. We define the adjointpolytope P(s) as the set of those points in P, whose latticedistance to every facet of P is at least s. We prove astructure theorem for lattice polytopes with highQ-codegree. . . . Moreover, we illustrate how classificationresults in adjunction theory can be translated into newclassification results for lattice polytopes.
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
Mirror Symmetry
Survey Papers
The 21st Century
May 2011
Conclusion
18 May 2011 math.AC (math.CO)
OlteanuMonomial cut ideals
Abstract:B. Sturmfels and S. Sullivant associated to any graph a toricideal, called the cut ideal. We consider monomial cut idealsand we show that their algebraic properties such as theminimal primary decomposition, the property of having alinear resolution or being Cohen–Macaulay may be derivedfrom the combinatorial structure of the graph.
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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 integermatrices is introduced. We study the problem when the toricring of a centrally symmetric configuration is normal as wellas is Gorenstein. In addition, Gröbner bases of toric idealsof centrally symmetric configurations will be discussed.Special attentions will be given to centrally symmetricconfigurations of unimodular matrices and those ofincidence matrices of finite graphs.
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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 canbe used for computations in toric geometry. More precisely,counter-examples to conjectures related to A-determinantsand defect polytopes are constructed.
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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, measuresand heights
Abstract:We show that the height of a toric variety with respect to atoric metrized line bundle can be expressed as the integralover a polytope of a certain adelic family of concavefunctions. . . . We also present a closed formula for theintegral over a polytope of a function of one variablecomposed with a linear form. This allows us to compute theheight of toric varieties with respect to some interestingmetrics arising from polytopes. We also compute the heightof toric projective curves with respect to the Fubini-Studymetric, and of some toric bundles.
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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 andcomplex geometry. In symplectic geometry, the constructionof a symplectic toric manifold from a smooth polytope is dueto Delzant. . . . For rational but not necessarily smoothpolytopes the Delzant construction was refined by Lermanand Tolman, leading to symplectic toric orbifolds or moregenerally, symplectic toric DM stacks (Lerman and Malkin).. . . we hope that this text serves as an example drivenintroduction to symplectic toric geometry for thealgebraically minded reader.
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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 withToric Geometry and Applications!
Snapshotsfrom the
History ofToric
Geometry
David A. Cox
1970–1988Demazure
KKMS
Other Early Papers
The Russian School
Polytopes
Other Developments
Some Quotes
Since 1988Secondary Fan
HomogeneousCoordinates
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 withToric Geometry and Applications!
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