Artin fans. AMS special session on Combinatorics and Algebraic Geometry. Dan Abramovich. Brown University. October 24, 2014

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Artin fans AMS special session on Combinatorics and Algebraic Geometry Dan Abramovich Brown University October 24, 2014 Abramovich (Brown) Artin fans October 24, 2014 1 / 13

Heros: Abramovich (Brown) Artin fans October 24, 2014 2 / 13

Heros: Martin Olsson Abramovich (Brown) Artin fans October 24, 2014 2 / 13

Heros: Martin Olsson Jonathan Wise Qile Chen, Steffen Marcus, Mark Gross, Bernd Siebert Abramovich (Brown) Artin fans October 24, 2014 2 / 13

Heros: Martin Olsson Jonathan Wise Qile Chen, Steffen Marcus, Mark Gross, Bernd Siebert Martin Ulirsch Abramovich (Brown) Artin fans October 24, 2014 2 / 13

Superabundance Mikhalkin-Speyer: there is a tropical cubic curve C of genus 1 in TP 3 which does not lift to an algebraic curve (Speyer, Tropical Geometry, Berkeley thesis 2005, Figure 5.1). Abramovich (Brown) Artin fans October 24, 2014 3 / 13

Superabundance (continued) I want to understand this phenomenon. Abramovich (Brown) Artin fans October 24, 2014 4 / 13

Superabundance (continued) I want to understand this phenomenon. Principles: Abramovich (Brown) Artin fans October 24, 2014 4 / 13

Superabundance (continued) I want to understand this phenomenon. Principles: Tropical curves in TP 3 encode degenerations of curves in P 3 Abramovich (Brown) Artin fans October 24, 2014 4 / 13

Superabundance (continued) I want to understand this phenomenon. Principles: Tropical curves in TP 3 encode degenerations of curves in P 3 They encode in detail the manner in which they degenerate Abramovich (Brown) Artin fans October 24, 2014 4 / 13

Superabundance (continued) I want to understand this phenomenon. Principles: Tropical curves in TP 3 encode degenerations of curves in P 3 They encode in detail the manner in which they degenerate They encode logarithmic stable maps in P 3. Abramovich (Brown) Artin fans October 24, 2014 4 / 13

Superabundance (continued) I want to understand this phenomenon. Principles: Tropical curves in TP 3 encode degenerations of curves in P 3 They encode in detail the manner in which they degenerate They encode logarithmic stable maps in P 3. But logarithmic stable maps are obstructed. Abramovich (Brown) Artin fans October 24, 2014 4 / 13

Superabundance (continued) I want to understand this phenomenon. Principles: Tropical curves in TP 3 encode degenerations of curves in P 3 They encode in detail the manner in which they degenerate They encode logarithmic stable maps in P 3. But logarithmic stable maps are obstructed. Question Is there a world in which they are not obstructed? Abramovich (Brown) Artin fans October 24, 2014 4 / 13

Logarithmic structures Definition A pre logarithmic structure is X = (X, M α O X ) Abramovich (Brown) Artin fans October 24, 2014 5 / 13

Logarithmic structures Definition A pre logarithmic structure is X = (X, M α O X ) or just (X, M) Abramovich (Brown) Artin fans October 24, 2014 5 / 13

Logarithmic structures Definition A pre logarithmic structure is such that X = (X, M α O X ) or just (X, M) X is a scheme - the underlying scheme Abramovich (Brown) Artin fans October 24, 2014 5 / 13

Logarithmic structures Definition A pre logarithmic structure is such that X = (X, M α O X ) or just (X, M) X is a scheme - the underlying scheme M is a sheaf of monoids on X, and Abramovich (Brown) Artin fans October 24, 2014 5 / 13

Logarithmic structures Definition A pre logarithmic structure is such that X = (X, M α O X ) or just (X, M) X is a scheme - the underlying scheme M is a sheaf of monoids on X, and α is a monoid homomorphism, where the monoid structure on O X is the multiplicative structure. Abramovich (Brown) Artin fans October 24, 2014 5 / 13

Logarithmic structures Definition A pre logarithmic structure is such that X = (X, M α O X ) or just (X, M) X is a scheme - the underlying scheme M is a sheaf of monoids on X, and α is a monoid homomorphism, where the monoid structure on O X is the multiplicative structure. Definition It is a logarithmic structure if α : α 1 O X O X is an isomorphism. Abramovich (Brown) Artin fans October 24, 2014 5 / 13

Examples Examples (X, O X O X ), the trivial logarithmic structure. Abramovich (Brown) Artin fans October 24, 2014 6 / 13

Examples Examples (X, O X O X ), the trivial logarithmic structure. Let X, D X be a variety with a divisor. We define M D O X : Abramovich (Brown) Artin fans October 24, 2014 6 / 13

Examples Examples (X, O X O X ), the trivial logarithmic structure. Let X, D X be a variety with a divisor. We define M D O X : M D (U) = {f O X (U) f UD O X (U D) }. Abramovich (Brown) Artin fans October 24, 2014 6 / 13

Examples Examples (X, O X O X ), the trivial logarithmic structure. Let X, D X be a variety with a divisor. We define M D O X : M D (U) = {f O X (U) f UD O X (U D) }. Let k be a field, N k k (n, z) z 0 n defined by sending 0 1 and n 0 otherwise. Abramovich (Brown) Artin fans October 24, 2014 6 / 13

The magic of logarithmic geomery Any toric variety X is logarithmically smooth T X O dim X X. Abramovich (Brown) Artin fans October 24, 2014 7 / 13

The magic of logarithmic geomery Any toric variety X is logarithmically smooth T X O dim X X. A nodal curve is logarithmically smooth over a logarithmic point. Abramovich (Brown) Artin fans October 24, 2014 7 / 13

Here be monsters! Logarithmic obstructions to deforming a logarithmic map C P 3 lie in H 1 (C, O 3 C ). Abramovich (Brown) Artin fans October 24, 2014 8 / 13

Here be monsters! Logarithmic obstructions to deforming a logarithmic map C P 3 lie in H 1 (C, O 3 C ). These can be nonzero on a broken cubic curve! Abramovich (Brown) Artin fans October 24, 2014 8 / 13

Artin fans Olsson: {Logarithmic structures X on X } {X Log}. Abramovich (Brown) Artin fans October 24, 2014 9 / 13

Artin fans Olsson: {Logarithmic structures X on X } {X Log}. The stack Log is huge and does not specify combinatorial data. Abramovich (Brown) Artin fans October 24, 2014 9 / 13

Artin fans Olsson: {Logarithmic structures X on X } {X Log}. The stack Log is huge and does not specify combinatorial data. Proposition (Wise; ℵ, Chen, Marcus) There is an initial factorization X A X Log such that A X Log is étale, representable, strict. Abramovich (Brown) Artin fans October 24, 2014 9 / 13

Artin fans Olsson: {Logarithmic structures X on X } {X Log}. The stack Log is huge and does not specify combinatorial data. Proposition (Wise; ℵ, Chen, Marcus) There is an initial factorization X A X Log such that A X Log is étale, representable, strict. The stack A X is small, totally combinatorial. Abramovich (Brown) Artin fans October 24, 2014 9 / 13

P 3 and A P 3 P 3 = (A 4 {0})/G m. So {C P 3 } {(L, s 0,..., s 3 ) s i do not vanish together}. Abramovich (Brown) Artin fans October 24, 2014 10 / 13

P 3 and A P 3 P 3 = (A 4 {0})/G m. So Now So {C P 3 } {(L, s 0,..., s 3 ) s i do not vanish together}. A P 3 = (A 4 {0})/G 4 m. {C A P 3} {((L 0, s 0 ),..., (L 3, s 3 )) s i do not vanish together}. Abramovich (Brown) Artin fans October 24, 2014 10 / 13

The monsters evaporate! T P 3 = O 3, Abramovich (Brown) Artin fans October 24, 2014 11 / 13

The monsters evaporate! T P 3 = O 3, but T AP 3 = 0. Abramovich (Brown) Artin fans October 24, 2014 11 / 13

The monsters evaporate! T P 3 = O 3, but T AP 3 = 0. Logarithmic obstructions to deforming a logarithmic map C A P 3 lie in H 1 (C, 0). Abramovich (Brown) Artin fans October 24, 2014 11 / 13

The monsters evaporate! T P 3 = O 3, but T AP 3 = 0. Logarithmic obstructions to deforming a logarithmic map C A P 3 lie in H 1 (C, 0). The obstructions are gone! Abramovich (Brown) Artin fans October 24, 2014 11 / 13

Sample theorem Theorem (ℵ-Wise) If Y X is a toric modification, then Logarithmic Gromov Witten invariants of X coincide with those of Y. Abramovich (Brown) Artin fans October 24, 2014 12 / 13

Sample theorem Theorem (ℵ-Wise) If Y X is a toric modification, then Logarithmic Gromov Witten invariants of X coincide with those of Y. Reason: M(A Y ) M(A X ) is birational. So M(Y ) M(X ) is virtually birational. Abramovich (Brown) Artin fans October 24, 2014 12 / 13

Tropicalization Things are connected in Martin Ulirsch s fundamental commutative diagram: X ℶ A ℶ X Σ X ρ X r X ρ A r A ρ Σ X A X F X r Σ Abramovich (Brown) Artin fans October 24, 2014 13 / 13

Tropicalization Things are connected in Martin Ulirsch s fundamental commutative diagram: X ℶ A ℶ X Σ X ρ X r X ρ A r A ρ Σ X A X F X r Σ F P 3 = P 3 F 1 Abramovich (Brown) Artin fans October 24, 2014 13 / 13

Tropicalization Things are connected in Martin Ulirsch s fundamental commutative diagram: X ℶ A ℶ X Σ X ρ X r X ρ A r A ρ Σ X A X F X r Σ F P 3 = P 3 F 1 Σ P 3 = TP 3. Abramovich (Brown) Artin fans October 24, 2014 13 / 13

Tropicalization Things are connected in Martin Ulirsch s fundamental commutative diagram: X ℶ A ℶ X Σ X ρ X r X ρ A r A ρ Σ X A X F X r Σ F P 3 = P 3 F 1 Σ P 3 = TP 3. X ℶ - Berkovich analytic formal fiber Abramovich (Brown) Artin fans October 24, 2014 13 / 13

Tropicalization Things are connected in Martin Ulirsch s fundamental commutative diagram: X ℶ A ℶ X Σ X ρ X r X ρ A r A ρ Σ X A X F X r Σ F P 3 = P 3 F 1 Σ P 3 = TP 3. X ℶ - Berkovich analytic formal fiber P 3 and A P 3 share their tropicalization TP 3. Abramovich (Brown) Artin fans October 24, 2014 13 / 13

Tropicalization Things are connected in Martin Ulirsch s fundamental commutative diagram: X ℶ A ℶ X Σ X ρ X r X ρ A r A ρ Σ X A X F X r Σ F P 3 = P 3 F 1 Σ P 3 = TP 3. X ℶ - Berkovich analytic formal fiber P 3 and A P 3 share their tropicalization TP 3. A ℶ X Σ X is a homeomorphism. Abramovich (Brown) Artin fans October 24, 2014 13 / 13

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