Minimalfreeresolutions,Bettinumbers,andcombinatoricsEdinburgh,June2,2015 p , 1 A K conjecturefortoricsurfaces

p,1

(joint work in progress with Filip Cools and Alexander Lemmens)

Wouter Castryck (Universiteit Gent)

Minimal free resolutions, Betti numbers, and combinatorics Edinburgh, June 2, 2015

June 2, 2015 AK conjecture for toric surfaces 1/38

June 2, 2015 AK conjecture for toric surfaces 2/38

k : algebraically closed field of characteristic 0

∆ : two-dimensional lattice polygon T² = (k\ {0})² = Speck[x^±1,y^±1] N_∆ = ](∆∩Z²)

S_∆ = k[X_i,j|(i,j)∈∆∩Z²]

We consider the morphism ϕ_∆:T²,→P^N∆−1

| {z }

: (x,y)7→(xⁱy^j)_{(i,j)∈(∆∩Z}2) ProjS∆

and defineX(∆)as the Zariski closure of the image.

What does the graded Betti table ofX(∆)look like?

How does it depend on the combinatorics of∆?

June 2, 2015 AK conjecture for toric surfaces 3/38

k : algebraically closed field of characteristic 0

∆ : two-dimensional lattice polygon T² = (k\ {0})² = Speck[x^±1,y^±1] N_∆ = ](∆∩Z²)

S_∆ = k[X_i,j|(i,j)∈∆∩Z²]

We consider the morphism ϕ_∆:T²,→P^N∆−1

| {z }

: (x,y)7→(xⁱy^j)_{(i,j)∈(∆∩Z}2) ProjS∆

and defineX(∆)as the Zariski closure of the image.

What does the graded Betti table ofX(∆)look like?

How does it depend on the combinatorics of∆?

June 2, 2015 AK conjecture for toric surfaces 3/38

k : algebraically closed field of characteristic 0

∆ : two-dimensional lattice polygon T² = (k\ {0})² = Speck[x^±1,y^±1] N_∆ = ](∆∩Z²)

S_∆ = k[X_i,j|(i,j)∈∆∩Z²]

We consider the morphism ϕ_∆:T²,→P^N∆−1

| {z }

: (x,y)7→(xⁱy^j)_{(i,j)∈(∆∩Z}2) ProjS∆

and defineX(∆)as the Zariski closure of the image.

What does the graded Betti table ofX(∆)look like?

How does it depend on the combinatorics of∆?

June 2, 2015 AK conjecture for toric surfaces 3/38

∆ =

(1,1)

(1,0) (0,1)

(0,0)

Here

ϕ∆:T²,→P³: (x,y)7→(1,x,y,xy), so

X(∆) :X_0,0X_1,1−X_1,0X_0,1=:f. The Betti diagram reads

0 1 0 1 0 1 0 1

because of

0→S_∆(−2)^g−→^7→gf S_∆→S_∆

(f)→0.

June 2, 2015 AK conjecture for toric surfaces 4/38

∆ =

(1,1)

(1,0) (0,1)

(0,0)

Here

ϕ∆:T²,→P³: (x,y)7→(1,x,y,xy), so

X(∆) :X_0,0X_1,1−X_1,0X_0,1=:f. The Betti diagram reads

0 1 0 1 0 1 0 1

because of

0→S_∆(−2)^g−→^7→gf S_∆→S_∆

(f)→0.

June 2, 2015 AK conjecture for toric surfaces 4/38

∆ = Υ :=

(−1,−1)

(1,0) (0,1)

Here

ϕ∆:T²,→P³: (x,y)7→(x⁻¹y⁻¹,1,x,y), so

X(∆) :X_−1,−1X_1,0X_0,1−X_0,0³ =:f. The Betti diagram reads

0 1 0 1 0 1 0 0 2 0 1

0→S_∆(−3)^g−→^7→gf S_∆→S_∆

(f)→0.

June 2, 2015 AK conjecture for toric surfaces 5/38

∆ = Υ :=

(−1,−1)

(1,0) (0,1)

Here

ϕ∆:T²,→P³: (x,y)7→(x⁻¹y⁻¹,1,x,y), so

X(∆) :X_−1,−1X_1,0X_0,1−X_0,0³ =:f. The Betti diagram reads

0 1 0 1 0 1 0 0 2 0 1

0→S_∆(−3)^g−→^7→gf S_∆→S_∆

(f)→0.

June 2, 2015 AK conjecture for toric surfaces 5/38

∆ =

(0,1)

(0,0)

(b,1)

(a,0)

ϕ_∆:T²,→P^a+b+1: (x,y)7→(1,x, . . . ,x^a,y,xy, . . . ,x^by) HereX(∆)is cut out by

rank

X0,0 X1,0 . . . Xa−1,0 X0,1 X1,1 . . . Xb−1,1

X1,0 X2,0 . . . Xa,0 X1,1 X2,1 . . . Xb,1

≤1, i.e. it is arational normal surface scrollof type(a,b).

Betti diagram:

0 1 2 3 . . . N∆−4 N∆−3

0 1 0 0 0 . . . 0 0

1 0 ^N∆₂⁻²

2 ^N∆₃⁻²

3 ^N∆₄⁻²

. . . (N∆−4) ^N_N^∆−2

∆−3

N∆−3

because the Eagon-Northcott complex is exact.

June 2, 2015 AK conjecture for toric surfaces 6/38

∆ =

(0,1)

(0,0)

(b,1)

(a,0)

ϕ_∆:T²,→P^a+b+1: (x,y)7→(1,x, . . . ,x^a,y,xy, . . . ,x^by) HereX(∆)is cut out by

rank

X0,0 X1,0 . . . Xa−1,0 X0,1 X1,1 . . . Xb−1,1

X1,0 X2,0 . . . Xa,0 X1,1 X2,1 . . . Xb,1

≤1, i.e. it is arational normal surface scrollof type(a,b).

Betti diagram:

0 1 2 3 . . . N∆−4 N∆−3

0 1 0 0 0 . . . 0 0

1 0 ^N∆₂⁻²

2 ^N∆₃⁻²

3 ^N∆₄⁻²

. . . (N∆−4) ^N_N^∆−2

∆−3

N∆−3

because the Eagon-Northcott complex is exact.

June 2, 2015 AK conjecture for toric surfaces 6/38

∆ =dΣ :=

(0,0) (d,0) (0, d)

ϕ_∆:T²,→P(^d+22 )⁻¹: (x,y)7→(xⁱy^j)_i+j≤d HereX(∆)is the image of thed-uple embedding

ρ_d :P²,→P(^d+2₂ )⁻¹ (Veronese surface).

Ford =2: classical Veronese surface, with Betti diagram

0 1 2 3

0 1 0 0 0

1 0 6 8 3

(= Eagon-Northcott withN_∆=6).

In general: unknown (computed ford ≤5 byGreco–Martino).

June 2, 2015 AK conjecture for toric surfaces 7/38

∆ =dΣ :=

(0,0) (d,0) (0, d)

ϕ_∆:T²,→P(^d+22 )⁻¹: (x,y)7→(xⁱy^j)_i+j≤d HereX(∆)is the image of thed-uple embedding

ρ_d :P²,→P(^d+2₂ )⁻¹ (Veronese surface).

Ford =2: classical Veronese surface, with Betti diagram

0 1 2 3

0 1 0 0 0

1 0 6 8 3

(= Eagon-Northcott withN_∆=6).

In general: unknown (computed ford ≤5 byGreco–Martino).

June 2, 2015 AK conjecture for toric surfaces 7/38

∆ =dΣ :=

(0,0) (d,0) (0, d)

ϕ_∆:T²,→P(^d+22 )⁻¹: (x,y)7→(xⁱy^j)_i+j≤d HereX(∆)is the image of thed-uple embedding

ρ_d :P²,→P(^d+2₂ )⁻¹ (Veronese surface).

Ford =2: classical Veronese surface, with Betti diagram

0 1 2 3

0 1 0 0 0

1 0 6 8 3

(= Eagon-Northcott withN_∆=6).

In general: unknown (computed ford ≤5 byGreco–Martino).

June 2, 2015 AK conjecture for toric surfaces 7/38

Preliminary remark: call∆and∆⁰ (unimodularly) equivalentif there exists

ψ:R²→R²: i

j

7→A i

j

+B, A∈GL₂(Z),B ∈Z² such that∆⁰ =ψ(∆).

Notation: ∆∼= ∆⁰.

ϕ

Fact: ∆∼= ∆⁰ ⇒ X(∆)∼=_{proj. eq.}X(∆⁰) ⇒ same Betti table.

Therefore: interested in polygons up to equivalence only.

June 2, 2015 AK conjecture for toric surfaces 8/38

Preliminary remark: call∆and∆⁰ (unimodularly) equivalentif there exists

ψ:R²→R²: i

j

7→A i

j

+B, A∈GL₂(Z),B ∈Z² such that∆⁰ =ψ(∆).

Notation: ∆∼= ∆⁰.

ϕ

Fact: ∆∼= ∆⁰ ⇒ X(∆)∼=_{proj. eq.}X(∆⁰) ⇒ same Betti table.

Therefore: interested in polygons up to equivalence only.

June 2, 2015 AK conjecture for toric surfaces 8/38

Hering: shape of the Betti table ofX(∆)is

0 1 2 3 . . . N∆−4 N∆−3

0 1 0 0 0 . . . 0 0

1 0 b1 b2 b3 . . . bN_∆−4 bN_∆−3

2 0 cN_∆−3 cN_∆−4 cN_∆−5 . . . c2 c1

wherec-row (cubic strand) is zero iff

∆⁽¹⁾:=conv

∆^int∩Z²

=∅ which holds iff∆is among

(0,1)

(0,0)

(b,1)

(a,0) (0,0) (0,2)

(2,0)

up to unimodular equivalence.

(Note: this is iffX(∆)is a surface of minimal degree.)

June 2, 2015 AK conjecture for toric surfaces 9/38

Hering: shape of the Betti table ofX(∆)is

0 1 2 3 . . . N∆−4 N∆−3

0 1 0 0 0 . . . 0 0

1 0 b1 b2 b3 . . . bN_∆−4 bN_∆−3

2 0 cN_∆−3 cN_∆−4 cN_∆−5 . . . c2 c1

wherec-row (cubic strand) is zero iff

∆⁽¹⁾:=conv

∆^int∩Z²

=∅ which holds iff∆is among

(0,1)

(0,0)

(b,1)

(a,0) (0,0) (0,2)

(2,0)

up to unimodular equivalence.

(Note: this is iffX(∆)is a surface of minimal degree.)

June 2, 2015 AK conjecture for toric surfaces 9/38

What else is known?

0 1 2 3 . . . N∆−4 N∆−3

0 1 0 0 0 . . . 0 0

1 0 b1 b2 b3 . . . bN_∆−4 bN_∆−3

2 0 cN_∆−3 cN_∆−4 cN_∆−5 . . . c2 c1

By Pick’s theorem the Hilbert function ofX(∆)equals d 7→Vol(∆)d²+](∂∆∩Z²)

2 d+1, which implies that

b_i−c_N−1−i =i

N−1 i+1

−2

N−3 i−1

Vol(∆) for alli.

June 2, 2015 AK conjecture for toric surfaces 10/38

What else is known?

0 1 2 3 . . . N∆−4 N∆−3

0 1 0 0 0 . . . 0 0

1 0 b1 b2 b3 . . . bN_∆−4 bN_∆−3

2 0 cN_∆−3 cN_∆−4 cN_∆−5 . . . c2 c1

By Pick’s theorem the Hilbert function ofX(∆)equals d 7→Vol(∆)d²+](∂∆∩Z²)

2 d+1, which implies that

b_i−c_N−1−i =i

N−1 i+1

−2

N−3 i−1

Vol(∆) for alli.

June 2, 2015 AK conjecture for toric surfaces 10/38

What else is known?

0 1 2 3 . . . N∆−4 N∆−3

0 1 0 0 0 . . . 0 0

1 0 ^N∆₂⁻¹

−2Vol(∆) b2 b3 . . . bN_∆−4 bN_∆−3

2 0 cN_∆−3 cN_∆−4 cN_∆−5 . . . c2 ](∆⁽¹⁾∩Z²)

By Pick’s theorem the Hilbert function ofX(∆)equals d 7→Vol(∆)d²+](∂∆∩Z²)

2 d+1, which implies that

b_i−c_N−1−i =i

N−1 i+1

−2

N−3 i−1

Vol(∆) for alli.

June 2, 2015 AK conjecture for toric surfaces 10/38

What else is known?

0 1 2 3 . . . N∆−4 N∆−3

0 1 0 0 0 . . . 0 0

1 0 ^N∆₂⁻¹

−2Vol(∆) b2 b3 . . . bN_∆−4 bN_∆−3

2 0 cN_∆−3 cN_∆−4 cN_∆−5 . . . c2 ](∆⁽¹⁾∩Z²)

Koelman: ideal ofX(∆)is generated by quadrics, i.e.

c_N∆−3=0,

if and only if](∂∆∩Z²)>3.If not then in addition one needs c_N∆−3=

1 if∆⁽¹⁾is two-dimensional, N_∆−3 if not

cubics (C.–Cools).

June 2, 2015 AK conjecture for toric surfaces 11/38

What else is known?

0 1 2 3 . . . N∆−4 N∆−3

0 1 0 0 0 . . . 0 0

1 0 ^N∆₂⁻¹

−2Vol(∆) b2 b3 . . . bN_∆−4 bN_∆−3

2 0 cN_∆−3 cN_∆−4 cN_∆−5 . . . c2 ](∆⁽¹⁾∩Z²)

Koelman: ideal ofX(∆)is generated by quadrics, i.e.

c_N∆−3=0,

if and only if](∂∆∩Z²)>3. If not then in addition one needs c_N∆−3=

1 if∆⁽¹⁾is two-dimensional, N_∆−3 if not

cubics (C.–Cools).

June 2, 2015 AK conjecture for toric surfaces 11/38

What else is known?

0 1 2 3 . . . N∆−4 N∆−3

0 1 0 0 0 . . . 0 0

1 0 ^N∆₂⁻¹

−2Vol(∆) b2 b3 . . . bN_∆−4 bN_∆−3

2 0 cN_∆−3 cN_∆−4 cN_∆−5 . . . c2 ](∆⁽¹⁾∩Z²)

Koelman: ideal ofX(∆)is generated by quadrics, i.e.

c_N∆−3=0,

if and only if](∂∆∩Z²)>3. If not then in addition one needs c_N∆−3=

1 if∆⁽¹⁾is two-dimensional, N_∆−3 if not

cubics (C.–Cools).

June 2, 2015 AK conjecture for toric surfaces 11/38

Vast generalization ofKoelman’s result:

0 1 2 3 . . . N∆−5 N∆−4 N∆−3

0 1 0 0 0 . . . 0 0 0

1 0 b1 b2 b3 . . . bN_∆−5 bN_∆−4 bN_∆−3

2 0 cN_∆−3 cN_∆−4 cN_∆−5 . . . c3 c2 c1

The number of leadingc_i’s that are zero equals ](∂∆∩Z²)−3

bySchenck,Gallego–Purnaprajna,Hering.

(0,2)

(2,0) (−2,−2)

Example:

0 1 2 3 4 5 6 7

0 1 0 0 0 0 0 0 0

1 0 24 84 126 84 20 0 0

2 0 0 0 0 20 36 21 4

June 2, 2015 AK conjecture for toric surfaces 12/38

At the other end:Green’sK_p,1theorem.

0 1 2 3 . . . N∆−5 N∆−4 N∆−3

0 1 0 0 0 . . . 0 0 0

1 0 b1 b2 b3 . . . bN_∆−5 bN_∆−4 bN_∆−3

2 0 cN_∆−3 cN_∆−4 cN_∆−5 . . . c3 c2 c1

One hasb_N∆−36=0 iffX(∆)is a variety of minimal degree m^(recall)

∆⁽¹⁾=∅

(0,1) m

(0,0)

(b,1)

(a,0) (0,0) (0,2)

(2,0)

If not thenb_N∆−3=N_∆−3 by Eagon-Northcott.

June 2, 2015 AK conjecture for toric surfaces 13/38

At the other end:Green’sK_p,1theorem.

0 1 2 3 . . . N∆−5 N∆−4 N∆−3

0 1 0 0 0 . . . 0 0 0

1 0 b1 b2 b3 . . . bN_∆−5 bN_∆−4 bN_∆−3

2 0 cN_∆−3 cN_∆−4 cN_∆−5 . . . c3 c2 c1

One hasb_N∆−36=0 iffX(∆)is a variety of minimal degree m^(recall)

∆⁽¹⁾=∅

(0,1) m

(0,0)

(b,1)

(a,0) (0,0) (0,2)

(2,0)

If not thenb_N∆−3=N_∆−3 by Eagon-Northcott.

June 2, 2015 AK conjecture for toric surfaces 13/38

At the other end:Green’sK_p,1theorem.

0 1 2 3 . . . N∆−5 N∆−4 N∆−3

0 1 0 0 0 . . . 0 0 0

1 0 b1 b2 b3 . . . bN_∆−5 bN_∆−4 bN_∆−3

2 0 cN_∆−3 cN_∆−4 cN_∆−5 . . . c3 c2 c1

One hasb_N∆−36=0 iffX(∆)is a variety of minimal degree m^(recall)

∆⁽¹⁾=∅

(0,1) m

(0,0)

(b,1)

(a,0) (0,0) (0,2)

(2,0)

If not thenb_N∆−3=N_∆−3 by Eagon-Northcott.

June 2, 2015 AK conjecture for toric surfaces 13/38

Similar generalization ofK_p,1theorem (for toric surfaces)?

0 1 2 3 . . . N∆−5 N∆−4 N∆−3

0 1 0 0 0 . . . 0 0 0

1 0 b1 b2 b3 . . . bN_∆−5 bN_∆−4 bN_∆−3

2 0 cN_∆−3 cN_∆−4 cN_∆−5 . . . c3 c2 c1

Goal: find a combinatorial interpretation for the number of concludingb_i’s that are zero.

Remainder of this talk:

I state a concrete guess/conjecture,

I give a partial proof,

I digress on Green’s canonical syzygy conjecture forcurves.

June 2, 2015 AK conjecture for toric surfaces 14/38

Similar generalization ofK_p,1theorem (for toric surfaces)?

0 1 2 3 . . . N∆−5 N∆−4 N∆−3

0 1 0 0 0 . . . 0 0 0

1 0 b1 b2 b3 . . . bN_∆−5 bN_∆−4 bN_∆−3

2 0 cN_∆−3 cN_∆−4 cN_∆−5 . . . c3 c2 c1

Goal: find a combinatorial interpretation for the number of concludingb_i’s that are zero.

Remainder of this talk:

I state a concrete guess/conjecture,

I give a partial proof,

I digress on Green’s canonical syzygy conjecture forcurves.

June 2, 2015 AK conjecture for toric surfaces 14/38

p,1

June 2, 2015 AK conjecture for toric surfaces 15/38

Thelattice widthof a lattice polygon∆is

lw(∆) =min{d| ∃unimodular transf.ϕ:ϕ(∆)⊂R×[0,d]}, i.e. it is the minimal possible height of an equivalent polygon.

ϕ d

Has been studied since the 70s (in a.o. complexity theory!).

Easy to compute in practice:

lw(∆) =

lw(∆⁽¹⁾) +3 if∆∼=dΣfor somed ≥2, lw(∆⁽¹⁾) +2 if not,

where lw(∅) =−1 (Lubbes–Schicho,C.–Cools).

June 2, 2015 AK conjecture for toric surfaces 16/38

Thelattice widthof a lattice polygon∆is

lw(∆) =min{d| ∃unimodular transf.ϕ:ϕ(∆)⊂R×[0,d]}, i.e. it is the minimal possible height of an equivalent polygon.

ϕ d

Has been studied since the 70s (in a.o. complexity theory!).

Easy to compute in practice:

lw(∆) =

lw(∆⁽¹⁾) +3 if∆∼=dΣfor somed ≥2, lw(∆⁽¹⁾) +2 if not,

where lw(∅) =−1 (Lubbes–Schicho,C.–Cools).

June 2, 2015 AK conjecture for toric surfaces 16/38

Examples:

∆ =

(0,1)

(0,0)

(b,1)

(a,0)

lw(∆) =1

∆ =dΣ =

(0,0) (d,0) (0, d)

lw(∆) =d

∆ =2Υ =

(0,2)

(2,0) (−2,−2)

lw(∆) =4

June 2, 2015 AK conjecture for toric surfaces 17/38

p,1

Betti table ofX(∆):

0 1 2 3 . . . N∆−5 N∆−4 N∆−3

0 1 0 0 0 . . . 0 0 0

1 0 b1 b2 b3 . . . bN_∆−5 bN_∆−4 bN_∆−3

2 0 cN_∆−3 cN_∆−4 cN_∆−5 . . . c3 c2 c1

Guess/conjecture: the number of concludingb_i’s that are zero equals

lw(∆)−1 unless

I ∆∼=dΣfor somed ≥2, then it is lw(∆)−2=d−2,

I ∆∼=2Υ, then it islw(∆)−2=4−2=2.

Note: this value is 0 iff∆⁽¹⁾=∅iffX(∆)has minimal degree.

June 2, 2015 AK conjecture for toric surfaces 18/38

p,1

Betti table ofX(∆):

0 1 2 3 . . . N∆−5 N∆−4 N∆−3

0 1 0 0 0 . . . 0 0 0

1 0 b1 b2 b3 . . . bN_∆−5 bN_∆−4 bN_∆−3

2 0 cN_∆−3 cN_∆−4 cN_∆−5 . . . c3 c2 c1

Guess/conjecture: the number of concludingb_i’s that are zero equals

lw(∆)−1 unless

I ∆∼=dΣfor somed ≥2, then it is lw(∆)−2=d−2,

I ∆∼=2Υ, then it islw(∆)−2=4−2=2.

Note: this value is 0 iff∆⁽¹⁾=∅iffX(∆)has minimal degree.

June 2, 2015 AK conjecture for toric surfaces 18/38

p,1

(0,0) (3,0) (0,3)

0 1 2 3 4 5 6 7

0 1 0 0 0 0 0 0 0

1 0 27 105 189 189 105 27 0

2 0 0 0 0 0 0 0 1

(0,1)

(3,0) (4,0) (4,1) (0,3)

0 1 2 3 4 5 6 7 8

0 1 0 0 0 0 0 0 0 0

1 0 32 136 266 280 140 12 0 0

2 0 0 0 0 0 20 49 24 4

(1,0)

(4,1) (0,4)

0 1 2 3 4 5 6

0 1 0 0 0 0 0 0

1 0 15 35 21 0 0 0

2 0 1 6 36 55 30 6

June 2, 2015 AK conjecture for toric surfaces 19/38

p,1

(0,2)

(2,0) (−2,−2)

0 1 2 3 4 5 6 7

0 1 0 0 0 0 0 0 0

1 0 24 84 126 84 20 0 0

2 0 0 0 0 20 36 21 4

(0,1)

(5,0) (2,5)

0 1 2 3 4 5

0 1 0 0 0 0 0

1 0 56 330 936 1528 1405

2 0 0 1 10 85 609

. . .

. . .

6 7 8 9 10 11

0 0 0 0 0 0

741 175 0 0 0 0

1330 1540 1056 430 99 10

June 2, 2015 AK conjecture for toric surfaces 20/38

Using Magma intrinsic, we have verified the conjecture for several hundreds of polygons.

Problem:N_∆grows at least quadratically with lw(∆), so hard to gather data for large lattice widths.

Current goal: write an algorithm for computing Betti tables of toric surfaces that exploits

I known facts about Betti table,

I the action ofT²on Koszul cohomology.

June 2, 2015 AK conjecture for toric surfaces 21/38

Using Magma intrinsic, we have verified the conjecture for several hundreds of polygons.

Problem:N_∆grows at least quadratically with lw(∆), so hard to gather data for large lattice widths.

Current goal: write an algorithm for computing Betti tables of toric surfaces that exploits

I known facts about Betti table,

I the action ofT²on Koszul cohomology.

June 2, 2015 AK conjecture for toric surfaces 21/38

Recall: if∆is

(0,1)

(0,0)

(b,1)

(a,0)

thenX(∆)is cut out by rank

X0,0 X1,0 . . . Xa−1,0 X0,1 X1,1 . . . Xb−1,1

X1,0 X2,0 . . . Xa,0 X1,1 X2,1 . . . Xb,1

≤1, i.e. it is arational normal surface scrollof type(a,b).

June 2, 2015 AK conjecture for toric surfaces 22/38

If∆has a more general shape

(id,d), . . . ,(id+ad,d)

...

(i1,1), . . . ,(i1+a1,1) (i0,0), . . . ,(i0+a0,0)

then this is no longer true, but . . .the ideal ofX(∆)still contains the rank≤1 locus of

_X

i0,0 . . . X_i

0+a0−1,0 X_i

1,1 . . . X_i

1+a1−1,1 X_i

0+1,0 . . . X_i

0+a0,0 X_i

1+1,1 . . . X_i

1+a1,1 . . . X^X_id^id_+1,d^{,d . . .}. . . ^XidX_id^+ad_+ad^−1,d_,d

,

⇒X(∆)contained in rational normal scroll of type(a₀, . . . ,a_d).

June 2, 2015 AK conjecture for toric surfaces 23/38

If∆has a more general shape

(id,d), . . . ,(id+ad,d)

...

(i1,1), . . . ,(i1+a1,1) (i0,0), . . . ,(i0+a0,0)

then this is no longer true, but . . . the ideal ofX(∆)stillcontains the rank≤1 locus of

_X

i0,0 . . . X_i

0+a0−1,0 X_i

1,1 . . . X_i 1+a1−1,1 X_i

0+1,0 . . . X_i

0+a0,0 X_i

1+1,1 . . . X_i

1+a1,1 . . . X^X_id^id_+1,d^{,d . . .}. . . ^XidX_id^+ad_+ad^−1,d_,d

,

⇒X(∆)contained inrational normal scrollof type(a₀, . . . ,a_d).

June 2, 2015 AK conjecture for toric surfaces 23/38

Betti table of rational normal scroll is

0 1 2 3 . . . f−2 f−1

0 1 0 0 0 . . . 0 0

1 0 ^f₂ 2 ₃^f

3 ₄^f

. . . (f−2) _f−1^f

(f−1) ^f_f

withf =N_∆−d−1, because of Eagon-Northcott.

Scroll containsX(∆)⇒the above is asubtableof the Betti table ofX(∆):

0 1 2 3 . . . N∆−5 N∆−4 N∆−3

0 1 0 0 0 . . . 0 0 0

1 0 b1 b2 b3 . . . bN_∆−5 bN_∆−4 bN_∆−3

2 0 cN_∆−3 cN_∆−4 cN_∆−5 . . . c3 c2 c1

We find: number of zerob_i’s is at most

N_∆−3−(f −1) =N_∆−3−(N_∆−d −1) =d−1.

June 2, 2015 AK conjecture for toric surfaces 24/38

Betti table of rational normal scroll is

0 1 2 3 . . . f−2 f−1

0 1 0 0 0 . . . 0 0

1 0 ^f₂ 2 ₃^f

3 ₄^f

. . . (f−2) _f−1^f

(f−1) ^f_f

withf =N_∆−d−1, because of Eagon-Northcott.

Scroll containsX(∆)⇒the above is asubtableof the Betti table ofX(∆):

0 1 2 3 . . . N∆−5 N∆−4 N∆−3

0 1 0 0 0 . . . 0 0 0

1 0 b1 b2 b3 . . . bN_∆−5 bN_∆−4 bN_∆−3

2 0 cN_∆−3 cN_∆−4 cN_∆−5 . . . c3 c2 c1

We find: number of zerob_i’s is at most

N_∆−3−(f −1) =N_∆−3−(N_∆−d −1) =d−1.

June 2, 2015 AK conjecture for toric surfaces 24/38

Betti table of rational normal scroll is

0 1 2 3 . . . f−2 f−1

0 1 0 0 0 . . . 0 0

1 0 ^f₂ 2 ₃^f

3 ₄^f

. . . (f−2) _f−1^f

(f−1) ^f_f

withf =N_∆−d−1, because of Eagon-Northcott.

Scroll containsX(∆)⇒the above is asubtableof the Betti table ofX(∆):

0 1 2 3 . . . N∆−5 N∆−4 N∆−3

0 1 0 0 0 . . . 0 0 0

1 0 b1 b2 b3 . . . bN_∆−5 bN_∆−4 bN_∆−3

2 0 cN_∆−3 cN_∆−4 cN_∆−5 . . . c3 c2 c1

We find: number of zerob_i’s is at most

N_∆−3−(f −1) =N_∆−3−(N_∆−d −1) =d−1.

June 2, 2015 AK conjecture for toric surfaces 24/38

Betti table ofX(∆):

0 1 2 3 . . . N∆−5 N∆−4 N∆−3

0 1 0 0 0 . . . 0 0 0

1 0 b1 b2 b3 . . . bN_∆−5 bN_∆−4 bN_∆−3

2 0 c_N∆−3 c_N∆−4 c_N∆−5 . . . c3 c2 c1

For the optimal choiced =lw(∆)we find:

The number of concluding zerob_i’s is at most lw(∆)−1.

Alternatively: possible to construct explicit non-zero classes in Koszul cohomology.

Finer analysis then moreover gives:

I ∆∼=dΣfor somed ≥2, then at most lw(∆)−2=d−2,

I ∆∼=2Υ, then at mostlw(∆)−2=4−2=2.

June 2, 2015 AK conjecture for toric surfaces 25/38