non-reductive git and curve...

Non-reductive GIT and curve counting

Gergely Bérczi – OxfordBased on joint work with F. Kirwan, B. Doran, T. Hawes, J. Jackson and A.

Szenes

Oxford Algebraic and Sympkectic Geometry Seminar

May 31, 2016

Gergely Bérczi – Oxford Based on joint work with F. Kirwan, B. Doran, T. Hawes, J. Jackson and A. SzenesNon-reductive GIT and curve counting

Mumford’s reductive GIT in a nutshell

X complex projective variety, on which linear algebraic group G acts.

Linearisation: ample line bundle L on X and a lift of the action of G to L.

Replacing L with L⊗k ⇒ we can assume that X ⊆ Pn, the action of G on Xextends to an action on Pn given ρ : G → GL(n+ 1), and L is the hyperplane linebundle on Pn.

OL(X ) = ⊕∞k=0H0(X , L⊗k ) has an induced G -action. If G is reductive, OL(X )G

is finitely generated graded algebra, and OL(X )G ↪→ OL(X ) induces

X 99K X//G = ProjOL(X )G

∪ =

X ss � X//G∪ ∪X s → X s/G Geometric quotient

Topologically: X//G = X ss/ ∼ where x ∼ y ⇔ Gx ∩ Gy ∩ X ss 6= ∅.Hilbert-Mumford numerical criterion: T ⊂ G max torus acting diagonally viat · [x0 : . . . : xn] = [tα0x0 : . . . : tαnxn] for some αi ∈ Cr = t∗C.

x = [x0 : . . . : xn] ∈ Xs(ss)T ⇔ {0} sits inside(in the closure of) Conv(αi : xi 6= 0)

x ∈ X is (semi)stable for G iff gx is T -(semi)stable for all g ∈ G .


What if G is not reductive?

Cornerstones of bad behaviour:1 OL(X )G is not necessarily finitely generated (Nagata 1957) so Proj(OL(X )G ) is

not a projective variety.2 Even if OL(X )G fin. gen. the quotient map q : X → Proj(OL(X )G ) is not nec.

surjective, the image is just constructible.3 Recall: every linear algebraic group can be written as H = U o R where R is

reductive and U is unipotent. Unipotent orbits cannot necessarily be separatedwith invariants.

Example: C+ = {(

1 t0 1

): t ∈ C} acts on X = Sym2C2. Let x0, x1, x2 be

basis of X . Then A(X )C+= C[x0, x2

1 − x0x2] and the orbit space looks like:

Q: Can we define a sensible ’quotient’ variety X//G when G is not reductive?


Graded algebraic groups

Definition

U (unipotent) group is graded if ∃ λ : C∗ → Aut(U) with the weights of the C∗action on Lie(U) all positive. This defines the group extension

U = U o C∗ with (u, t) · (u′, t′) = (u · λ(t)(u′), tt′)

We say that H = U o R is graded if there is a central C∗ ⊂ R such that

U = U o C∗ ⊂ H is graded

In most applications the acting non-reductive group is graded.1) Moduli spaces of toric hypersurfaces

Aut(P(1, 1, 2)) = R n U where R = GL(2) and U = (C+)3

(x , y , z) 7→ (x , y , z + λx2 + µxy + νy2) for (λ, µ, ν) ∈ (C+)3

Aut(P(1, 1, 2)) =

ν

Sym2GL(2) µλ

0 0 0 1

The central one-parameter subgroup C∗ =

t2 0

t2 0t2 0

0 0 0 1

of R ∼= GL(2) acts

on Lie(U) with weight 2.Gergely Bérczi – Oxford Based on joint work with F. Kirwan, B. Doran, T. Hawes, J. Jackson and A. SzenesNon-reductive GIT and curve counting

Reparametrisation groups

2) Jets of reparametrisation germsJk (1, n) = {k-jets of germs of holomorphic maps (C, 0)→ (Cn, 0)} == {(f ′, f ′′, . . . , f (k)) : f ′ 6= 0}Jk (1, 1) is the group of k-jets of germs of biholomorphisms of (C, 0). Jk (1, 1)acts on Jk (1, n) via reparametrisation:

f (z) = zf ′(0) + z22! f′′(0) + . . . + zk

k! f(k)(0) ∈ Jk (1, n) and

ϕ(z) = α1z + α2z2 + . . . + αkz

k ∈ Jk (1, 1)Then

f ◦ ϕ(z) = (f ′(0)α1)z + (f ′(0)α2 +f ′′(0)2!

α21)z

2 + . . .

= (f ′, . . . , f (k)/k!) ·

α1 α2 α3 . . . αk

0 α21 2α1α2 . . . 2α1αk−1 + . . .

0 0 α31 . . . 3α2

1αk−2 + . . .0 0 0 . . . ·· · · . . . αk

1

= C∗ o Uk

This group plays a central role in many applications: Enumerative geometry of Hilbertschemes of points, singularities of maps, hyperbolicity questions and in particular theGreen-Griffiths-Lang conjecture...


The U Theorem

Let U be a graded unipotent lin. alg. group and U = U o C∗. Let X be an irreduciblenormal U-variety and L→ X an very ample linearisation of the U action.

XC∗min := X ∩ P(H0(X , L)∗min) =

{x ∈ X

x is a C∗-fixed point andC∗ acts on L∗|x with min weight

}Xmin := {x ∈ X | limt→0 t · x ∈ XC∗

min} (t ∈ Gm ⊆ U) the Bialynicki-Birula stratum.

Theorem (B.,Doran, Hawes, Kirwan 2012-16)

Let L→ X be an irreducible normal U-variety with ample linearisation L. If

(*) x ∈ XC∗min ⇒ StabU (x) = 1

(or more generally dim StabU (x) = miny∈X dim StabU (y)) holds then

1 Xmin ⊆ X s(U,L), so we get a locally trivial U-quotient Xmin → Xmin/U.

2 Twist the linearisation by a character χ : U → C∗ of U/U such that 0 lies in the lowestbounded chamber for the C∗ action on X . Then for suitably divisible integers r > 0Oχ(X )U = O(Lχ)⊗r (X )U is finitely generated and for the quotient

X//χU := Proj(Oχ(X )U )

we have X//U = X ss,U/ ∼ where x ∼ y ⇔ Ux ∩ Uy ∩ X ss 6= ∅.

3 Xmin \ (U · XC∗min) = X s,U = X ss,U and X//χU = X s,U/U is a geometric quotient of X s,U

by U. X s,U ,X ss,U are given by Hilbert-Mumford type criterion.


Remarks1 Similar construction for any graded non-reductive group H = U o R.2 If (*) does not hold then there is a sequence of blow-ups of X along U-invariant

projective subvarieties resulting in a projective variety X with a linear action of U(with respect to a power of an ample line bundle given by perturbing the pullbackof L by small multiples of the exceptional divisors for the blow-ups) which satisfies(*). Namely, let d maximal s.t

X dmin = {z ∈ XC∗

min | dim StabU(z) = d} 6= ∅.

Then U · X dmin = U · X d

min is a closed subvariety of X 0min and we blow-up this and

perturb the pull-back of L with a small multiple of the ex. divisor to get ampleU-linearisation on the blown-up.

3 (joint with J. Jackson and F. Kirwan): Nice VGIT picture for H = U o R actionwhen R contains higher dimensional grading torus, that is, the grading C∗ can bemoved inside R.

4 (joint with F. Kirwan) Cohomology of U-quotients: symplectic implosion forU-quotients and Kirwan surjectivity for H∗

U(X ss)→ H∗(X//U) and residue

formulae for cohomological pairings (intersection numbers) on X//U.


Hilbert schemes and tautological bundles

X/C smooth variety of dimension n.

X [k] = {ξ ⊂ X : dim(ξ) = 0, l(ξ) = dimH0(ξ,Oξ) = k} Hilbert scheme of kpoints on X

X[k]p = {ξ ∈ X [k] : supp(ξ) = p} Punctual Hilbert scheme at p ∈ X

Ultimate goal: Determine the geometric and topological invariants of X [n].For surfaces these were extensively studied such as Betti numbers (Ellingsrud,Stromme, Göttsche), Hodge numbers (Sörgel, Göttsche), cohomology ring(Nakajima, Grojnowski,Lehn), Chern numbers of tautological bundles(Kleiman-Piene, Lehn, Rennemo, Marian-Oprea-Pandharipande).When dim(X ) > 2 not much is known:

Rennemo, Tzeng: see in a minute

motivic DT invariants (Behrend-Bryan-Szendröi).

Tautological vector bundles:F–rank r bundle (loc. free sheaf) on X F [k]–rank rk bundle over X [k]:Fibre over ξ ∈ X [k] is F ⊗Oξ = H0(ξ,F |ξ).

Equivalently: F [k] = q∗p∗(F ) where X [k] × X ⊃ Zq //

p

��

X [k]

X

.


Geometric subsets of Hilbert schemesFollowing Rennemo (2014) we define the geometric subset of type Q = (Q1, . . . ,Qs ) as

P(Q) = {ξ ∈ X [k] : ξ = ξ1 ∪ . . . ∪ ξs where ξi ∈ X [ki ]pi

is of type Qi}.

A type Q ⊂ X [k]p is a closed subset which is union of isomorphism classes:

ξ ∈ Q, ξ ' ξ′ ⇒ ξ′ ∈ Q.

Definition: For an algebra A of dimension k let QA = {ξ ∈ X [k] : ξ ' A}.Fix monomial ideals Iλ1 , . . . , Iλs corresponding to the partitions (Young tableau) λ1, . . . , λs

and letQλi = {ξ : Oξ ' C[z1, . . . , zn]/Iλi }.

Let P(λ1, . . . , λs ) = P(Qλ1 , . . . ,Qλs ) denote the corresponding monomial geometric subset.

Simplest case: s = 1 and λ = · · ·︸︷︷︸k

then A = C[z]/zk and

Qλ = CX [k]p = {ξ ∈ X [k]

p : ξ ⊂ Cp for some smooth curve C ⊂ X}

P( · · ·︸︷︷︸k

) = CX [k] = ∪p∈XCX[k]p is the curvilinear component.

Most famous case (Göttsche): X = S surface, λ1 = . . . = λs = . Then

Iλ = (x2, xy , y2) and

P(s · λ) = P

, . . . ,︸︷︷︸s

⊂ S [3s].

is 2s-dimensional closed variety and the count of s-nodal curves in the 5s-ample linearsystem P(|L|) is

∫P(s·λ) c2s (L

[3s]).


Tautological integrals and curve counting

P(Q1, . . . ,Qs) = {ξ ∈ X [k] : ξ = ξ1 t . . . t ξs where ξi ∈ X[ki ]pi is of type Qi}

Motivation: counting hypersurfaces with prescribed singularities on X .Fact: Let T1, . . . ,Ts be analytic singularity types with expected codimensiond =

∑si=1 di . Then there is a k and a geometric set W = P(T1, . . . ,Ts) ⊂ X [k] such

that a generic hypersurface containing a Z ∈W has the specified singularities.Therefore a general Pd ⊆ |L| the number of hypersurfaces containing a subschemeZ ∈W equals

∫W cdim(W )(L

[k]).

Theorem (Existence of universal polynomial, Rennemo, Tzeng 2013-14)

For any geometric subset P(Q) ⊂ X [k] and Chern monomial R(c1, c2, . . .) of degreedim(P(Q)) the tautological integral∫

P(Q)R(ci (F

[k])) = R(ci (F ), ci (X ))

is given by a universal polynomial of the Chern numbers of F and X .

Rest of the talk: Show that non-reductive moduli description combined withequivariant localisation results in closed iterated residue formulae for

∫P(Q) R(ci (F

[k])).


s = 1: Curvilinear Hilbert schemes

CX[k]p = {ξ ∈ X

[k]p : ξ ⊂ Cp for some smooth curve C ⊂ X} =

{ξ ∈ X[k]p : Oξ ' zC[z]/zk+1} Curvilinear locus at p

Def: CX[k]p is the curvilinear component of the Hilbert scheme. This is a singular

irreducible projective variety of dimension (n − 1)(k − 1).

On surfaces (n = 2) CX[k]p = X

[k]p , for n > 2 CX

[k]p is a component of the punctual

Hilbert scheme.Non-reductive quotient model of curvilinear Hilbert schemes:ξ ∈ CX

[k+1]p ξ ⊂ Cp ⊂ X smooth curve germ fξ : (C, 0)→ (X , p) k-jet of germ

parametrising Cp . fξ is determined up to polynomial reparametrisation germsφ : (C, 0)→ (C, 0). Let

Jk (1, n) = {k-jets f : (C, 0)→ (Cn, 0)} = {(f ′, f ′′, . . . , f [k]) : f ′ 6= 0}

CX[k+1]p = {k-jets (C, 0)→ (Cn, 0)}/{k-jets (C, 0)→ (C, 0)} = Jk (1, n)/Jk (1, 1).


Observation: If g ∈ Jk (n, 1) then g ◦ fξ = 0⇒ g ◦ (fξ ◦ φ) = 0In other words: Ann(fξ) ⊂ Jk (n, 1) is invariant under Jk (1, 1). The map fξ 7→ Ann(fξ)defines an embedding

ρ : CX[k+1]p = Jk (1, n)/Jk (1, 1) ↪→ Grass(codim = k, Jk (n, 1))

If fξ = f1z + . . .+ fkzk with fi ∈ Cn;

g(v) = Av + Bv2 + . . . with A ∈ Hom(Cn,C), B ∈ Hom(Sym2(Cn),C), . . .then g ◦ fξ = 0 has the form:

A(f1) = 0,

A(f2) + B(f1, f1) = 0,

A(f3) + 2B(f1, f2) + C(f1, f1, f1) = 0,

Theorem (B-Szenes 2012, B 2015)

1 The dual of ρ can be written as

ρ : Jk (1, n)/Jk (1, 1) ↪→ Grass(k, Jk (n, 1)∗)

(f ′, . . . , f (k)) 7→ Span(f ′, f ′′ + (f ′)2, . . . ,∑

a1+a2+...+ai=j

f (a1)f (a2). . . f (ai ), . . .),

2 For k ≤ n ρ(Jk (1, n)) = GL(n) · ρ(e1, . . . , en) where {e1, . . . , en} is a basis of Cn.


Theorem (Integrals over the curvilinear Hilbert schemes, B. 2015)

∫CX

[k+1]R(ci (F )) =

∫X

Resz=∞

∏i<j (zi − zj )Qk (z)R(ci (zi + θj , θj ))dz∏

i+j≤l≤k (zi + zj − zl )(z1 . . . zk )n

k∏i=1

sX (1zi)

Features of the residue formulae:1 The residue gives a degree n symmetric polynomial in Chern roots of F and Segre

classes of X reflecting the Gottsche conjecture on curvilinear components.2 For fixed k the formula gives a universal generating series for the integrals as the

dimension increases.3 The geometry of Qk . The GLk -module of 3-tensors Hom(Ck , Sym2Ck ) has a

diagonal decomposition

Hom(Ck , Sym2Ck ) =⊕

Cqmrl , 1 ≤ m, r , l ≤ k,

qmrl has weight (zm + zr − zl ). Let

ε =k∑

m=1

k−m∑r=1

qmrm+r ⊂W =

⊕1≤m+r≤l≤k

Cqmrl ⊂ Hom(Ck , Sym2Ck )

Then Qk (z) = mdeg(Bkε,W ). In particular Q1,2,3 = 1,Q4 = 2z1 + z2 − z4.


s=1: Monomial geometric subsets supported at one point

Next step Integration on P(λ). We demonstrate this on λ = (s, 2s) = ︸︷︷︸2s

.

Two dimensional algebra ⇒ test surface to get non-reductive quotient model.ξ ∈ P(λ)p fξ : (C2, 0)→ (X , p) λ-jet of germ parametrising the surface germ Sξfor which ξ ∈ Sξ.

Jλ(C2,Cn) := {∂f

∂ ix∂jy|(i , j) ∈ ︸︷︷︸

2δ

}

fξ is determined up to λ-jets of reparametrisation germs φ : (C2, 0)→ (C2, 0). So

P(λ)p = Jλ(2, n)/Jλ(2, 2).

Observation: If g ∈ J2s(n, 2) then g ◦ fξ = 0⇒ g ◦ (fξ ◦ φ) = 0If fξ =

∑(i,j)∈λ fijx

iy j with fij ∈ Cn and g(v) = Av + Bv2 + . . . withA ∈ Hom(Cn,C2), B ∈ Hom(Sym2(Cn),C), . . .then g ◦ fξ = 0 has the form:

A(f10) = 0 A(f01) = 0

A(f20) + 2B(f10, f10) = 0 A(f11) + B(f10, f01) = 0

. . .

We have 3s equations and induces an embedding

P(λ) ↪→ Grass(3s, Sym≤2sCn)

Then we can apply equivariant localisation, turn the formula into iterated residue andprove a vanishing theorem. (HARD WORK!!!)


Theorem (Tautological integrals on punctual monomial geometric subsets, B. 2016)

Introduce the variables indexed by boxes of λ:

λ =

z01 z11 zs1

· z10 zδ0 z2s,0

Let X be a smooth projective variety, F a bundle over X , R(ci (F[3s]) a Chern

polynomial of degree dim(P(λ)) = s + 3. Then

∫P(λ)

R(ci (F[3s]) = Resz=∞

∏(a,b)<(a′,b′)

(zab − za′b′ )R(ci (θj , θj + zab : (a, b) ∈ B))∏

(a,b)∈BsS

(1

zab

)dz

∏a+b≤c≤2s

(za0 − zb0)∏

(a,b)+(a′,b′)≤(c,1)≤(s,1)(a,b)+(a′,b′)≤(c+1,0)≤(s+1,0)

(zab + za′b′ − za′b′ )(∏

(a,b)∈Bzab )

2

Similar formulae hold for other Young tableau λ’s. Non-reductive quotient model isderived using equations of test m-folds if λ is m-dimensional partition.


s ≥ 1: Monomial geometric subsets with general support

Goal: Study P(s · ) = P( , . . . ,︸︷︷︸s

). Monomial ideals can come together

different ways:

1) along a line: ← ← =

2) along different axes:

��

oo =

.

Observations:

P( ) ⊂ P( ).

codim(P( ) ⊂ P( , . . . , )) = 2.

codim(P( , ) ⊂ P( , . . . , )) = 1.


P( , , )

P( )P( , )12

P( , )23

P( , )13

Observations: If S is toric with number of T -fixed points ≥ s (we can assume this bytaking appropriate blow-ups..) then

P( , )ij ,P( ) are T -invariant subvarieties of

P( , , ).

The pair (S [s]0 ⊂ S [s]) is T -equivariant to (P(︸︷︷︸

2s

) ⊂ P(s · )). Due to

Haiman S[s]0 is complete intersection in S [s]


Lemma (Localisation at invariant subvarieties)

Suppose that M is a compact manifold and T is a complex torus acting smoothly onM, and the fixed point set MT of the T -action on M is finite. Let M1, . . . ,Mr

torus-invariant pairwise disjoint submanifolds such that MT ⊂ ∪ri=1Mi . Then for foran equivariantly closed form α ∈ H•T (M)∫

Mα =

r∑i=1

∫Mi

α[dim(Mi )]|Mi

EulerT (N(Mi )).

Here EulerT (N(Mi ) is the T -equivariant Euler class of the T -equivariant normalbundle N(Mi ) of Mi in M and α[dim(Mi )] is the differential-form-degree-dim(Mi ) partof α.

According to Kleiman and Piene we can write∫P(s· )

c2s(L[3s]) = Ns = Ps/s! where

Ps satisfy the formal identity∑

s≥0Ps t

s

rs!= exp(

∑q≥1

aq tq

q!) for some integers

a0, a1, . . .. In particular,

P0 = 1,P1 = a1,P2 = a21 + a2,P3 = a3

1 + 3a2a1 + a3, . . . .

Observation:Terms of Ps ↔ partitions of s ↔ Torus-invariant submanifolds in the

localisation formula. In particular: as =∫P(s,2s)

c2s (L[3s])

EulerT (P(︸︷︷︸2s

)⊂(P(s· )

.


The residue formula

Let S be a nonsingular projective surface and L a 5s-ample line bundle on S . LetNs(L) denote the count of s-nodal hypersurfaces in a generic linear system Ps ⊂ |L|.According to Kleiman and Piene we can write Ns = Ps/s! where Ps satisfy the formalidentity

∑s≥0

Ps ts

rs!= exp(

∑q≥1

aq tq

q!) for some integers a0, a1, . . .. In particular,

P0 = 1,P1 = a1,P2 = a21 + a2,P3 = a3

1 + 3a2a1 + a3, . . . .

Theorem (New formula for counts of s-nodal curves, B.-Szenes 2016)

Introduce the variables indexed by boxes

B =

z01 z11 zs1

· z10 zs0 z2s,0

Then for s > 1 we have

as = Resz=∞

(z10 · · · zs0)−1 ∏(a,b)<(a′,b′)

(zab − za′b′ )c2s (L, L + zab : (a, b) ∈ B)

∏(a,b)∈B

sS

(1

zab

)dz

∏a+b≤c≤2s

(za0 − zb0)∏

(a,b)+(a′,b′)≤(c,1)≤(s,1)(a,b)+(a′,b′)≤(c+1,0)≤(s+1,0)

(zab + za′b′ − z

a′b′ )(∏

(a,b)∈Bzab )

2


Examples: a1, a2

s = 1. Variables:z01

· z10

a1 = 3L2+2Lc1(S)+c2(S) = Resz=∞

(z10 − z01)2c2(L, L+ z10, L+ z01)dz2(z10z01)2

s(1/z10)s(1/z01)

where:c2(L, L + z10, L + z01) denotes the second elementary symmetric polynomial formesfrom the formal Chern roots L, L + z10, L + z01 andsS = 1/cS is the total Segre class of S and in particulars0 = 1, s1 = c1(S), s2 = c2

1 − c2.s = 2. Variables:

z01 z11

· z10 z20 z30

The following identity is now computer-checked

− 42L2 − 39Lc1(S) − 6c21 (S) − 7c2(S) = a2 =

Resz=∞

∏(a,b)<(a′,b′)∈B(zab − za′b′ )c4(L, L + z10, L + z20, L + z30, L + z01, L + z11)

∏(a,b)∈B sS

(1

zab

)dz

z10(2z10 − z20)(z10 + z20 − z30)(2z10 − z30)(z10 + z01 − z30)(z10 + z01 − z11)(2z10 − z11)(z10z20z30z01z11)2

where againc4(L, L + z10, L + z20, L + z30, L + z01, L + z11) denotes the fourth elementarysymmetric polynomial formes from the formal Chern roots L, L + zab, (a, b) ∈ B.sS = 1/cS is the total Segre class of S and in particulars0 = 1, s1 = c1(S), s2 = c2

1 − c2.


non-reductive git and curve...

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