[PPT] - Recursions and Colored Hilbert Schemes Anna Brosowsky [Joint Work PowerPoint Presentation

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1/26 Recursions and Colored Hilbert Schemes

Recursions and Colored Hilbert Schemes

Anna Brosowsky [Joint Work With: Nathaniel Gillman, Murray Pendergrass] [Mentor: Dr. Amin Gholampour]

Cornell University

24 September 2016 WiMiN Smith College

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2/26 Recursions and Colored Hilbert Schemes

Outline

1

Background

2

Recursion

3

Future Work

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3/26 Recursions and Colored Hilbert Schemes Background

Outline

1

Background

2

Recursion

3

Future Work

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4/26 Recursions and Colored Hilbert Schemes Background

The Problem

Object of Study: The Hilbert scheme of type (m0, m1). Long Term Goal: Find the Poincar´ e polynomial of the punctual Hilbert scheme of type (m0, m1). Why? The Poincar` e polynomial is a topological invariant, meaning it doesn’t change with stretching and bending. Short Term Goal: Count the points of the punctual Hilbert scheme of type (m0, m1). Why? We can use this to find a generating function, which we can then use in the Weil conjectures, to find the Poincar´ e polynomial.

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5/26 Recursions and Colored Hilbert Schemes Background

What is the punctual Hilbert scheme of type (m0, m1)?

We’ll first introduce some background:

1 Monomial ideals 2 Young diagrams 3 Group actions 4 Colored Young diagrams

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6/26 Recursions and Colored Hilbert Schemes Background

Ideals

Definition An ideal I ⊂ k[x, y] for a field k is a set of polynomials, but with a few rules attached: α, β ∈ I implies α + β ∈ I α ∈ I and m ∈ k[x, y] implies α · m ∈ I Theorem Every polynomial ideal can be written as g1, . . . , gn = {f1g1 + · · · + fngn | fi ∈ k[x, y]}, which is the set of all k[x, y] linear combinations of the gi. These gi are called the generators.

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7/26 Recursions and Colored Hilbert Schemes Background

Monomial ideals

Definition A monomial ideal is an ideal generated by monomials. Example x = {Ax |A ∈ k[x, y]}

x3, y3

= {Ax3 + By3 | A, B ∈ k[x, y]} are monomial ideals within the polynomial ring k[x, y]. x + y = {A(x + y) | A ∈ k[x, y]} is not a monomial ideal.

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8/26 Recursions and Colored Hilbert Schemes Background

Young diagrams

A Young diagram is a visual representation of a monomial ideal. Example We’ll construct a Young diagram for the monomial ideal

x4, x2y, xy3, y4

⊂ k[x, y]

✲

x

✻

y

♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣

y4 xy3 x2y x4

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9/26 Recursions and Colored Hilbert Schemes Background

Group actions

Our Z2 group action is defined by x → −x, y → −y Example Under the given transformation: 1 = x0y0 → (−x)0(−y)0 = 1, ∴ 1 → 1 x3y5 → (−x)3(−y)5 = x3y5, ∴ x3y5 → x3y5 y5 = x0y5 → (−x)0(−y)5 = −y5, ∴ y5 → −y5 and in general, xayb → (−1)a+bxayb

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10/26 Recursions and Colored Hilbert Schemes Background

Colored Young diagrams

We can combine the Young diagram and the group action to “color the Young diagram.” Procedure:

1 Draw the Young diagram as before 2 If the monomial for a box maps to itself, we “color” the box

with a 0

3 If the monomial for a box maps to the negative of itself, we

“color” the box with a 1

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11/26 Recursions and Colored Hilbert Schemes Background

Example: coloring a Young diagram

Example Recall our previous ideal,

x4, x2y, xy3, y4

⊂ k[x, y]. We can see that 1 → 1, x → −x, y → −y, xy → xy, etc.

✲

x

✻

y

♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣

→

✲

x

✻

y

♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣

1 1 1 1 1

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12/26 Recursions and Colored Hilbert Schemes Background

And finally... the punctual Hilbert scheme

The punctual Hilbert scheme of type (m0, m1) is defined as Hilb(m0,m1) k2 =

I ⊆ k[x, y]
k[x, y]

I ≃ m0ρ0 + m1ρ1, V (I) = 0

where k = Fq is a finite field of order q. But the subset of

Hilb(m0,m1) k2 made of monomial ideals looks like {Young diagrams with m0 0’s and m1 1’s}

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13/26 Recursions and Colored Hilbert Schemes Background

Example: punctual Hilbert scheme of type (4, 5)

Example Recall the monomial ideal

x4, x2y, xy3, y4

⊂ k[x, y].

✲

x

✻

y

♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣

1 1 1 1 1 By definition, the ideal

x4, x2y, xy3, y4

is in Hilb(4,5) k2, since the ideal has 4 0’s and 5 1’s when represented as a colored Young diagram under our group action.

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14/26 Recursions and Colored Hilbert Schemes Recursion

Outline

1

Background

2

Recursion

3

Future Work

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15/26 Recursions and Colored Hilbert Schemes Recursion

Stratified Hilbert schemes

The stratified punctual Hilbert scheme of type (m0, m1), (d0, d1), written Hilb(m0,m1),(d0,d1) k2, is a subset of Hilb(m0,m1)

k2. The

strata is determined by the first box outside the Young diagram in each row. d0 = number of those boxes containing zero d1 = number of those boxes containing one.

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16/26 Recursions and Colored Hilbert Schemes Recursion

Stratified Hilbert scheme example

Example y4, xy3, x2y, x4 ∈ Hilb(4,5),(3,1) k2

✲

x

✻

y

♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣

1 1 1 1 1 1 Hilb(m0,m1) k2 =

d0,d1≥0

Hilb(m0,m1),(d0,d1) k2 ⇒ #Hilb(m0,m1) k2 =

d0,d1≥0

#Hilb(m0,m1),(d0,d1) k2

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17/26 Recursions and Colored Hilbert Schemes Recursion

Example

✲

x

✻

y

♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣

1 1 1 1 1 1

→

✲

x

✻

y

♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣

1 1 0 1 1

→

✲

x

✻

y

♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣

0 1 0

→

✲

x

✻

y

♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣

0 1

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18/26 Recursions and Colored Hilbert Schemes Recursion

Recursion

#Hilb(m0,m1),(d0,d1) k2 =

0≤d′

0≤d1

0≤d′

1≤d0

d′

1−d′ 0=d0−d1+(−1)(d′ 0+d′ 1)((d′ 0+d′ 1+d0+d1)%2)

qr·#Hilb(m0−d1,m1−d0),(d′

0,d′ 1)

k2 where r =

d′

if d0 + d1 ≡ 0 mod 2 d′

1

if d0 + d1 ≡ 1 mod 2 Let a, b, c, d ∈ Z≥0. The base cases are #Hilb(0,b>0),(c,d) k2 = 0 #Hilb(a,b),(c>b,d) k2 = 0 #Hilb(a,b),(c,d>a) k2 = 0 #Hilb(a=0,b),(0,0) k2 = 0 #Hilb(0,0),(0,0) k2 = 1

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19/26 Recursions and Colored Hilbert Schemes Recursion

Why (m0 − d1, m1 − d0)?

Recurse by “chopping off” first column and sliding diagram

ver, then counting which ideals give same diagram

Same as removing last block in each row 0 outside diagram ⇒ 1 in last box 1 outside diagram ⇒ 0 in last box Also requires d′

0 ≤ d1 and d′ 1 ≤ d0 in smaller scheme

1 0 0 1 0 1 1 0 1 0 0 1 0 1 0 1 0 1 − → 0 1 0 1 0 1 0 1 0 1 0 1 0

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20/26 Recursions and Colored Hilbert Schemes Recursion

Recursion

#Hilb(m0,m1),(d0,d1) k2 =

0≤d′

0≤d1

0≤d′

1≤d0

d′

1−d′ 0=d0−d1+(−1)(d′ 0+d′ 1)((d′ 0+d′ 1+d0+d1)%2)

qr·#Hilb(m0−d1,m1−d0),(d′

0,d′ 1)

k2 where r =

d′

if d0 + d1 ≡ 0 mod 2 d′

1

if d0 + d1 ≡ 1 mod 2 Let a, b, c, d ∈ Z≥0. The base cases are #Hilb(0,b>0),(c,d) k2 = 0 #Hilb(a,b),(c>b,d) k2 = 0 #Hilb(a,b),(c,d>a) k2 = 0 #Hilb(a=0,b),(0,0) k2 = 0 #Hilb(0,0),(0,0) k2 = 1

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21/26 Recursions and Colored Hilbert Schemes Recursion

Some Special Cases

#Hilb(k,k+1),(k+1,k−1) k2 = 1 #Hilb(k,k+1),(k+1,k−1) k2 = 1 #Hilb(k,k),(1,0) k2 = qk #Hilb(k+1,k),(0,1) k2 = qk #Hilb(k,k+1),(2,0) k2 = k

2

qk−1

#Hilb(k+1,k),(0,2) k2 = k

2

qk

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22/26 Recursions and Colored Hilbert Schemes Future Work

Outline

1

Background

2

Recursion

3

Future Work

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23/26 Recursions and Colored Hilbert Schemes Future Work

Generating Function

Prove Dr. Gholampour’s conjectured generating function for the number of points in the punctual Hilbert scheme of n points

n0,n1≥0

(#Hilbn0,n1) tn0

0 tn1 1 =

j≥1

1 (1 − qj−1(t0t1)j)(1 − qj(t0t1)j) ·

m∈Z

tm2

0 tm2+m 1

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24/26 Recursions and Colored Hilbert Schemes Future Work

Future Work

Current future tasks and questions include Complete long term task

Prove generating function for #Hilb(m0,m1) k2 Use this to prove generating function for the non-punctual version of the Hilbert scheme Apply Weil conjectures to generating function; get Poincar´ e polynomial

Study different group actions

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25/26 Recursions and Colored Hilbert Schemes Appendix For Further Reading

References I

K. Yoshioka.

The Betti numbers of the moduli space of stable sheaves of rank 2 on P2.

J. reine angew. Math., 453:193–220, 1994.
A. Weil

Numbers of Solutions of Equations in Finite Fields.

Bull. Amer. Math. Soc., 55:497-508, 1949.

´

A. Gyenge, A. N´

emethi, and B. Szendr˝

i.

Euler characteristics of Hilbert schemes of points on simple surface singularities. http://arxiv.org/abs/1512.06848, 2015.

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26/26 Recursions and Colored Hilbert Schemes Appendix For Further Reading

References II

S.M. Gusein-Zade, I. Luengo, and A. Melle-Hern´ andez. On generating series of classes of equivariant Hilbert schemes

f fat points.

Moscow Mathematical Journal, 10(3):593–602, 2010.

L. G´
ttsche.

The Betti numbers of the Hilbert scheme of points on a smooth projective surface.

Math. Ann. 286, 193–207, 1990.