what is a moduli space? - rebecca tramel · 2019. 2. 14. · projective space in general, the...

What is a Moduli Space?

Becca Tramel

Mount Holyoke College

Smith College Lunch Talk

February 14, 2019

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Overview

1 Definitions

2 Projective SpaceLines in R2

P(V )

3 GrassmanniansDefinitionGr(2, 4)Gr(r ,V )

4 Other Moduli Spaces

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Algebraic Geometry

In the field of Algebraic Geometry, we study the geometry ofsolution sets of polynomial equations.

An affine variety is the set of solutions to a set of polynomialequations in a vector space V . (Rn, Cn, (Z/pZ)n, . . . )

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Examples

y = x2

-10 -5 5 10

20

40

60

80

100

y2 = x3 − 2x + 1 Elliptic Curve

-4 -2 2 4

-5

5

Torus

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Moduli Space: Intuitive Definition

Goal: Study a class of geometric objects all at once.

Intuitive Definition

A moduli space is a geometric space M satisying that:

{Points in M} ↔ {Objects in the class}Moving “continuously” in M corresponds to “continuous”deformation of the objects being studied.

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First Example: Lines in R2

To construct a moduli space of lines through the origin in R2, we need tofind a way to specify a line.

-3 -2 -1 1 2 3

-6

-4

-2

2

4

6

Ideas

Equation y = mx + b (b = 0).

Slope m.

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Slope as a parameter

All lines through the origin can be specified by one real number, the slope,. . . except for one.

-3 -2 -1 1 2 3

-6

-4

-2

2

4

6

0 mR

We need to add a point at ∞ for the vertical line. But we must gotowards this same ∞ traveling on R in both directions.

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P(R2)

So the moduli space of lines through the origin in R2 is a circle!

0

∞

This circle is called P(R2), the projectivization of R2, or the realprojective line.

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Coordinates on P(R2)

To specify a point on P(R2), we could give the slope, but this makes∞ special.

Another idea: specify a point on the line (x , y).

Enough since two points define a line, and (0, 0) is on all the lines.But then get repeats: (2, 1) and (4, 2) specify the same line.

Homogeneous coordinate system: Specify points as [x : y ] where[x : y ] = [kx : ky ] for any constant k 6= 0.

P(R2) = (R2 − (0, 0))/ ∼

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Coordinates on P(R2)

(1, 0)

(0, 1)

(x , y)

[1 : 0]

[0 : 1]

[x : y ]

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P(V )

Projective space

In general, the moduli space of lines through the origin in a vector space Vis called P(V ), the projectivization of V . P(V ) is called a projectivespace.

Note: If V is n-dimensional, then P(V ) is (n − 1)-dimensional.

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Other projectives spaces

P(C2) - the Riemann sphere P((Z/2Z)3) - the Fano plane

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Grassmannians

Definition

The moduli space of r -dimensional subspaces of a vector space V is calleda Grassmannian, and denoted Gr(r ,V ).

Gr(1,V ) ∼= P(V ). If V is n-dimensional, Gr(1,V ) is(n − 1)-dimensional.

If V is n-dimensional, Gr(n − 1,V ) ∼= P(V ) too!

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Gr(n − 1,V )

In R3, every plane has a normal vector. This gives a one-to-onecorrespondence between lines and planes through the origin.In general, every n − 1-dimensional subspace of an n-dimensionalvector space V has a normal vector.

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Gr(2, 4)

The first “interesting” example is Gr(2,V ), where V is 4-dimensional(sometimes written Gr(2, 4)).

To construct Gr(2, 4) we need to specify a plane in V .

Would need 2 linear equations!OR: give two vectors which are not colinear. Their span is a planethrough the origin.

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Plucker coordinates

Given two pairs of vectors: ~u, ~v and ~a, ~b, how can we tell if theydefine the same plane?

If ~u = (u0, u1, u2, u3) and ~v = (v0, v1, v2, v3), look at the matrixu0 v0u1 v1u2 v2u3 v3

.

The Plucker coordinate pij is the determinant of the submatrix(ui viuj vj

).

Get 6 coordinates: p01, p02, p03, p12, p23, p31.

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Plucker coordinates

For example, the vectors (2, 1, 0, 1) and (0, 1, 1, 0) define a plane whosePlucker coordinates are calculated as follows:

For p01:

2 01 10 11 0

,

p01 = (2)(1)− (0)(1) = 2.

p01 = 2

p02 = 2

p03 = 0

p12 = 1

p23 = −1

p31 = 0.

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Plucker coordinates

What if we pick two other vectors in the same plane, for example(2, 1, 0, 1) and (2, 3, 2, 1)?

For p01:

2 21 30 21 1

,

p01 = (2)(3)− (2)(1) = 4.

p01 = 4

p02 = 4

p03 = 0

p12 = 2

p23 = −2

p31 = 0.

Comparison: (2, 2, 0, 1,−1, 0) versus (4, 4, 0, 2,−2, 0).

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Gr(2,V ) in projective space

Plucker coordinates for Gr(2,V ) define a point[p01 : p02 : p03 : p12 : p23 : p31] in a projective space (5-dimensional).

But not all points in this projective space correspond to a plane.

The Plucker coordinates of a plane satisfy a (homogeneous) equation:

p01p23 + p02p31 + p03p12 = 0.

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Plucker relation

Where does this equation, called a Plucker relation come from?

p01p23 + p02p31 + p03p12 = (u0v1 − u1v0)(u2v3 − u3v2)+

(u0v2 − u2v2)(u3v1 − u1v3) + (u0v3 − u3v0)(u1v2 − u2v1).

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Gr(2,V ) as a projective variety

Definition

A projective variety is the solution set to a set of homogeneouspolynomials (polynomials in which every term has the same degree) in aprojective space.

Gr(2,V ) is a projective variety!

It is a 4 dimensional space inside of a 5-dimensional projective space.

It is the solution set to a single degree 2 homogeneous polynomial.

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Geometry of Gr(2,V )

We can ask questions about the geometry of the set of planes in V asquestions about Gr(2,V ).

For example, if we fix one plane P in V , can we find the other pointsin Gr(2,V ) corresponding to planes that intersect P in a line?

If [P01,P02,P03,P12,P23,P31] are the coordinates corresponding to P,then another plane with coordinates [p01, p02, p03, p12, p23, p31] willintersect it in a line if and only if the rank of the matrix whosecolumns span both planes is 3.

This becomes the equation

P01p23 + P02p31 + P03p23 = 0.

This extra requirement means there is a 3-dimensional subspace ofGr(2,V ) (a cross-section) corresponding to such planes.

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Gr(r ,V )

More generally, Gr(r ,V ) the space of r -dimensional subspaces of avector space V , is a projective variety.

An r -dimensional subspace is defined by r linearly independentvectors. If we organize them as columns in a matrix(~v1 ~v2 · · · ~vr

)we can get Plucker coordinates as the

determinant of the r × r submatrices.

These coordinates will satisfy equations called the Plucker relations.

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Difficulties in the construction of moduli spaces

In general, cannot construct a moduli space of a class of objects ifeach object has lots of automorphisms.

For example, if we try to construct a moduli space of spheres(P(C2)), we run into trouble because they can be “spun around”.

One solution: look instead at stable objects in your class. Differentchoices of stable objects give different moduli spaces and differentinsights about your class of objects!

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Thank you!!!

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