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What is a Moduli Space?
Becca Tramel
Mount Holyoke College
Smith College Lunch Talk
February 14, 2019
Becca Tramel (Mount Holyoke College) Moduli Spaces February 14, 2019 1 / 25
Overview
1 Definitions
2 Projective SpaceLines in R2
P(V )
3 GrassmanniansDefinitionGr(2, 4)Gr(r ,V )
4 Other Moduli Spaces
Becca Tramel (Mount Holyoke College) Moduli Spaces February 14, 2019 2 / 25
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, . . . )
Becca Tramel (Mount Holyoke College) Moduli Spaces February 14, 2019 3 / 25
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
Becca Tramel (Mount Holyoke College) Moduli Spaces February 14, 2019 4 / 25
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.
Becca Tramel (Mount Holyoke College) Moduli Spaces February 14, 2019 5 / 25
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.
Becca Tramel (Mount Holyoke College) Moduli Spaces February 14, 2019 6 / 25
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.
Becca Tramel (Mount Holyoke College) Moduli Spaces February 14, 2019 7 / 25
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.
Becca Tramel (Mount Holyoke College) Moduli Spaces February 14, 2019 8 / 25
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))/ ∼
Becca Tramel (Mount Holyoke College) Moduli Spaces February 14, 2019 9 / 25
Coordinates on P(R2)
(1, 0)
(0, 1)
(x , y)
[1 : 0]
[0 : 1]
[x : y ]
Becca Tramel (Mount Holyoke College) Moduli Spaces February 14, 2019 10 / 25
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.
Becca Tramel (Mount Holyoke College) Moduli Spaces February 14, 2019 11 / 25
Other projectives spaces
P(C2) - the Riemann sphere P((Z/2Z)3) - the Fano plane
Becca Tramel (Mount Holyoke College) Moduli Spaces February 14, 2019 12 / 25
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!
Becca Tramel (Mount Holyoke College) Moduli Spaces February 14, 2019 13 / 25
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.
Becca Tramel (Mount Holyoke College) Moduli Spaces February 14, 2019 14 / 25
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.
Becca Tramel (Mount Holyoke College) Moduli Spaces February 14, 2019 15 / 25
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.
Becca Tramel (Mount Holyoke College) Moduli Spaces February 14, 2019 16 / 25
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.
Becca Tramel (Mount Holyoke College) Moduli Spaces February 14, 2019 17 / 25
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).
Becca Tramel (Mount Holyoke College) Moduli Spaces February 14, 2019 18 / 25
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.
Becca Tramel (Mount Holyoke College) Moduli Spaces February 14, 2019 19 / 25
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).
Becca Tramel (Mount Holyoke College) Moduli Spaces February 14, 2019 20 / 25
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.
Becca Tramel (Mount Holyoke College) Moduli Spaces February 14, 2019 21 / 25
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.
Becca Tramel (Mount Holyoke College) Moduli Spaces February 14, 2019 22 / 25
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.
Becca Tramel (Mount Holyoke College) Moduli Spaces February 14, 2019 23 / 25
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!
Becca Tramel (Mount Holyoke College) Moduli Spaces February 14, 2019 24 / 25
Thank you!!!
Becca Tramel (Mount Holyoke College) Moduli Spaces February 14, 2019 25 / 25