Symmetries and Moduli Spaces

Question 1

Discuss plane conics over C - manifolds vs varieties vs schemes.

How is geometry reflected in algebra of ring?

Answer 1

irreducible - manifolds - integral domains

reducible - varieties - idempotents direct sums

nilpotent - schemes - nilpotent elements

pg 1

Question 2

Discuss 3 ways of thinking about smooth manifolds

Relationships?

Answer 2

1. Atlas - glue together open sets
2. Embedded in R^n
3. Specify ring of functions

pg 3-4

Question 3

Discuss relationship between subspaces in A^n and collections of polynomials

Answer 3

Z : {sets of polys} –> {Zero loci}
I : {Subsets} –> {Polys vanishing on subset}

I(Z(f1, … , fn)) = radical completion
Z(I(V)) = Zariski closure

pg 4-6

Question 4

Consider the collection of points V in A^2
(1,1), (2, 4), (3, 9), … , (n, n^2), …

Show if collection is finite, then V closure = V. If infinite, V closure = parabola.

Answer 4

pg 6

Question 5

Define: toric manifold

Answer 5

Take as charts copies of A^n and transition functions (gluings) in GL(n, Z)

pg 7

Question 6

Discuss affine varieties

going from X to k[X] and from k[X] to X?

Answer 6

pg 7

Question 7

Discuss projective space

Answer 7

1-dim subspaces

pg 8-9

Question 8

Discuss space of conics - sets inside this space?

Veronese embedding?

Answer 8

Double-lines = l(x)^2 = linear poly squared – Veronese embedding

vanishing of 2x2 minors

degenerate conics - vanishing of determinant <— very large group of symmetries

Question 9

Toric variety? Show how Veronese embedding a toric variety. Relations?

Answer 9

pg 12-13

Question 10

Pencil, base locus conics in P^5?

Answer 10

pg 14

Question 11

Discuss topology of algebraic variety defined by polynomials in C[x1, … , xn]

Answer 11

If coeff embed into C, then X(C) has Hausdorff topology as a subspace of C^n. However topology of X(C) does depend on embedding. For example Serre showed fundamental group no invariant <– so we can read many things about variety off from ring but Hausdorff topology not one of them.

Thought: Zariski topology too weak to determine this finer topology

pg 17-18

Question 12

Discuss geometry algebra dictionary for affine and projective varieties

Answer 12

pg 19

Question 13

State and prove what the fundamental pieces of projective varieties are.

Answer 13

Any projective variety is isomorphic to a linear section of Veronese varieties

pg 20

Question 14

State and prove what the building blocks for maps between projective varieties are

Answer 14

Any map between projective varieties is a composition of Veronese maps and projections
pg 20

Question 15

Discuss products of projective spaces

Answer 15

Segre embedding: Consider the canonical inclusion of V x W –> V (x) W. This behaves well w.r.t. scaling so descends to a map from P(V) x P(W) –> P(V (x) W)

This is a projective variety

pg 21

Question 16

Discuss fiber products and graphs and pullback

Answer 16

Define via universal propery

Construct using graphs

Example: pullback

pg 22-24

Question 17

Define: trivializable bundle, locally trivial bundle, trivialization data, section

Answer 17

g: Y –> X a bundle

Y = X x F <– trivializable

Locally trivial if Y can be covered by open subspaces s.t. restricition of bundle to open subspace is trivializable

trivialization data: the maps phi_i from g^-1(U_i) –> U_i x F

A section of g: Y –> X is a map s:X –> Y s.t. gos = id_X

pg 24, 26

Question 18

Show projective space is a locally trivial bundle in Zariski topology

Bundle associated to projective variety? Importance?

Answer 18

Fiber = A^1 \ 0 <– line without a point

The bundle picture is the same as the def of coordinate patches on P(V)

Every projective variety comes equiped with bundle coming from pulling back the bundle over P(V) via X –> P(V)

pg 25

Question 19

Discuss defining bundle via open sets and gluings. Consistency conditions? Transition functions? Retrivializations?

Answer 19

Consider x in intersection of 2 open sets. Look at map between local trivialization. Intuit consistency conditions

pg 26-27

Question 20

Define: Principal G-bundles

Answer 20

I THINK HE INTENDS SOMETHING MORE GENERAL THEN THE FOLLOWING WIKI: A principal G-bundle, where G denotes any topological group, is a fiber bundle pi :P –> X together with a continuous right action PxG –> P such that G preserves the fibers of P and acts freely and transitively (meaning each fiber is a G-torsor) on them in such a way that for each x\in X and y\in P_{x}, the map G\to P_{x} sending g to yg is a homeomorphism. In particular each fiber of the bundle is homeomorphic to the group G itself.

Since the group action preserves the fibers of \pi :P\to X and acts transitively, it follows that the orbits of the G-action are precisely these fibers and the orbit space P/G is homeomorphic to the base space X. Because the action is free and transitive, the fibers have the structure of G-torsors. A G-torsor is a space that is homeomorphic to G but lacks a group structure since there is no preferred choice of an identity element.

An equivalent definition of a principal G-bundle is as a G-bundle \pi :P\to X with fiber G where the structure group acts on the fiber by left multiplication. Since right multiplication by G on the fiber commutes with the action of the structure group, there exists an invariant notion of right multiplication by G on P. The fibers of \pi then become right G-torsors for this action.

pg 27

Question 21

Discuss Grothendick view on points

Answer 21

Points = morphisms into space

pg 28-29

Question 22

Define algebraic groups in 2 ways:
1. representable functors
2. classical

Answer 22

Group object in category of varieties (or schemes)

pg 30

Question 23

Is Z an algebraic group?

Answer 23

pg 31-32

Question 24

Define: vector bundles, operations on vector bundles?

Discuss Tautological bundle on P(V)

Answer 24

Locally trivial bundle, fibers = vector spaces, maps fiberwise linear

([v], u)

(+) and (x)
pg 32-33

Question 25

Define: Picard group Pic(X)

Answer 25

{Line bundles on X} under tensor product

pg 34-35

Question 26

Discuss maps into projective space. Compare to maps into Grassmanian

Answer 26

pg 37-38