Symmetries and Moduli Spaces Flashcards
Discuss plane conics over C - manifolds vs varieties vs schemes.
How is geometry reflected in algebra of ring?
irreducible - manifolds - integral domains
reducible - varieties - idempotents direct sums
nilpotent - schemes - nilpotent elements
pg 1
Discuss 3 ways of thinking about smooth manifolds
Relationships?
- Atlas - glue together open sets
- Embedded in R^n
- Specify ring of functions
pg 3-4
Discuss relationship between subspaces in A^n and collections of polynomials
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
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.
pg 6
Define: toric manifold
Take as charts copies of A^n and transition functions (gluings) in GL(n, Z)
pg 7
Discuss affine varieties
going from X to k[X] and from k[X] to X?
pg 7
Discuss projective space
1-dim subspaces
pg 8-9
Discuss space of conics - sets inside this space?
Veronese embedding?
Double-lines = l(x)^2 = linear poly squared – Veronese embedding
vanishing of 2x2 minors
degenerate conics - vanishing of determinant <— very large group of symmetries
Toric variety? Show how Veronese embedding a toric variety. Relations?
pg 12-13
Pencil, base locus conics in P^5?
pg 14
Discuss topology of algebraic variety defined by polynomials in C[x1, … , xn]
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
Discuss geometry algebra dictionary for affine and projective varieties
pg 19
State and prove what the fundamental pieces of projective varieties are.
Any projective variety is isomorphic to a linear section of Veronese varieties
pg 20
State and prove what the building blocks for maps between projective varieties are
Any map between projective varieties is a composition of Veronese maps and projections
pg 20
Discuss products of projective spaces
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
Discuss fiber products and graphs and pullback
Define via universal propery
Construct using graphs
Example: pullback
pg 22-24
Define: trivializable bundle, locally trivial bundle, trivialization data, section
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
Show projective space is a locally trivial bundle in Zariski topology
Bundle associated to projective variety? Importance?
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
Discuss defining bundle via open sets and gluings. Consistency conditions? Transition functions? Retrivializations?
Consider x in intersection of 2 open sets. Look at map between local trivialization. Intuit consistency conditions
pg 26-27
Define: Principal G-bundles
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
Discuss Grothendick view on points
Points = morphisms into space
pg 28-29
Define algebraic groups in 2 ways:
1. representable functors
2. classical
Group object in category of varieties (or schemes)
pg 30
Is Z an algebraic group?
pg 31-32
Define: vector bundles, operations on vector bundles?
Discuss Tautological bundle on P(V)
Locally trivial bundle, fibers = vector spaces, maps fiberwise linear
([v], u)
(+) and (x)
pg 32-33
Define: Picard group Pic(X)
{Line bundles on X} under tensor product
pg 34-35
Discuss maps into projective space. Compare to maps into Grassmanian
pg 37-38
