# GIT lecture 12

This lecture ( GIT11-12) was given on 9/23/09

### Excerpts

In the notes , where is the vector space of all homogeneous forms in variables of degree . There is a smooth and singular part: . The former is characterized when all partials of the given form only vanish at the origin.

Recall where is a hypersurface and is a singular point of . sits inside a product so there are two projections.

Now because iff all partials of vanish at which is conditions.

There is also a claim that . For a generic point should have so . The point is we want to say the fibers of are generically 0-dimensional so and from the previous inequality conclude that and is in fact a hypersurface cut out by a polynomial, the descriminant . By why is ?

Essentially because powers of the determinant are the only nontrivial characters on , and because the action of preserves the smooth and singular locus, it follows that is a semi invariant which, in particular, will be fixed by .

Lemma 12.2 implies the stabilizer of any in is finite.

the proof uses a proposition in lecture 8 about reducing to lie algebras (among other things).

Proposition 12.3 Every is properly stable with respect to the action of .

### Classical Binary Invariants

This next bit is about finding invariants. An important tool is the resultant; its a good thing to look up on wikipedia. The important point is that the resultant of two polynomials vanishes when they have common zeros. So the resultant applied to will determine if is singular or not.

Note about . Say it acts on a vector space , then so their is an action by elements ; let’s represent this as . Then gives a character on . This is a function on .

Taking th symmetric powers we also get an inducted action . It can be checked

Now . In general, is a linear combination of (why is this so?) This can be used to determine . An element of determines a trivial 1 dimensional representation of , so the dimension can be determined by counting the number of trivial representations that appear in , that is the number of . This is exactly

Note there are some error in the posted notes, basically a lot of places things like are written where really it should be . This makes a HUGE difference, be cautious.

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- Published:
- September 24, 2009 / 8:00 pm

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