How to blow up orbit closures real good (Orbit closures and rational - - PDF document

How to blow up orbit closures real good (Orbit closures and rational surfaces)

Ted Chinburg November 19, 2012

Joint work with Birge Huisgen-Zimmermann and Frauke Bleher. Question: How to bound the geometry of orbit closures? Set-up:

• k = k
• Λ = finite dimensional k-algebra, basic
• P = projective finite dimensional Λ-module, P = Λǫ, with simple top T
• d < dimk P
• GrassT

d = Grassmannian of all Λ-submodules C of P so that d = dimk P/C

• d′ = dimk C = dimk P − d
• g ∈ AutΛ(P) acts on GrassT

d as C → g.C

• Y = orbit of C under AutΛ(P), Y = AutΛ(P).C ⊆ Y = closure of Y in GrassT

d

Question: How can we bound the geometry of Y using the algebra data Λ, P, C? Different Approaches:

• 1. Stratify Y into nice pieces, consider particular pieces, e.g. maximal degenerations (Huisgen-

Zimmermann).

• 2. Bound the singularities of Y (e.g. Bongartz, Zwara, Bender, Bobinski, Skowronski, . . .).
• 3. Study desingularizations of Y , classify these globally.

Hypothesis: Suppose Y = AutΛ(P).C is two-dimensional. Theorem: (Huisgen-Zimmermann) Under this hypothesis, Y ∼ = A2

k

C(ǫ + t1w1 + t2w2) ← − (t1, t2) 1

where C ⊂ P = Λǫ, t1, t2 ∈ k and w1, w2 ∈ ǫJǫ are appropriate elements when J = rad(Λ). Corollary: Y and Y are integral rational surfaces in GrassT

d .

Theorem: (Castelnuovo, Zariski, . . .) We have a diagram of birational morphisms: Y

smooth

• π
• Y

#

finite

• X

Y where

• Y

= orbit closure of Y

• Y

#

= normalization of Y ( finite over Y )

• Y

smooth = minimal desingularization of Y #

• X

= relatively minimal smooth projective surface

• π

= given by a finite sequence of “good” blow ups (at reduced closed points) Here, X is isomorphic to one of: P2, P1 × P1, Xe for e > 1. Definition of Xe’s:

• 1. These are ruled surfaces: Xe

π

− → P1, fibers ∼ = P1.

• 2. Xe = Proj(OP1 ⊕ OP1(−e)).
• 3. Xe = {(z0 : z1) × (y0 : · · · : ye+1) ∈ P1 × Pe+1 | z0yi+1 − z1yi = 0 for i = 0, . . . , e − 1}, i.e.

Xe ⊆ P1 × Pe+1.

• 4. To get Xe from X0 = P1 × P1 :

this is P1 × P1

A2

P1 × ∞ ∞ × P1

s blow up

• ✠
• ✟✟✟✟

✟

• P1 × ∞

PPPPP P E ❇ ❇ ❇ ❇ ❇ ❇

• ∞ × P1

blow down ✻

• P1 × ∞

′

E′ this is X1

(not rel. min.)

Continue same way

(blow up crossing pt, blow down new exceptional)

to get X2, . . .

2

Theorem: (Bleher, Huisgen-Zimmermann, C.) There is a bound which depends only on dimk P for the e such that X could be isomorphic to Xe, and for the number of good blow ups needed to factor π : Y

smooth → X.

Comments:

• 1. If dimk C = 1, must have X = P2.
• 2. If dimk C = 2, have examples with X = P2, P1 × P1, X2.
• 3. Can dimk P be replaced by dimk C in Theorem? Can all Xe arise?

Ideas in Proof:

• 1. Start with the orbit Y .

Have ψ : A2

∼ =

− → Y ⊆ Y (t1, t2) → C(ǫ + t1w1 + t2w2)

• 2. General theory:

ψ can be extended to a birational map ψ : P1 × P1

Y

A2 = A1 × A1

• defined outside a finite set D of closed points:

A2

P1 × ∞ × × × ∞ × P1 × ×

One has to bound #D using only dimk P. For each q ∈ D = bad points, choose local parameters z1, z2 at q so A = OP1×P1,q = k[z1, z2](z1,z2) Show that near q (but not at q), ψ : Spec(A) − q

• GrassT

d

• PN

is defined by homogeneous coordinates (f0(z1, z2) : f1(z1, z2) : · · · : fN(z1, z2)) where fi(z1, z2) ∈ k[z1, z2] have degree bounded by function of dimk P. 3

• 3. Focus on q ∈ D. Let I = ideal in A generated by f0(z1, z2), . . . , fN(z1, z2).

BI

• ψI
• Blow-upI(A)

Spec(A) − q

ψ

• Y ⊆ PN

P1 × P1 where ψI is an actual morphism. But BI is a “bad” blow up since I is complicated. Theorem of Zariski: There exists a finite sequence of “good” blow ups of Spec(A) giving: WI

(∗)

• BI
• Spec(A)

Spec(A) − q

• ψ

Y ⊆ PN

where (∗) is a finite sequence of “good” blow ups (i.e. successive blow ups of reduced closed points). Must show there exists uniform bound on number of “good” blow ups needed, independent

• f I, dependent only on dimk P.

(For this, look at Grassmannian of all possible ideals I, look at generic point, then use Noetherian induction.) Stitch together all “good” blow ups at q ∈ D to produce a bounded smooth projective blow up of P1 × P1 which dominates Y . Example: Let Λ = kQ/I, where Q = 1

ω

• α
• β

2

I = ω3, βω2 ⊆ kQ P = Λe1 ⊇ C = k(α + β) + kαω 8-dim. 2-dim. Y = AutΛ(P).C = {C · (e1 + t1ω + t2ω2) | (t1, t2) ∈ A2

k}

X2

• Y

=

• P1 × ∞

′′

E′

1

X2 is constructed as in #4. of the definition

• f the Xe’s

blow down

• P1 × ∞

′′

to obtain Y

4