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 = Λ ǫ , with simple top T • d < dim k P • Grass T d = Grassmannian of all Λ-submodules C of P so that d = dim k P/C • d ′ = dim k C = dim k P − d • g ∈ Aut Λ ( P ) acts on Grass T d as C → g.C • Y = orbit of C under Aut Λ ( P ), Y = Aut Λ ( P ) .C ⊆ Y = closure of Y in Grass T 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 = Aut Λ ( P ) .C is two-dimensional . Theorem: (Huisgen-Zimmermann) Under this hypothesis, ∼ A 2 Y = k C ( ǫ + t 1 w 1 + t 2 w 2 ) ← − ( t 1 , t 2 ) 1

� � � where C ⊂ P = Λ ǫ , t 1 , t 2 ∈ k and w 1 , w 2 ∈ ǫJǫ are appropriate elements when J = rad(Λ). Corollary: Y and Y are integral rational surfaces in Grass T d . Theorem: (Castelnuovo, Zariski, . . . ) We have a diagram of birational morphisms: smooth Y # π Y finite X Y where • Y = orbit closure of Y # • Y = normalization of Y ( finite over Y ) smooth = minimal desingularization of Y # • 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: P 1 × P 1 , P 2 , X e for e > 1 . Definition of X e ’s: π → P 1 , fibers ∼ = P 1 . 1. These are ruled surfaces: X e − 2. X e = Proj( O P 1 ⊕ O P 1 ( − e )). 3. X e = { ( z 0 : z 1 ) × ( y 0 : · · · : y e +1 ) ∈ P 1 × P e +1 | z 0 y i +1 − z 1 y i = 0 for i = 0 , . . . , e − 1 } , i.e. X e ⊆ P 1 × P e +1 . 4. To get X e from X 0 = P 1 × P 1 : blow up PPPPP ✟ ′ � P 1 × ∞ P 1 × ∞ E � ✟✟✟✟ � ✠ Continue P s ❇ same way ❇ � A 2 P 1 × ∞ ❇ E ′ � � (blow up ❇ crossing pt, ❇ ❇ blow down ∞ × P 1 � new exceptional) ∞ × P 1 ✻ this is X 1 to get X 2 , . . . this is P 1 × P 1 (not rel. min.) blow down 2

� � � � � � � � Theorem: (Bleher, Huisgen-Zimmermann, C.) There is a bound which depends only on dim k P for the e such that X could be isomorphic to X e , smooth → X . and for the number of good blow ups needed to factor π : Y Comments: 1. If dim k C = 1, must have X = P 2 . 2. If dim k C = 2, have examples with X = P 2 , P 1 × P 1 , X 2 . 3. Can dim k P be replaced by dim k C in Theorem? Can all X e arise? Ideas in Proof: 1. Start with the orbit Y . Have ∼ = A 2 ψ : − → Y ⊆ Y ( t 1 , t 2 ) → C ( ǫ + t 1 w 1 + t 2 w 2 ) 2. General theory: ψ can be extended to a birational map P 1 × P 1 � Y ψ : A 2 = A 1 × A 1 defined outside a finite set D of closed points: P 1 × ∞ × × × A 2 × × ∞ × P 1 One has to bound # D using only dim k P . For each q ∈ D = bad points, choose local parameters z 1 , z 2 at q so A = O P 1 × P 1 ,q = k [ z 1 , z 2 ] ( z 1 ,z 2 ) Show that near q (but not at q ), Grass T ψ : Spec( A ) − q d P N is defined by homogeneous coordinates ( f 0 ( z 1 , z 2 ) : f 1 ( z 1 , z 2 ) : · · · : f N ( z 1 , z 2 )) where f i ( z 1 , z 2 ) ∈ k [ z 1 , z 2 ] have degree bounded by function of dim k P . 3

� � � � � � � � � � � � 3. Focus on q ∈ D . Let I = ideal in A generated by f 0 ( z 1 , z 2 ) , . . . , f N ( z 1 , z 2 ). B I Blow-up I ( A ) ψ I ψ Y ⊆ P N Spec( A ) − q P 1 × P 1 where ψ I is an actual morphism. But B I 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: � B I W I ( ∗ ) ψ � Y ⊆ P N Spec( A ) Spec( A ) − q � � 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 of I , dependent only on dim k 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 P 1 × P 1 which dominates Y . Example: Let Λ = kQ/I , where α Q = 1 � 2 ω β � ω 3 , βω 2 � ⊆ kQ I = P = Λ e 1 ⊇ C = k ( α + β ) + kαω 8-dim. 2-dim. Aut Λ ( P ) .C = { C · ( e 1 + t 1 ω + t 2 ω 2 ) | ( t 1 , t 2 ) ∈ A 2 Y = k } ′′ � � X 2 = P 1 × ∞ X 2 is constructed as in #4. of the definition blow of the X e ’s E ′ Y 1 down ′′ � � P 1 × ∞ to obtain Y 4