Arithmetic Kolchin Irreducibility

Transcription

1 Arithmetic Kolchin Irreducibility Taylor Dupuy (with James Freitag and Lance E. Miller)

2 Kolchin irreducible =) J 1 (X) irreducible (singular)

3 Let A be a C-algebra X variety over C J n (X),J 1 (X) = new varieties over C = higher order tangent spaces J n (X)(A) =X(A[T ]/(t n+1 )) J 1 (X)(A) =X(A[[T ]])

4 Gillet, Mustata, de Fernex, Loeser-Sebag, Kolchin, Nicaise- Sebag, Ishii-Kollar, (Chambert-Loir)-Nicaise-Sebag

5 Kolchin irreducible =) J 1 (X) irreducible (singular) Gillet, Mustata, de Fernex, Loeser-Sebag, Kolchin, Nicaise- Sebag, Ishii-Kollar, (Chambert-Loir)-Nicaise-Sebag

6 Example 1. J 1 (X) X

7 Example 1. J 1 (X) X

8 Example 2. X : x 4 + y 4 + z 4 =0 expected dimension of J 3 (X) =2 4=8 dimension above (0, 0, 0) = 9 proof: J 3 (X) =pluginc[t]/(t 4 ) valued points x = x 0 + x 1 t + x 2 t 2 + x 3 t 3 mod t 4 y = y 0 + y 1 t + y 2 t 2 + y 3 t 3 mod t 4 z = z 0 + z 1 t + z 2 t 3 + z 3 t 3 mod t 4

9 Example 2. X : x 4 + y 4 + z 4 =0 expected dimension of J 3 (X) =2 4=8 dimension above (0, 0, 0) = 9 proof: J 3 (X) =pluginc[t]/(t 4 ) valued points x = x 0 + x 1 t + x 2 t 2 + x 3 t 3 mod t 4 y = y 0 + y 1 t + y 2 t 2 + y 3 t 3 mod t 4 z = z 0 + z 1 t + z 2 t 3 + z 3 t 3 mod t 4 (x 1 t + x 2 t 2 + x 3 t 3 ) 4 +(y 1 t + y 2 t 2 + y 3 t 3 ) 4 +(z 1 t + z 2 t 3 + z 3 t 3 ) 4 0

10 Mustata: lct(x, D) =dim(x) sup r 0 dim J r (D) r +1 x 5 + y 5 + z 5 x 3 + y 3 + z x y 99

11 Proof of Kolchin Irreducibility Step 1: Deformations = Irreducibility. Step 2: Smooth case. Step 3: Reduction to Smooth Case

12 Step 1: Deforming Arcs = Irreducibility J 1 (X) P e Arc Deformability: P X 0 deforms e Y (= X sing ) 0 generically outside Y

13 Step 2: Smooth Case (Classical) Theorem. smooth, irreducible =) J r (X) irreducible J r (X) X

14 Step 2: Smooth Case (Classical) Theorem. smooth, irreducible =) J r (X) irreducible J r (X) proof assuming lemma: X

15 Step 2: Smooth Case (Classical) Theorem. smooth, irreducible =) J r (X) irreducible J r (X) proof assuming lemma: r 1 (U) = U A (r+1) dim(x) X

16 Step 2: Smooth Case (Classical) Theorem. smooth, irreducible =) J r (X) irreducible J r (X) proof assuming lemma: r 1 (U) = U A (r+1) dim(x) O( 1 r (U)) = O(U)[ variables ] X

17 Step 2: Smooth Case (Classical) Theorem. smooth, irreducible =) J r (X) irreducible J r (X) proof assuming lemma: r 1 (U) = U A (r+1) dim(x) O( 1 r (U)) = O(U)[ variables ] domain X

18 Step 3: Reduction to Smooth Case (classical) J 1 (Sm(X)) J 1 (X)

19 Step 3: Reduction to Smooth Case (classical) J 1 (Sm(X)) J 1 (X) irreducible

20 Step 3: Reduction to Smooth Case (classical) J 1 (Sm(X)) J 1 (X) irreducible

21 Step 3: Reduction to Smooth Case (classical) = J 1 (Sm(X)) J 1 (X) irreducible

22 Step 3: Reduction to Smooth Case (classical) = J 1 (Sm(X)) J 1 (X) irreducible X 1 Sing(X) =Y 1 2 J 1 (Y ) (Y )

23 Step 3: Reduction to Smooth Case (classical) = J 1 (Sm(X)) J 1 (X) irreducible X 1 Sing(X) =Y 1 2 J 1 (Y ) (Y )

24 ex Step 3: Reduction to Smooth Case (classical) J 1 (Sm(X)) J 1 (X) X h 1 Sing(X) =Y 1 2 J 1 (Y ) (Y )

25 Step 3: Reduction to Smooth Case (classical) ex X h 1 Sing(X) =Y

26 Step 3: Reduction to Smooth Case (classical) ex X h 1 Sing(X) =Y

27 Step 3: Reduction to Smooth Case (classical) ex X h 1 Sing(X) =Y

28 Step 3: Reduction to Smooth Case (classical) ex X h Sing(X) =Y

29 Step 3: Reduction to Smooth Case (classical) X 0 1 1

30 Step 3: Reduction to Smooth Case (classical) X J 1 (Y )

31 Step 3: Reduction to Smooth Case (classical) X J 1 (Y ) J 1 (Sm(X)) J 1 (X)

32 Step 3: Reduction to Smooth Case (classical) X J 1 (Y ) 1 2 J 1 (Sm(X)) J 1 (X)

33 Step 3: Reduction to Smooth Case (classical) X J 1 (Y ) 1 2 J 1 (Sm(X)) = J 1 (X)

34 Recap of Classical Step 1: Deformations = Irreducibility J r (X) Step 2: Smooth case Step 3: Reduction to Smooth Case X ex X h Sing(X) =Y 1 2J 1 (Y )

35 Arithmetic Jet Spaces 1. work over b Z ur p 2. replace power series with Witt vectors J p,r (X)(A) =X(W p.r (A))

36 Theorem (Buium) bx smooth and integral =) b J p,1 ( b X) integral Theorems (Dupuy-Frietag-Miller) X smooth and a bx integral ne =) J p,1 (X) irreducible Y! X (weak) a ne smoothening by integral =) J p,1 (X) (weakly) irreducible

37 Example of a conditional result: S1 X smooth and b X integral =) J p,1 (X) irreducible. S2 Y! X (weak) smoothening by integral =) J p,1 (X) (weakly) irreducible S1 =) S2

38 Step 2: Smooth Case Theorem. (Buium) bj p,r (X) X/R smooth R = W p,1 (F alg p ) bj p,r (X)! b X an affine bundle bx Corollary. smooth bx irreducible b Jp,r (X) =) irreducible

39 Smoothenings Alterations?? (Introduces Ramification) Spec(K) / e X Spec(R) / X Neron Smoothenings (Sebag-Loeser,Nicaise-(Chambert- Loir)):

40 Smoothenings Alterations?? (Introduces Ramification) Spec(K) / e X Spec(R) / X Neron Smoothenings (Sebag-Loeser,Nicaise-(Chambert- Loir)): 9h : Y! X Y smooth, b Y irreducible. Y (W p,1 (F alg p ))! X(W p,1 (F alg p )) surjective

41 THANK YOU