ZARISKI CANCELLATION FOR SURFACES Mikhail ZAIDENBERG (joint with - - PowerPoint PPT Presentation

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ZARISKI CANCELLATION FOR SURFACES Mikhail ZAIDENBERG (joint with Hubert FLENNER and Shulim KALIMAN) May 15, 2018 Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES ZARISKI CANCELLATION QUESTION (Oskar Zariski, Algebra Congress, Paris

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SLIDE 1

ZARISKI CANCELLATION FOR SURFACES

Mikhail ZAIDENBERG

(joint with Hubert FLENNER and Shulim KALIMAN)

May 15, 2018

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 2

ZARISKI CANCELLATION

QUESTION (Oskar Zariski, Algebra Congress, Paris 1949): K(t) ∼ = K ′(t) ⇒ K ∼ = K ′ ?

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 3

ZARISKI CANCELLATION

QUESTION (Oskar Zariski, Algebra Congress, Paris 1949): K(t) ∼ = K ′(t) ⇒ K ∼ = K ′ ? where K, K ′ are fields, usually rational function fields

  • f algebraic varieties.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 4

ZARISKI CANCELLATION

QUESTION (Oskar Zariski, Algebra Congress, Paris 1949): K(t) ∼ = K ′(t) ⇒ K ∼ = K ′ ? where K, K ′ are fields, usually rational function fields

  • f algebraic varieties.

The biregular version asks as to when X × An ∼ = X ′ × An ⇒ X ∼ = X ′ where X, X ′ are affine algebraic varieties.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 5

ZCP - FIRST ANSWERS

"YES" for curves (dim X = dim X ′ = 1) (Abhyankar-Eakin-Heinzer 1972)

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 6

ZCP - FIRST ANSWERS

"YES" for curves (dim X = dim X ′ = 1) (Abhyankar-Eakin-Heinzer 1972) "NO" for surfaces (dim X = dim X ′ = 2) (Danielewski 1988)

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 7

ZCP - FIRST ANSWERS

"YES" for curves (dim X = dim X ′ = 1) (Abhyankar-Eakin-Heinzer 1972) "NO" for surfaces (dim X = dim X ′ = 2) (Danielewski 1988) "YES" if X = A2 and char k = 0, ¯ k = k (Miyanishi-Sugie and Fujita 1979-1980)

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 8

ZCP - FIRST ANSWERS

"YES" for curves (dim X = dim X ′ = 1) (Abhyankar-Eakin-Heinzer 1972) "NO" for surfaces (dim X = dim X ′ = 2) (Danielewski 1988) "YES" if X = A2 and char k = 0, ¯ k = k (Miyanishi-Sugie and Fujita 1979-1980) "NO" if X = A3

k and char k = p > 0

(N. Gupta 2013, using an Asanuma’s construction)

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 9

ZCP – DEFINITIONS

We call X a

  • ZARISKI FACTOR

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 10

ZCP – DEFINITIONS

We call X a

  • ZARISKI FACTOR if the cancellation holds for X

with any X ′ and n;

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 11

ZCP – DEFINITIONS

We call X a

  • ZARISKI FACTOR if the cancellation holds for X

with any X ′ and n;

  • ZARISKI 1-FACTOR

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 12

ZCP – DEFINITIONS

We call X a

  • ZARISKI FACTOR if the cancellation holds for X

with any X ′ and n;

  • ZARISKI 1-FACTOR if the cancellation holds for

X with any X ′ and n = 1;

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 13

ZCP – DEFINITIONS

We call X a

  • ZARISKI FACTOR if the cancellation holds for X

with any X ′ and n;

  • ZARISKI 1-FACTOR if the cancellation holds for

X with any X ′ and n = 1;

  • STRONG ZARISKI FACTOR

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 14

ZCP – DEFINITIONS

We call X a

  • ZARISKI FACTOR if the cancellation holds for X

with any X ′ and n;

  • ZARISKI 1-FACTOR if the cancellation holds for

X with any X ′ and n = 1;

  • STRONG ZARISKI FACTOR if any isomorphism

Φ: X × An

∼ =

− → X ′ × An restricts to an isomorphism φ: X

∼ =

− → X ′

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 15

ZCP – DEFINITIONS

We call X a

  • ZARISKI FACTOR if the cancellation holds for X

with any X ′ and n;

  • ZARISKI 1-FACTOR if the cancellation holds for

X with any X ′ and n = 1;

  • STRONG ZARISKI FACTOR if any isomorphism

Φ: X × An

∼ =

− → X ′ × An restricts to an isomorphism φ: X

∼ =

− → X ′ yielding a commutative diagram X × An Φ ✲ X ′ × An X

❄

φ

✲ X ′ ❄

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 16

ZCP - a survey

THEOREM (Iitaka-Fujita 1977) ¯ k(X) ≥ 0 ⇒ X is a strong Zariski factor.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 17

ZCP - a survey

THEOREM (Iitaka-Fujita 1977) ¯ k(X) ≥ 0 ⇒ X is a strong Zariski factor.

  • 3. THEOREM (Bandman–Makar-Limanov 2005)

X does not admit any Ga-action ⇒ X is a strong Zariski 1-factor.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 18

ZCP - a survey

THEOREM (Iitaka-Fujita 1977) ¯ k(X) ≥ 0 ⇒ X is a strong Zariski factor.

  • 3. THEOREM (Bandman–Makar-Limanov 2005)

X does not admit any Ga-action ⇒ X is a strong Zariski 1-factor. REMARK There are examples ([BML05], [KMMR08])

  • f smooth affine surfaces X such that ¯

k(X) = −∞ and X does not admit any A1–fibration over an affine base.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 19

ZCP - a survey

THEOREM (Iitaka-Fujita 1977) ¯ k(X) ≥ 0 ⇒ X is a strong Zariski factor.

  • 3. THEOREM (Bandman–Makar-Limanov 2005)

X does not admit any Ga-action ⇒ X is a strong Zariski 1-factor. REMARK There are examples ([BML05], [KMMR08])

  • f smooth affine surfaces X such that ¯

k(X) = −∞ and X does not admit any A1–fibration over an affine base. These are strong Zariski 1-factors.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 20

ZCP - a survey

THEOREM (Iitaka-Fujita 1977) ¯ k(X) ≥ 0 ⇒ X is a strong Zariski factor.

  • 3. THEOREM (Bandman–Makar-Limanov 2005)

X does not admit any Ga-action ⇒ X is a strong Zariski 1-factor. REMARK There are examples ([BML05], [KMMR08])

  • f smooth affine surfaces X such that ¯

k(X) = −∞ and X does not admit any A1–fibration over an affine base. These are strong Zariski 1-factors. However, some of them are not Zariski 2-factors (Dubouloz ′15).

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 21

ZCP - a survey

THEOREM (Iitaka-Fujita 1977) ¯ k(X) ≥ 0 ⇒ X is a strong Zariski factor.

  • 3. THEOREM (Bandman–Makar-Limanov 2005)

X does not admit any Ga-action ⇒ X is a strong Zariski 1-factor. REMARK There are examples ([BML05], [KMMR08])

  • f smooth affine surfaces X such that ¯

k(X) = −∞ and X does not admit any A1–fibration over an affine base. These are strong Zariski 1-factors. However, some of them are not Zariski 2-factors (Dubouloz ′15). Conjecturally, none of them is a Zariski 2-factor except for the line bundles (Dubouloz ′16).

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 22

ZCP for surfaces - EXAMPLES

There are families of counterexamples to 1-cancellation among affine surfaces in A3 given by

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 23

ZCP for surfaces - EXAMPLES

There are families of counterexamples to 1-cancellation among affine surfaces in A3 given by zmu − p(z, v) = 0 .

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 24

ZCP for surfaces - EXAMPLES

There are families of counterexamples to 1-cancellation among affine surfaces in A3 given by zmu − p(z, v) = 0 . Such a surface X carries a A1-fibration z|X : X → A1.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 25

ZCP for surfaces - EXAMPLES

There are families of counterexamples to 1-cancellation among affine surfaces in A3 given by zmu − p(z, v) = 0 . Such a surface X carries a A1-fibration z|X : X → A1. In Danielewski-Fieseler examples (′88 -′92) p(z, v) = v 2 − 1 .

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 26

ZCP for surfaces - EXAMPLES

There are families of counterexamples to 1-cancellation among affine surfaces in A3 given by zmu − p(z, v) = 0 . Such a surface X carries a A1-fibration z|X : X → A1. In Danielewski-Fieseler examples (′88 -′92) p(z, v) = v 2 − 1 . Other examples of this kind are due to tom Dieck ′92, Wilkens ′98, Miyanishi-Masuda ′05.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 27

ZCP for surfaces - EXAMPLES

There are families of counterexamples to 1-cancellation among affine surfaces in A3 given by zmu − p(z, v) = 0 . Such a surface X carries a A1-fibration z|X : X → A1. In Danielewski-Fieseler examples (′88 -′92) p(z, v) = v 2 − 1 . Other examples of this kind are due to tom Dieck ′92, Wilkens ′98, Miyanishi-Masuda ′05. REMARK In higher dimensions, examples were constructed by Dubouloz (′07), Finston and Maubach (2008), Jelonek (′09-′10), Dubouloz, Moser-Jauslin and Poloni (′11), K. Masuda (′17).

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 28

Parabolic Gm-surfaces

DEFINITION A PARABOLIC Gm-SURFACE is a normal affine surface with an A1-fibration π: X → B

  • ver a smooth affine curve B equipped with an

effective Gm-action along the fibers of π.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 29

Parabolic Gm-surfaces

DEFINITION A PARABOLIC Gm-SURFACE is a normal affine surface with an A1-fibration π: X → B

  • ver a smooth affine curve B equipped with an

effective Gm-action along the fibers of π. For a parabolic Gm-surface one has:

  • π has only irreducible smooth fibers;

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 30

Parabolic Gm-surfaces

DEFINITION A PARABOLIC Gm-SURFACE is a normal affine surface with an A1-fibration π: X → B

  • ver a smooth affine curve B equipped with an

effective Gm-action along the fibers of π. For a parabolic Gm-surface one has:

  • π has only irreducible smooth fibers;

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 31

Parabolic Gm-surfaces

DEFINITION A PARABOLIC Gm-SURFACE is a normal affine surface with an A1-fibration π: X → B

  • ver a smooth affine curve B equipped with an

effective Gm-action along the fibers of π. For a parabolic Gm-surface one has:

  • π has only irreducible smooth fibers;
  • the fixed points of Gm form a section s of π;

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 32

Parabolic Gm-surfaces

DEFINITION A PARABOLIC Gm-SURFACE is a normal affine surface with an A1-fibration π: X → B

  • ver a smooth affine curve B equipped with an

effective Gm-action along the fibers of π. For a parabolic Gm-surface one has:

  • π has only irreducible smooth fibers;
  • the fixed points of Gm form a section s of π;
  • s meets any multiple fiber in a cyclic quotient

singularity of X;

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 33

Parabolic Gm-surfaces

DEFINITION A PARABOLIC Gm-SURFACE is a normal affine surface with an A1-fibration π: X → B

  • ver a smooth affine curve B equipped with an

effective Gm-action along the fibers of π. For a parabolic Gm-surface one has:

  • π has only irreducible smooth fibers;
  • the fixed points of Gm form a section s of π;
  • s meets any multiple fiber in a cyclic quotient

singularity of X;

  • X has no other singular point.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 34

ZCP for surfaces - MAIN THEOREM 1

THEOREM Let X → B be an A1–fibration on a normal affine surface X over a smooth affine curve B. Then TFAE:

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 35

ZCP for surfaces - MAIN THEOREM 1

THEOREM Let X → B be an A1–fibration on a normal affine surface X over a smooth affine curve B. Then TFAE:

  • X is a Zariski factor;

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 36

ZCP for surfaces - MAIN THEOREM 1

THEOREM Let X → B be an A1–fibration on a normal affine surface X over a smooth affine curve B. Then TFAE:

  • X is a Zariski factor;
  • X is a Zariski 1-factor;

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 37

ZCP for surfaces - MAIN THEOREM 1

THEOREM Let X → B be an A1–fibration on a normal affine surface X over a smooth affine curve B. Then TFAE:

  • X is a Zariski factor;
  • X is a Zariski 1-factor;
  • X is a quotient of a line bundle L over an affine

curve by a finite cyclic group of automorphisms of L;

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 38

ZCP for surfaces - MAIN THEOREM 1

THEOREM Let X → B be an A1–fibration on a normal affine surface X over a smooth affine curve B. Then TFAE:

  • X is a Zariski factor;
  • X is a Zariski 1-factor;
  • X is a quotient of a line bundle L over an affine

curve by a finite cyclic group of automorphisms of L;

  • X is a parabolic Gm-surface.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 39

ZCP for surfaces - MAIN THEOREM 1

THEOREM Let X → B be an A1–fibration on a normal affine surface X over a smooth affine curve B. Then TFAE:

  • X is a Zariski factor;
  • X is a Zariski 1-factor;
  • X is a quotient of a line bundle L over an affine

curve by a finite cyclic group of automorphisms of L;

  • X is a parabolic Gm-surface.

REMARK For a smooth X the result was also

  • btained by Adrien Dubouloz (′16) with a different
  • proof. A smooth parabolic Gm-surface is a line bundle
  • ver an affine curve.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 40

ZCP for surfaces - MAIN THEOREM 1

COROLLARY A normal affine surface X is a Zariski 1-factor if and only if

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 41

ZCP for surfaces - MAIN THEOREM 1

COROLLARY A normal affine surface X is a Zariski 1-factor if and only if either

  • X does not admit any Ga-action,

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 42

ZCP for surfaces - MAIN THEOREM 1

COROLLARY A normal affine surface X is a Zariski 1-factor if and only if either

  • X does not admit any Ga-action, or
  • X is a parabolic Gm-surface.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 43

ZCP for surfaces - MAIN THEOREM 1

COROLLARY A normal affine surface X is a Zariski 1-factor if and only if either

  • X does not admit any Ga-action, or
  • X is a parabolic Gm-surface.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 44

DANIELEWSKI-FIESELER SURFACES

DEFINITION A normal affine surface π: X → B A1-fibered over a smooth affine curve B is called

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 45

DANIELEWSKI-FIESELER SURFACES

DEFINITION A normal affine surface π: X → B A1-fibered over a smooth affine curve B is called a GENERALIZED DANIELEWSKI-FIESELER SURFACE (GDF)

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 46

DANIELEWSKI-FIESELER SURFACES

DEFINITION A normal affine surface π: X → B A1-fibered over a smooth affine curve B is called a GENERALIZED DANIELEWSKI-FIESELER SURFACE (GDF) if all fibers of π are reduced.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 47

DANIELEWSKI-FIESELER SURFACES

DEFINITION A normal affine surface π: X → B A1-fibered over a smooth affine curve B is called a GENERALIZED DANIELEWSKI-FIESELER SURFACE (GDF) if all fibers of π are reduced. A GDF surface X is automatically smooth.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 48

DANIELEWSKI-FIESELER SURFACES

DEFINITION A normal affine surface π: X → B A1-fibered over a smooth affine curve B is called a GENERALIZED DANIELEWSKI-FIESELER SURFACE (GDF) if all fibers of π are reduced. A GDF surface X is automatically smooth. LEMMA (COVERING TRICK) Let Y → C be an A1-fibration on a normal affine surface Y over a smooth affine curve C.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 49

DANIELEWSKI-FIESELER SURFACES

DEFINITION A normal affine surface π: X → B A1-fibered over a smooth affine curve B is called a GENERALIZED DANIELEWSKI-FIESELER SURFACE (GDF) if all fibers of π are reduced. A GDF surface X is automatically smooth. LEMMA (COVERING TRICK) Let Y → C be an A1-fibration on a normal affine surface Y over a smooth affine curve C. Then there exists a cyclic branched covering B → C such that the induced A1-fibration ˜ X → B yields, after passing to a normalization X → ˜ X,

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 50

DANIELEWSKI-FIESELER SURFACES

DEFINITION A normal affine surface π: X → B A1-fibered over a smooth affine curve B is called a GENERALIZED DANIELEWSKI-FIESELER SURFACE (GDF) if all fibers of π are reduced. A GDF surface X is automatically smooth. LEMMA (COVERING TRICK) Let Y → C be an A1-fibration on a normal affine surface Y over a smooth affine curve C. Then there exists a cyclic branched covering B → C such that the induced A1-fibration ˜ X → B yields, after passing to a normalization X → ˜ X, a GDF surface X → B.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 51

DANIELEWSKI-FIESELER SURFACES

DEFINITION A normal affine surface π: X → B A1-fibered over a smooth affine curve B is called a GENERALIZED DANIELEWSKI-FIESELER SURFACE (GDF) if all fibers of π are reduced. A GDF surface X is automatically smooth. LEMMA (COVERING TRICK) Let Y → C be an A1-fibration on a normal affine surface Y over a smooth affine curve C. Then there exists a cyclic branched covering B → C such that the induced A1-fibration ˜ X → B yields, after passing to a normalization X → ˜ X, a GDF surface X → B. Thus, Y → C is the quotient of X → B by the cyclic group action on X and on B.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 52

TOM DIECK REDUCTION

The covering trick allows to reduce the Zariski Cancellation Problem for general affine surfaces A1-fibered over affine curves to its Z/dZ equivariant version for GDF surfaces, due to the following commutative diagram.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 53

TOM DIECK REDUCTION

X0 × A1 ∼ =G

✲ X1 × A1 ❅ ❅ ❅ ❅ ❅ ❅

/G

❘ ❅ ❅ ❅ ❅ ❅ ❅

/G

❘

Y0 × A1 ∼ =

✲ Y1 × A1

B

❄

id

✲ B ❄ ❅ ❅ ❅ ❅ ❅ ❅

/G

❘ ❅ ❅ ❅ ❅ ❅ ❅

/G

❘

C

❄

id

✲ C ❄

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 54

DANIELEWSKI-FIESELER QUOTIENT

DEFINITIONS Let π: X → B be a GDF surface.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 55

DANIELEWSKI-FIESELER QUOTIENT

DEFINITIONS Let π: X → B be a GDF surface. Consider on X the equivalence relation: x ∼ x′ if π(x) = π(x′) = b and x, x′ belong to the same component of π−1(b).

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 56

DANIELEWSKI-FIESELER QUOTIENT

DEFINITIONS Let π: X → B be a GDF surface. Consider on X the equivalence relation: x ∼ x′ if π(x) = π(x′) = b and x, x′ belong to the same component of π−1(b). The quotient ˘ B = X/ ∼ is a non-separated scheme over B called the DANIELEWSKI-FIESELER QUOTIENT.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 57

DANIELEWSKI-FIESELER QUOTIENT

DEFINITIONS Let π: X → B be a GDF surface. Consider on X the equivalence relation: x ∼ x′ if π(x) = π(x′) = b and x, x′ belong to the same component of π−1(b). The quotient ˘ B = X/ ∼ is a non-separated scheme over B called the DANIELEWSKI-FIESELER QUOTIENT. The map π factorizes into morphisms π: X → ˘ B → B ,

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 58

DANIELEWSKI-FIESELER QUOTIENT

DEFINITIONS Let π: X → B be a GDF surface. Consider on X the equivalence relation: x ∼ x′ if π(x) = π(x′) = b and x, x′ belong to the same component of π−1(b). The quotient ˘ B = X/ ∼ is a non-separated scheme over B called the DANIELEWSKI-FIESELER QUOTIENT. The map π factorizes into morphisms π: X → ˘ B → B , where ˘ B → B is an isomorphism outside the points b1, . . . , bt that correspond to degenerate fibers of π,

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 59

DANIELEWSKI-FIESELER QUOTIENT

DEFINITIONS Let π: X → B be a GDF surface. Consider on X the equivalence relation: x ∼ x′ if π(x) = π(x′) = b and x, x′ belong to the same component of π−1(b). The quotient ˘ B = X/ ∼ is a non-separated scheme over B called the DANIELEWSKI-FIESELER QUOTIENT. The map π factorizes into morphisms π: X → ˘ B → B , where ˘ B → B is an isomorphism outside the points b1, . . . , bt that correspond to degenerate fibers of π, and ˘ B has ni points {bij} over bi if π−1(bi) consists of ni (reduced) A1-components.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 60

THE PICARD GROUP Pic( ˘ B)

DEFINITION A divisor D on ˘ B is a formal sum D =

N

  • j=1

mj ˘ bj , where ˘ bj ∈ ˘ B and mj ∈ Z.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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SLIDE 61

THE PICARD GROUP Pic( ˘ B)

DEFINITION A divisor D on ˘ B is a formal sum D =

N

  • j=1

mj ˘ bj , where ˘ bj ∈ ˘ B and mj ∈ Z. A principal divisor is the divisor D = div(ϕ) of a rational function ϕ on ˘ B, actually, on B.

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SLIDE 62

THE PICARD GROUP Pic( ˘ B)

DEFINITION A divisor D on ˘ B is a formal sum D =

N

  • j=1

mj ˘ bj , where ˘ bj ∈ ˘ B and mj ∈ Z. A principal divisor is the divisor D = div(ϕ) of a rational function ϕ on ˘ B, actually, on B. Indeed, one has O( ˘ B) = O(B) and Frac(O( ˘ B)) = Frac(O(B)).

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SLIDE 63

THE PICARD GROUP Pic( ˘ B)

DEFINITION A divisor D on ˘ B is a formal sum D =

N

  • j=1

mj ˘ bj , where ˘ bj ∈ ˘ B and mj ∈ Z. A principal divisor is the divisor D = div(ϕ) of a rational function ϕ on ˘ B, actually, on B. Indeed, one has O( ˘ B) = O(B) and Frac(O( ˘ B)) = Frac(O(B)). The Picard group of ˘ B is the quotient group Pic( ˘ B) = Div( ˘ B)/Princ( ˘ B) .

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SLIDE 64

ZCP for surfaces - MAIN THEOREM 2

THEOREM There is a one-to-one correspondence between

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SLIDE 65

ZCP for surfaces - MAIN THEOREM 2

THEOREM There is a one-to-one correspondence between Pic( ˘ B) and the set of B-isomorphism classes

  • f cylinders X × A1 → B over the GDF surfaces X → B

with a given DF-quotient ˘ B.

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SLIDE 66

ZCP for surfaces - MAIN THEOREM 2

THEOREM There is a one-to-one correspondence between Pic( ˘ B) and the set of B-isomorphism classes

  • f cylinders X × A1 → B over the GDF surfaces X → B

with a given DF-quotient ˘ B. REMARK If ˘ B = B then π: X → B has no degenerate fiber, and so, is a line bundle.

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SLIDE 67

ZCP for surfaces - MAIN THEOREM 2

THEOREM There is a one-to-one correspondence between Pic( ˘ B) and the set of B-isomorphism classes

  • f cylinders X × A1 → B over the GDF surfaces X → B

with a given DF-quotient ˘ B. REMARK If ˘ B = B then π: X → B has no degenerate fiber, and so, is a line bundle. Given line bundles L (π: X → B) and L′ (π′ : X ′ → B) one has

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SLIDE 68

ZCP for surfaces - MAIN THEOREM 2

THEOREM There is a one-to-one correspondence between Pic( ˘ B) and the set of B-isomorphism classes

  • f cylinders X × A1 → B over the GDF surfaces X → B

with a given DF-quotient ˘ B. REMARK If ˘ B = B then π: X → B has no degenerate fiber, and so, is a line bundle. Given line bundles L (π: X → B) and L′ (π′ : X ′ → B) one has X × A1 ∼ =B X ′ × A1 ⇔ L ∼ = L′ . In fact, X is a Zariski factor.

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SLIDE 69

ZCP for surfaces - MAIN THEOREM 2

REMARK If B ∼ = A1 then any morphism A1 → B is constant.

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SLIDE 70

ZCP for surfaces - MAIN THEOREM 2

REMARK If B ∼ = A1 then any morphism A1 → B is

  • constant. Hence, for two A1-fibered surfaces X → B

and X ′ → B over the same base one has X × A1 ∼ = X ′ × A1 ⇔ X × A1 ∼ =B X ′ × A1 up to an automorphism of B.

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SLIDE 71

ZCP for surfaces - MAIN THEOREM 2

REMARK If B ∼ = A1 then any morphism A1 → B is

  • constant. Hence, for two A1-fibered surfaces X → B

and X ′ → B over the same base one has X × A1 ∼ = X ′ × A1 ⇔ X × A1 ∼ =B X ′ × A1 up to an automorphism of B.

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SLIDE 72

FIBER TREES

DEFINITION Consider a GDF surface π: X → B.

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SLIDE 73

FIBER TREES

DEFINITION Consider a GDF surface π: X → B. Let ¯ π: ¯ X → ¯ B be a P1-fibration which results from a smooth completion of X by a simple normal crossing divisor D = ¯ X \ X.

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SLIDE 74

FIBER TREES

DEFINITION Consider a GDF surface π: X → B. Let ¯ π: ¯ X → ¯ B be a P1-fibration which results from a smooth completion of X by a simple normal crossing divisor D = ¯ X \ X. Then D has a unique component S which is the section "at infinity" of ¯ π.

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SLIDE 75

FIBER TREES

DEFINITION Consider a GDF surface π: X → B. Let ¯ π: ¯ X → ¯ B be a P1-fibration which results from a smooth completion of X by a simple normal crossing divisor D = ¯ X \ X. Then D has a unique component S which is the section "at infinity" of ¯ π. For b ∈ ¯ B different from the points bi the fiber ¯ π−1(b) is reduced and irreducible, while ¯ π−1(bi) is a tree of P1-curves.

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SLIDE 76

FIBER TREES

DEFINITION Consider a GDF surface π: X → B. Let ¯ π: ¯ X → ¯ B be a P1-fibration which results from a smooth completion of X by a simple normal crossing divisor D = ¯ X \ X. Then D has a unique component S which is the section "at infinity" of ¯ π. For b ∈ ¯ B different from the points bi the fiber ¯ π−1(b) is reduced and irreducible, while ¯ π−1(bi) is a tree of P1-curves. Its dual graph Ti is called a FIBER TREE.

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SLIDE 77

FIBER TREES

DEFINITION Consider a GDF surface π: X → B. Let ¯ π: ¯ X → ¯ B be a P1-fibration which results from a smooth completion of X by a simple normal crossing divisor D = ¯ X \ X. Then D has a unique component S which is the section "at infinity" of ¯ π. For b ∈ ¯ B different from the points bi the fiber ¯ π−1(b) is reduced and irreducible, while ¯ π−1(bi) is a tree of P1-curves. Its dual graph Ti is called a FIBER TREE. The root Ri of Ti is the unique component of ¯ π−1(bi) meeting S.

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SLIDE 78

TYPE DIVISOR

DEFINITIONS The LEAVES Lij of Ti are its extremal vertices different from the root Ri.

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SLIDE 79

TYPE DIVISOR

DEFINITIONS The LEAVES Lij of Ti are its extremal vertices different from the root Ri. The LEVEL of Lij is the tree distance lij = dist (Lij, Ri).

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SLIDE 80

TYPE DIVISOR

DEFINITIONS The LEAVES Lij of Ti are its extremal vertices different from the root Ri. The LEVEL of Lij is the tree distance lij = dist (Lij, Ri). The leaves of Ti represent the components of π−1(bi), hence also the points bij of ˘ B over bi.

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SLIDE 81

TYPE DIVISOR

DEFINITIONS The LEAVES Lij of Ti are its extremal vertices different from the root Ri. The LEVEL of Lij is the tree distance lij = dist (Lij, Ri). The leaves of Ti represent the components of π−1(bi), hence also the points bij of ˘ B over bi. The TYPE DIVISOR is tp(π) = −

  • i,j

lijbij ∈ Div( ˘ B) .

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SLIDE 82

ZCP: MAIN THEOREM 3

THEOREM The cylinders over two GDF surfaces X → B and X ′ → B with the same DF-quotient ˘ B are isomorphic over B if and only if their type divisors are linearly equivalent on ˘ B.

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SLIDE 83

ZCP: MAIN THEOREM 3

THEOREM The cylinders over two GDF surfaces X → B and X ′ → B with the same DF-quotient ˘ B are isomorphic over B if and only if their type divisors are linearly equivalent on ˘ B. ˜ Γ : ˆ Γ :

Figure: The “bush” ˜ Γ and the “spring bush” ˆ Γ have the same type divisors

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SLIDE 84

MAIN INGREDIENTS OF THE PROOF

  • Flexibility (an equivariant relative version);

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SLIDE 85

MAIN INGREDIENTS OF THE PROOF

  • Flexibility (an equivariant relative version);
  • affine modifications, especially the Asanuma

modification;

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SLIDE 86

MAIN INGREDIENTS OF THE PROOF

  • Flexibility (an equivariant relative version);
  • affine modifications, especially the Asanuma

modification;

  • Cox rings technique.

Cox rings play an important role in the proof of Theorem 1. Next we stay on the two other tools.

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SLIDE 87

TRIVIALIZING SEQUENCE

DEFINITION Let X → B be a GDF surface. There exists a TRIVIALIZING SEQUENCE

  • f affine modifications

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SLIDE 88

TRIVIALIZING SEQUENCE

DEFINITION Let X → B be a GDF surface. There exists a TRIVIALIZING SEQUENCE

  • f affine modifications

X = Xm

ρm

− →B Xm−1

ρm−1

− →B . . .

ρ1

− →B X0 = B × A1 .

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SLIDE 89

TRIVIALIZING SEQUENCE

DEFINITION Let X → B be a GDF surface. There exists a TRIVIALIZING SEQUENCE

  • f affine modifications

X = Xm

ρm

− →B Xm−1

ρm−1

− →B . . .

ρ1

− →B X0 = B × A1 . The center of ρi is a reduced finite subcheme

  • f the exceptional divisor of ρi−1.

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SLIDE 90

TRIVIALIZING SEQUENCE

DEFINITION Let X → B be a GDF surface. There exists a TRIVIALIZING SEQUENCE

  • f affine modifications

X = Xm

ρm

− →B Xm−1

ρm−1

− →B . . .

ρ1

− →B X0 = B × A1 . The center of ρi is a reduced finite subcheme

  • f the exceptional divisor of ρi−1.

This sequence can be extended to a sequence of suitable SNC completions: ¯ X = ¯ Xm

¯ ρm

− →B ¯ Xm−1

¯ ρm−1

− →B . . .

¯ ρ1

− →B ¯ X0 = ¯ B × P1

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SLIDE 91

TRIVIALIZING SEQUENCE

DEFINITION Let X → B be a GDF surface. There exists a TRIVIALIZING SEQUENCE

  • f affine modifications

X = Xm

ρm

− →B Xm−1

ρm−1

− →B . . .

ρ1

− →B X0 = B × A1 . The center of ρi is a reduced finite subcheme

  • f the exceptional divisor of ρi−1.

This sequence can be extended to a sequence of suitable SNC completions: ¯ X = ¯ Xm

¯ ρm

− →B ¯ Xm−1

¯ ρm−1

− →B . . .

¯ ρ1

− →B ¯ X0 = ¯ B × P1 where ¯ ρi : ¯ Xi − →B ¯ Xi−1 is a blowup with a smooth reduced center.

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SLIDE 92

A GDF surface with a given fiber tree

Consider a rooted tree Γ with a root R and ni vertices

  • n level i, i = 1, . . . , m.

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SLIDE 93

A GDF surface with a given fiber tree

Consider a rooted tree Γ with a root R and ni vertices

  • n level i, i = 1, . . . , m.

Given a point b ∈ B we identify R with the fiber {b} × P1 of pr1 : ¯ X0 → ¯ B.

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SLIDE 94

A GDF surface with a given fiber tree

Consider a rooted tree Γ with a root R and ni vertices

  • n level i, i = 1, . . . , m.

Given a point b ∈ B we identify R with the fiber {b} × P1 of pr1 : ¯ X0 → ¯

  • B. Choose n1 points on R,

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SLIDE 95

A GDF surface with a given fiber tree

Consider a rooted tree Γ with a root R and ni vertices

  • n level i, i = 1, . . . , m.

Given a point b ∈ B we identify R with the fiber {b} × P1 of pr1 : ¯ X0 → ¯

  • B. Choose n1 points on R,

and let ¯ ρ1 : ¯ X1 − →B ¯ X0 be the blowup of these points with exceptional divisor E1 = ¯ F1 + . . . + ¯ Fn1.

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SLIDE 96

A GDF surface with a given fiber tree

Consider a rooted tree Γ with a root R and ni vertices

  • n level i, i = 1, . . . , m.

Given a point b ∈ B we identify R with the fiber {b} × P1 of pr1 : ¯ X0 → ¯

  • B. Choose n1 points on R,

and let ¯ ρ1 : ¯ X1 − →B ¯ X0 be the blowup of these points with exceptional divisor E1 = ¯ F1 + . . . + ¯

  • Fn1. We let

X1 = ¯ X1 \ (R′ ∪ S′

∞ ∪ ¯

F∞) where S∞ ⊂ ¯ X0 is the section at infinity and ¯ F∞ ⊂ ¯ X0 is the fiber at infinity.

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SLIDE 97

A GDF surface with a given fiber tree

Consider a rooted tree Γ with a root R and ni vertices

  • n level i, i = 1, . . . , m.

Given a point b ∈ B we identify R with the fiber {b} × P1 of pr1 : ¯ X0 → ¯

  • B. Choose n1 points on R,

and let ¯ ρ1 : ¯ X1 − →B ¯ X0 be the blowup of these points with exceptional divisor E1 = ¯ F1 + . . . + ¯

  • Fn1. We let

X1 = ¯ X1 \ (R′ ∪ S′

∞ ∪ ¯

F∞) where S∞ ⊂ ¯ X0 is the section at infinity and ¯ F∞ ⊂ ¯ X0 is the fiber at infinity. Then X1 → B is a GDF surface with a unique reducible fiber π∗

1(b) = F1 + . . . + Fn1.

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SLIDE 98

A GDF surface with a given fiber tree

The center of ¯ ρ2 : ¯ X2 − →B ¯ X1 consists of n2 points on E1 distributed between the components Fi of E1 \ S∞ according to the edges of Γ.

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SLIDE 99

A GDF surface with a given fiber tree

The center of ¯ ρ2 : ¯ X2 − →B ¯ X1 consists of n2 points on E1 distributed between the components Fi of E1 \ S∞ according to the edges of Γ. Continuing in this way we construct a GDF surface Xm → B where m is the height of Γ.

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SLIDE 100

A GDF surface with a given fiber tree

The center of ¯ ρ2 : ¯ X2 − →B ¯ X1 consists of n2 points on E1 distributed between the components Fi of E1 \ S∞ according to the edges of Γ. Continuing in this way we construct a GDF surface Xm → B where m is the height of Γ. This surface has a unique degenerate fiber π−1

m (b) whose fiber tree is Γ.

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SLIDE 101

GRAPH DIVISOR and the COARSE MODULI SPACE

Consider a GRAPH DIVISOR D = n

i=1 Γibi

where each Γi is a rooted tree.

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SLIDE 102

GRAPH DIVISOR and the COARSE MODULI SPACE

Consider a GRAPH DIVISOR D = n

i=1 Γibi

where each Γi is a rooted tree. Applying our procedure yields a GDF surface Xm → B with the degenerate fibers π−1

m (bi)

whose fiber trees are the Γi, i = 1, . . . , n.

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SLIDE 103

GRAPH DIVISOR and the COARSE MODULI SPACE

Consider a GRAPH DIVISOR D = n

i=1 Γibi

where each Γi is a rooted tree. Applying our procedure yields a GDF surface Xm → B with the degenerate fibers π−1

m (bi)

whose fiber trees are the Γi, i = 1, . . . , n. Any GDF surface along with its trivializing sequence appears in this way.

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SLIDE 104

GRAPH DIVISOR and the COARSE MODULI SPACE

Consider a GRAPH DIVISOR D = n

i=1 Γibi

where each Γi is a rooted tree. Applying our procedure yields a GDF surface Xm → B with the degenerate fibers π−1

m (bi)

whose fiber trees are the Γi, i = 1, . . . , n. Any GDF surface along with its trivializing sequence appears in this way. The centers of the ¯ ρi give continuous parameters. This leads to a coarse moduli space M(B, D)

  • f the GDF surfaces X → B with a given graph divisor

D.

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SLIDE 105

GRAPH DIVISOR and the COARSE MODULI SPACE

Consider a GRAPH DIVISOR D = n

i=1 Γibi

where each Γi is a rooted tree. Applying our procedure yields a GDF surface Xm → B with the degenerate fibers π−1

m (bi)

whose fiber trees are the Γi, i = 1, . . . , n. Any GDF surface along with its trivializing sequence appears in this way. The centers of the ¯ ρi give continuous parameters. This leads to a coarse moduli space M(B, D)

  • f the GDF surfaces X → B with a given graph divisor
  • D. The dimension dim M(B, D) roughly equals

the number of edges of D.

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SLIDE 106

ASANUMA TRICK

The cylinder X = X × A1 can also be obtained via a sequence of affine modifications X = Xm

˜ ρm

− →B Xm−1

˜ ρm−1

− →B . . .

˜ ρ1

− →B X0 = ¯ B × A2 .

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SLIDE 107

ASANUMA TRICK

The cylinder X = X × A1 can also be obtained via a sequence of affine modifications X = Xm

˜ ρm

− →B Xm−1

˜ ρm−1

− →B . . .

˜ ρ1

− →B X0 = ¯ B × A2 . These modifications have one-dimensional centers.

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SLIDE 108

ASANUMA TRICK

The cylinder X = X × A1 can also be obtained via a sequence of affine modifications X = Xm

˜ ρm

− →B Xm−1

˜ ρm−1

− →B . . .

˜ ρ1

− →B X0 = ¯ B × A2 . These modifications have one-dimensional centers. However, an elegant ASANUMA TRICK allows to replace this sequence by the one with the same cylinders and zero-dimensional centers of modifications.

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SLIDE 109

ASANUMA TRICK

The cylinder X = X × A1 can also be obtained via a sequence of affine modifications X = Xm

˜ ρm

− →B Xm−1

˜ ρm−1

− →B . . .

˜ ρ1

− →B X0 = ¯ B × A2 . These modifications have one-dimensional centers. However, an elegant ASANUMA TRICK allows to replace this sequence by the one with the same cylinders and zero-dimensional centers of modifications. Each component of the exceptional divisor Ei of ˜ ρi : Xi →B Xi−1 is isomorphic to A2.

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SLIDE 110

RELATIVE FLEXIBILITY and RIGIDITY OF CYLINDERS

The RELATIVE FLEXIBILITY in families of affine planes allows to move the center of each modification ˜ ρi+1 : Xi+1 →B Xi to a given position.

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SLIDE 111

RELATIVE FLEXIBILITY and RIGIDITY OF CYLINDERS

The RELATIVE FLEXIBILITY in families of affine planes allows to move the center of each modification ˜ ρi+1 : Xi+1 →B Xi to a given position. This leads to the following rigidity result.

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SLIDE 112

RELATIVE FLEXIBILITY and RIGIDITY OF CYLINDERS

The RELATIVE FLEXIBILITY in families of affine planes allows to move the center of each modification ˜ ρi+1 : Xi+1 →B Xi to a given position. This leads to the following rigidity result. THEOREM The cylinders X over the GDF surfaces X → B from M(B, D) are all pairwise isomorphic over B.

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SLIDE 113

CONCLUDING REMARKS

Passing from GDF surfaces to their cylinders one eliminates the continuous parameters.

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SLIDE 114

CONCLUDING REMARKS

Passing from GDF surfaces to their cylinders one eliminates the continuous parameters. Thus, the moduli of cylinders are zero-dimensional.

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SLIDE 115

CONCLUDING REMARKS

Passing from GDF surfaces to their cylinders one eliminates the continuous parameters. Thus, the moduli of cylinders are zero-dimensional. There is a natural Ga-action on X = X × A1 with quotient X. Variyng the GDF surface X within M(B, D) one gets a continuous family of pairwise non-conjugate Ga-actions on X parameterized by M(B, D).

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SLIDE 116

CONCLUDING REMARKS

Passing from GDF surfaces to their cylinders one eliminates the continuous parameters. Thus, the moduli of cylinders are zero-dimensional. There is a natural Ga-action on X = X × A1 with quotient X. Variyng the GDF surface X within M(B, D) one gets a continuous family of pairwise non-conjugate Ga-actions on X parameterized by M(B, D). All these actions preserve the A2-fibration X → B. The hard part of the proof consists to establish a bijection between the moduli space of cylinders and the Picard group Pic( ˘ B).

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SLIDE 117

CONCLUDING REMARKS

Passing from GDF surfaces to their cylinders one eliminates the continuous parameters. Thus, the moduli of cylinders are zero-dimensional. There is a natural Ga-action on X = X × A1 with quotient X. Variyng the GDF surface X within M(B, D) one gets a continuous family of pairwise non-conjugate Ga-actions on X parameterized by M(B, D). All these actions preserve the A2-fibration X → B. The hard part of the proof consists to establish a bijection between the moduli space of cylinders and the Picard group Pic( ˘ B). To do so we transform any graph divisor to a shruberry without changing the cylinder.

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SLIDE 118

OPEN QUESTIONS

QUESTION Classify all pairs of isomorphic cylinders which are not isomorphic over B.

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SLIDE 119

OPEN QUESTIONS

QUESTION Classify all pairs of isomorphic cylinders which are not isomorphic over B. QUESTION Classify all cylinders X over A1-fibered surfaces such that the group Aut (X) is transitive in codimension 2 on X.

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SLIDE 120

OPEN QUESTIONS

QUESTION Classify all pairs of isomorphic cylinders which are not isomorphic over B. QUESTION Classify all cylinders X over A1-fibered surfaces such that the group Aut (X) is transitive in codimension 2 on X. CONJECTURE Any such cylinder is the cylinder over a Gizatullin surface.

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SLIDE 121

OPEN QUESTIONS

QUESTION Classify all pairs of isomorphic cylinders which are not isomorphic over B. QUESTION Classify all cylinders X over A1-fibered surfaces such that the group Aut (X) is transitive in codimension 2 on X. CONJECTURE Any such cylinder is the cylinder over a Gizatullin surface. Notice that the cylinder over a Gizatullin surface has this property.

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SLIDE 122

OPEN QUESTIONS

QUESTION Classify all pairs of isomorphic cylinders which are not isomorphic over B. QUESTION Classify all cylinders X over A1-fibered surfaces such that the group Aut (X) is transitive in codimension 2 on X. CONJECTURE Any such cylinder is the cylinder over a Gizatullin surface. Notice that the cylinder over a Gizatullin surface has this property. Remember that a Gizatullin surface is A1-fibered over A1 with a unique reducible fiber.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-123
SLIDE 123

OPEN QUESTIONS

QUESTION Classify all pairs of isomorphic cylinders which are not isomorphic over B. QUESTION Classify all cylinders X over A1-fibered surfaces such that the group Aut (X) is transitive in codimension 2 on X. CONJECTURE Any such cylinder is the cylinder over a Gizatullin surface. Notice that the cylinder over a Gizatullin surface has this property. Remember that a Gizatullin surface is A1-fibered over A1 with a unique reducible fiber.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-124
SLIDE 124

FLEXIBLE CYLINDERS

The following theorem gives a partial confirmation of the latter conjecture.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-125
SLIDE 125

FLEXIBLE CYLINDERS

The following theorem gives a partial confirmation of the latter conjecture. THEOREM (Bandman-Makar-Limanov ′05) Let X → A1 be a smooth A1-fibered surface with a unique reducible fiber which is reduced.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-126
SLIDE 126

FLEXIBLE CYLINDERS

The following theorem gives a partial confirmation of the latter conjecture. THEOREM (Bandman-Makar-Limanov ′05) Let X → A1 be a smooth A1-fibered surface with a unique reducible fiber which is reduced. If Aut (X) is transitive in codimension 2 then the cylinder X over X is isomorphic to the cylinder over a Gizatullin surface.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-127
SLIDE 127

FLEXIBLE CYLINDERS

The following theorem gives a partial confirmation of the latter conjecture. THEOREM (Bandman-Makar-Limanov ′05) Let X → A1 be a smooth A1-fibered surface with a unique reducible fiber which is reduced. If Aut (X) is transitive in codimension 2 then the cylinder X over X is isomorphic to the cylinder over a Gizatullin surface. QUESTION Assume that Aut (X) is transitive in codimension 2. Is it true that the A1-fibration X → A1 has at most one reducible fiber?

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES