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

zariski cancellation for surfaces
SMART_READER_LITE
LIVE PREVIEW

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

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


slide-1
SLIDE 1

ZARISKI CANCELLATION FOR SURFACES

Mikhail ZAIDENBERG

(joint with Hubert FLENNER and Shulim KALIMAN)

May 15, 2018

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-2
SLIDE 2

ZARISKI CANCELLATION

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

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-3
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

slide-4
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

slide-5
SLIDE 5

ZCP - FIRST ANSWERS

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

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-6
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

slide-7
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

slide-8
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

slide-9
SLIDE 9

ZCP – DEFINITIONS

We call X a

  • ZARISKI FACTOR

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-10
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

slide-11
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

slide-12
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

slide-13
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

slide-14
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

slide-15
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

slide-16
SLIDE 16

ZCP - a survey

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

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-17
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

slide-18
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

slide-19
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

slide-20
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

slide-21
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

slide-22
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

slide-23
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

slide-24
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

slide-25
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

slide-26
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

slide-27
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

slide-28
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

slide-29
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

slide-30
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

slide-31
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

slide-32
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

slide-33
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

slide-34
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

slide-35
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

slide-36
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

slide-37
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

slide-38
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

slide-39
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

slide-40
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

slide-41
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

slide-42
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

slide-43
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

slide-44
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

slide-45
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

slide-46
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

slide-47
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

slide-48
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

slide-49
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

slide-50
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

slide-51
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

slide-52
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

slide-53
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

slide-54
SLIDE 54

DANIELEWSKI-FIESELER QUOTIENT

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

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-55
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

slide-56
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

slide-57
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

slide-58
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

slide-59
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

slide-60
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

slide-61
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-62
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)).

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-63
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) .

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-64
SLIDE 64

ZCP for surfaces - MAIN THEOREM 2

THEOREM There is a one-to-one correspondence between

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-65
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-66
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-67
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

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-68
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-69
SLIDE 69

ZCP for surfaces - MAIN THEOREM 2

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

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-70
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-71
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-72
SLIDE 72

FIBER TREES

DEFINITION Consider a GDF surface π: X → B.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-73
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-74
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-75
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-76
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-77
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-78
SLIDE 78

TYPE DIVISOR

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

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-79
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).

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-80
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-81
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) .

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-82
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-83
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

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-84
SLIDE 84

MAIN INGREDIENTS OF THE PROOF

  • Flexibility (an equivariant relative version);

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-85
SLIDE 85

MAIN INGREDIENTS OF THE PROOF

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

modification;

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-86
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-87
SLIDE 87

TRIVIALIZING SEQUENCE

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

  • f affine modifications

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-88
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 .

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-89
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-90
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

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-91
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-92
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-93
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-94
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,

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-95
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-96
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-97
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-98
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-99
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-100
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-101
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-102
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-103
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-104
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-105
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-106
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 .

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-107
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-108
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-109
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-110
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-111
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-112
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-113
SLIDE 113

CONCLUDING REMARKS

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

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-114
SLIDE 114

CONCLUDING REMARKS

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

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-115
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).

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-116
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).

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-117
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-118
SLIDE 118

OPEN QUESTIONS

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

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-119
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-120
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-121
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.

Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

slide-122
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