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

Parabolic G m -surfaces DEFINITION A PARABOLIC G m -SURFACE is a normal affine surface with an A 1 -fibration π : X → B over a smooth affine curve B equipped with an effective G m -action along the fibers of π . For a parabolic G m -surface one has: • π has only irreducible smooth fibers; • the fixed points of G m form a section s of π ; Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

Parabolic G m -surfaces DEFINITION A PARABOLIC G m -SURFACE is a normal affine surface with an A 1 -fibration π : X → B over a smooth affine curve B equipped with an effective G m -action along the fibers of π . For a parabolic G m -surface one has: • π has only irreducible smooth fibers; • the fixed points of G m form a section s of π ; • s meets any multiple fiber in a cyclic quotient singularity of X ; Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

Parabolic G m -surfaces DEFINITION A PARABOLIC G m -SURFACE is a normal affine surface with an A 1 -fibration π : X → B over a smooth affine curve B equipped with an effective G m -action along the fibers of π . For a parabolic G m -surface one has: • π has only irreducible smooth fibers; • the fixed points of G m 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

ZCP for surfaces - MAIN THEOREM 1 THEOREM Let X → B be an A 1 –fibration on a normal affine surface X over a smooth affine curve B . Then TFAE: Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

ZCP for surfaces - MAIN THEOREM 1 THEOREM Let X → B be an A 1 –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

ZCP for surfaces - MAIN THEOREM 1 THEOREM Let X → B be an A 1 –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

ZCP for surfaces - MAIN THEOREM 1 THEOREM Let X → B be an A 1 –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

ZCP for surfaces - MAIN THEOREM 1 THEOREM Let X → B be an A 1 –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 G m -surface. Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

ZCP for surfaces - MAIN THEOREM 1 THEOREM Let X → B be an A 1 –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 G m -surface. REMARK For a smooth X the result was also obtained by Adrien Dubouloz ( ′ 16) with a different proof. A smooth parabolic G m -surface is a line bundle over an affine curve. Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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

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 G a -action, Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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 G a -action, or • X is a parabolic G m -surface. Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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 G a -action, or • X is a parabolic G m -surface. Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

DANIELEWSKI-FIESELER SURFACES DEFINITION A normal affine surface π : X → B A 1 -fibered over a smooth affine curve B is called Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

DANIELEWSKI-FIESELER SURFACES DEFINITION A normal affine surface π : X → B A 1 -fibered over a smooth affine curve B is called a GENERALIZED DANIELEWSKI-FIESELER SURFACE (GDF) Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

DANIELEWSKI-FIESELER SURFACES DEFINITION A normal affine surface π : X → B A 1 -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

DANIELEWSKI-FIESELER SURFACES DEFINITION A normal affine surface π : X → B A 1 -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

DANIELEWSKI-FIESELER SURFACES DEFINITION A normal affine surface π : X → B A 1 -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 A 1 -fibration on a normal affine surface Y over a smooth affine curve C . Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

DANIELEWSKI-FIESELER SURFACES DEFINITION A normal affine surface π : X → B A 1 -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 A 1 -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 A 1 -fibration ˜ X → B yields, after passing to a normalization X → ˜ X , Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

DANIELEWSKI-FIESELER SURFACES DEFINITION A normal affine surface π : X → B A 1 -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 A 1 -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 A 1 -fibration ˜ X → B yields, after passing to a normalization X → ˜ X , a GDF surface X → B . Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

DANIELEWSKI-FIESELER SURFACES DEFINITION A normal affine surface π : X → B A 1 -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 A 1 -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 A 1 -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

TOM DIECK REDUCTION The covering trick allows to reduce the Zariski Cancellation Problem for general affine surfaces A 1 -fibered over affine curves to its Z / d Z equivariant version for GDF surfaces, due to the following commutative diagram. Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

TOM DIECK REDUCTION ∼ = G ✲ X 1 × A 1 X 0 × A 1 ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ / G / G ❅ ❅ ❘ ❅ ❅ ❘ ∼ = ✲ Y 1 × A 1 Y 0 × A 1 ❄ ❄ id ✲ B B ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ / G / G ❅ ❅ ❘ ❅ ❅ ❘ ❄ ❄ id ✲ C C Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

DANIELEWSKI-FIESELER QUOTIENT DEFINITIONS Let π : X → B be a GDF surface. Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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

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

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

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 b 1 , . . . , b t that correspond to degenerate fibers of π , Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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 b 1 , . . . , b t that correspond to degenerate fibers of π , and ˘ B has n i points { b ij } over b i if π − 1 ( b i ) consists of n i (reduced) A 1 -components. Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

THE PICARD GROUP Pic ( ˘ B ) DEFINITION A divisor D on ˘ B is a formal sum N m j ˘ � D = b j , j = 1 where ˘ b j ∈ ˘ B and m j ∈ Z . Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

THE PICARD GROUP Pic ( ˘ B ) DEFINITION A divisor D on ˘ B is a formal sum N m j ˘ � D = b j , j = 1 where ˘ b j ∈ ˘ B and m j ∈ 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

THE PICARD GROUP Pic ( ˘ B ) DEFINITION A divisor D on ˘ B is a formal sum N m j ˘ � D = b j , j = 1 where ˘ b j ∈ ˘ B and m j ∈ 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

THE PICARD GROUP Pic ( ˘ B ) DEFINITION A divisor D on ˘ B is a formal sum N m j ˘ � D = b j , j = 1 where ˘ b j ∈ ˘ B and m j ∈ 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

ZCP for surfaces - MAIN THEOREM 2 THEOREM There is a one-to-one correspondence between Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

ZCP for surfaces - MAIN THEOREM 2 THEOREM There is a one-to-one correspondence between Pic ( ˘ B ) and the set of B -isomorphism classes of cylinders X × A 1 → B over the GDF surfaces X → B with a given DF-quotient ˘ B . Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

ZCP for surfaces - MAIN THEOREM 2 THEOREM There is a one-to-one correspondence between Pic ( ˘ B ) and the set of B -isomorphism classes of cylinders X × A 1 → 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

ZCP for surfaces - MAIN THEOREM 2 THEOREM There is a one-to-one correspondence between Pic ( ˘ B ) and the set of B -isomorphism classes of cylinders X × A 1 → 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

ZCP for surfaces - MAIN THEOREM 2 THEOREM There is a one-to-one correspondence between Pic ( ˘ B ) and the set of B -isomorphism classes of cylinders X × A 1 → 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 × A 1 ∼ = B X ′ × A 1 ⇔ L ∼ = L ′ . In fact, X is a Zariski factor. Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

ZCP for surfaces - MAIN THEOREM 2 = A 1 then any morphism A 1 → B is REMARK If B �∼ constant. Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

ZCP for surfaces - MAIN THEOREM 2 = A 1 then any morphism A 1 → B is REMARK If B �∼ constant. Hence, for two A 1 -fibered surfaces X → B and X ′ → B over the same base one has X × A 1 ∼ = X ′ × A 1 ⇔ X × A 1 ∼ = B X ′ × A 1 up to an automorphism of B . Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

ZCP for surfaces - MAIN THEOREM 2 = A 1 then any morphism A 1 → B is REMARK If B �∼ constant. Hence, for two A 1 -fibered surfaces X → B and X ′ → B over the same base one has X × A 1 ∼ = X ′ × A 1 ⇔ X × A 1 ∼ = B X ′ × A 1 up to an automorphism of B . Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

FIBER TREES DEFINITION Consider a GDF surface π : X → B . Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

FIBER TREES DEFINITION Consider a GDF surface π : X → B . Let π : ¯ X → ¯ B be a P 1 -fibration which results from a ¯ smooth completion of X by a simple normal crossing divisor D = ¯ X \ X . Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

FIBER TREES DEFINITION Consider a GDF surface π : X → B . Let π : ¯ X → ¯ B be a P 1 -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

FIBER TREES DEFINITION Consider a GDF surface π : X → B . Let π : ¯ X → ¯ B be a P 1 -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 ∈ ¯ π − 1 ( b ) B different from the points b i the fiber ¯ π − 1 ( b i ) is a tree of is reduced and irreducible, while ¯ P 1 -curves. Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

FIBER TREES DEFINITION Consider a GDF surface π : X → B . Let π : ¯ X → ¯ B be a P 1 -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 ∈ ¯ π − 1 ( b ) B different from the points b i the fiber ¯ π − 1 ( b i ) is a tree of is reduced and irreducible, while ¯ P 1 -curves. Its dual graph T i is called a FIBER TREE. Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

FIBER TREES DEFINITION Consider a GDF surface π : X → B . Let π : ¯ X → ¯ B be a P 1 -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 ∈ ¯ π − 1 ( b ) B different from the points b i the fiber ¯ π − 1 ( b i ) is a tree of is reduced and irreducible, while ¯ P 1 -curves. Its dual graph T i is called a FIBER TREE. The root R i of T i is the unique component of ¯ π − 1 ( b i ) meeting S . Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

TYPE DIVISOR DEFINITIONS The LEAVES L ij of T i are its extremal vertices different from the root R i . Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

TYPE DIVISOR DEFINITIONS The LEAVES L ij of T i are its extremal vertices different from the root R i . The LEVEL of L ij is the tree distance l ij = dist ( L ij , R i ) . Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

TYPE DIVISOR DEFINITIONS The LEAVES L ij of T i are its extremal vertices different from the root R i . The LEVEL of L ij is the tree distance l ij = dist ( L ij , R i ) . The leaves of T i represent the components of π − 1 ( b i ) , hence also the points b ij of ˘ B over b i . Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

TYPE DIVISOR DEFINITIONS The LEAVES L ij of T i are its extremal vertices different from the root R i . The LEVEL of L ij is the tree distance l ij = dist ( L ij , R i ) . The leaves of T i represent the components of π − 1 ( b i ) , hence also the points b ij of ˘ B over b i . The TYPE DIVISOR is l ij b ij ∈ Div ( ˘ � tp ( π ) = − B ) . i , j Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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

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 . ˜ ˆ Γ : Γ : Figure: The “bush” ˜ Γ and the “spring bush” ˆ Γ have the same type divisors Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

MAIN INGREDIENTS OF THE PROOF • Flexibility (an equivariant relative version); Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

MAIN INGREDIENTS OF THE PROOF • Flexibility (an equivariant relative version); • affine modifications, especially the Asanuma modification; Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

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

TRIVIALIZING SEQUENCE DEFINITION Let X → B be a GDF surface. There exists a TRIVIALIZING SEQUENCE of affine modifications Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

TRIVIALIZING SEQUENCE DEFINITION Let X → B be a GDF surface. There exists a TRIVIALIZING SEQUENCE of affine modifications → B X 0 = B × A 1 . ρ m − 1 ρ m ρ 1 X = X m − → B X m − 1 − → B . . . − Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

TRIVIALIZING SEQUENCE DEFINITION Let X → B be a GDF surface. There exists a TRIVIALIZING SEQUENCE of affine modifications → B X 0 = B × A 1 . ρ m − 1 ρ m ρ 1 X = X m − → B X m − 1 − → B . . . − The center of ρ i is a reduced finite subcheme of the exceptional divisor of ρ i − 1 . Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

TRIVIALIZING SEQUENCE DEFINITION Let X → B be a GDF surface. There exists a TRIVIALIZING SEQUENCE of affine modifications → B X 0 = B × A 1 . ρ m − 1 ρ m ρ 1 X = X m − → B X m − 1 − → B . . . − The center of ρ i is a reduced finite subcheme of the exceptional divisor of ρ i − 1 . This sequence can be extended to a sequence of suitable SNC completions: ¯ X = ¯ ¯ ¯ → B ¯ ¯ → B ¯ X 0 = ¯ ρ m ρ m − 1 ρ 1 B × P 1 − − → B . . . − X m X m − 1 Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

TRIVIALIZING SEQUENCE DEFINITION Let X → B be a GDF surface. There exists a TRIVIALIZING SEQUENCE of affine modifications → B X 0 = B × A 1 . ρ m − 1 ρ m ρ 1 X = X m − → B X m − 1 − → B . . . − The center of ρ i is a reduced finite subcheme of the exceptional divisor of ρ i − 1 . This sequence can be extended to a sequence of suitable SNC completions: ¯ X = ¯ ¯ ¯ → B ¯ ¯ → B ¯ X 0 = ¯ ρ m ρ m − 1 ρ 1 B × P 1 − − → B . . . − X m X m − 1 ρ i : ¯ → B ¯ where ¯ X i − X i − 1 is a blowup with a smooth reduced center. Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

A GDF surface with a given fiber tree Consider a rooted tree Γ with a root R and n i vertices on level i , i = 1 , . . . , m . Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

A GDF surface with a given fiber tree Consider a rooted tree Γ with a root R and n i vertices on level i , i = 1 , . . . , m . Given a point b ∈ B we identify R with the fiber { b } × P 1 of pr 1 : ¯ X 0 → ¯ B . Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

A GDF surface with a given fiber tree Consider a rooted tree Γ with a root R and n i vertices on level i , i = 1 , . . . , m . Given a point b ∈ B we identify R with the fiber { b } × P 1 of pr 1 : ¯ X 0 → ¯ B . Choose n 1 points on R , Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

A GDF surface with a given fiber tree Consider a rooted tree Γ with a root R and n i vertices on level i , i = 1 , . . . , m . Given a point b ∈ B we identify R with the fiber { b } × P 1 of pr 1 : ¯ X 0 → ¯ B . Choose n 1 points on R , ρ 1 : ¯ → B ¯ and let ¯ X 1 − X 0 be the blowup of these points with exceptional divisor E 1 = ¯ F 1 + . . . + ¯ F n 1 . Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

A GDF surface with a given fiber tree Consider a rooted tree Γ with a root R and n i vertices on level i , i = 1 , . . . , m . Given a point b ∈ B we identify R with the fiber { b } × P 1 of pr 1 : ¯ X 0 → ¯ B . Choose n 1 points on R , ρ 1 : ¯ → B ¯ and let ¯ X 1 − X 0 be the blowup of these points with exceptional divisor E 1 = ¯ F 1 + . . . + ¯ F n 1 . We let X 1 \ ( R ′ ∪ S ′ X 1 = ¯ ∞ ∪ ¯ F ∞ ) where S ∞ ⊂ ¯ X 0 is the section at infinity and ¯ F ∞ ⊂ ¯ X 0 is the fiber at infinity. Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

A GDF surface with a given fiber tree Consider a rooted tree Γ with a root R and n i vertices on level i , i = 1 , . . . , m . Given a point b ∈ B we identify R with the fiber { b } × P 1 of pr 1 : ¯ X 0 → ¯ B . Choose n 1 points on R , ρ 1 : ¯ → B ¯ and let ¯ X 1 − X 0 be the blowup of these points with exceptional divisor E 1 = ¯ F 1 + . . . + ¯ F n 1 . We let X 1 \ ( R ′ ∪ S ′ X 1 = ¯ ∞ ∪ ¯ F ∞ ) where S ∞ ⊂ ¯ X 0 is the section at infinity and ¯ F ∞ ⊂ ¯ X 0 is the fiber at infinity. Then X 1 → B is a GDF surface with a unique reducible fiber π ∗ 1 ( b ) = F 1 + . . . + F n 1 . Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

A GDF surface with a given fiber tree ρ 2 : ¯ → B ¯ The center of ¯ X 2 − X 1 consists of n 2 points on E 1 distributed between the components F i of E 1 \ S ∞ according to the edges of Γ . Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

A GDF surface with a given fiber tree ρ 2 : ¯ → B ¯ The center of ¯ X 2 − X 1 consists of n 2 points on E 1 distributed between the components F i of E 1 \ S ∞ according to the edges of Γ . Continuing in this way we construct a GDF surface X m → B where m is the height of Γ . Mikhail ZAIDENBERG ZARISKI CANCELLATION FOR SURFACES

A GDF surface with a given fiber tree ρ 2 : ¯ → B ¯ The center of ¯ X 2 − X 1 consists of n 2 points on E 1 distributed between the components F i of E 1 \ S ∞ according to the edges of Γ . Continuing in this way we construct a GDF surface X m → 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