monomialization of strongly prepared morphisms from nonsingular n-folds to surfaces

monomialization of strongly prepared morphisms from nonsingular n-folds to surfaces Steven D Cutosy and Olga Kashcheyeva Introduction Monomialization of morphisms is the problem of transforming a mapping into a monomial mapping by blowing up a chain of nonsingular subvarieties in its domain and image. Consider the following basic example. Let Φ : A n Am of affine spaces over field. Then Φ is given by a collection of polynomials f,..., f m in n variables: y = f (x,..., x n ). y m = f m (x,..., x n ). be a morphism The simplest structure of Φ is obtained when f,..., f m are monomials and y = x a x a n n. y m = x a m x am n n. If, moreover, Φ is a dominant morphism the matrix ( j ) is forced to satisfy the nondegeneracy condition ran( j ) = m. In case of a general dominant morphism Φ : X Y between two - varieties X and Y we would lie to get such a nice description locally. The first author was partially supported by NSF

Definition.. Suppose that Φ : X Y is a dominant morphism of nonsingular varieties. Φ is called monomial if for all points p X there exist an étale neighborhood U of p, uniformizing parameters (x,..., x n ) on U, regular parameters y,..., y m in O Y, Φ(p) and a matrix (j ) of nonnegative integers with ran m such that y = x a x a n n. y m = x a m x amn n. The natural question arises. Question. Suppose that Φ : X Y is a dominant morphism of - varieties. Does there exist a monomialization of Φ? Or, more precisely, given a dominant morphism Φ : X Y does there exist a monomial morphism Φ : X Y such that the following diagram commutes X Φ X Y Φ Y and all vertical maps are products of blowups of nonsingular subvarieties in X and Y? The answer is yes over a characteristic zero field when Y is a curve or when Y is a surface and dim(x) 3. Suppose that is algebraically closed field of characteristic zero. If Φ : X C is a dominant morphism from a -variety to a curve the existence of monomialization follows from resolution of singularities. If Φ : P S is a dominant morphism of surfaces one proof of monomialization (over C) is given by Abulut and King [3]. And the last nown case, when Φ : X S is a dominant morphism from a 3-fold to a surface, is done by the first author in [4]. This proof of monomialization breas down into two ey steps: ) obtain a diagram X Φ X Y Φ Y 2

where Φ is a strongly prepared morphism and the vertical maps are the products of blowups of nonsingular subvarieties; 2) monomialize the strongly prepared morphism Φ : X Y. The natural next case to consider is monomialization of morphisms from n-folds to surfaces. A proof would follow from the two steps above when X is an n-fold and Y is a surface. In this paper we complete step 2). Our main result is Theorem.2. Suppose that Φ : X S is a strongly prepared morphism from a nonsingular n-fold X to a nonsingular surface S. Then there exists a finite sequence of quadratic transforms π : S S and monoidal transforms centered at nonsingular varieties of codimension 2 π 2 : X X such that the induced morphism Φ : X S is monomial. From here we deduce that it is possible to toroidalize a strongly prepared morphism. Theorem.3. Suppose that Φ : X S is a strongly prepared morphism from a nonsingular n-fold X to a nonsingular surface S. Then there exists a finite sequence of quadratic transforms π : S S and monoidal transforms centered at nonsingular varieties of codimension 2 π 2 : X X such that the induced morphism Φ : X S is toroidal. The definition of a strongly prepared morphism is given in Section 3, definition 3.2. The class of strongly prepared morphisms is rather restrictive. However, a natural example of such a morphism can be obtained as follows. Let Φ : X S be a monomial mapping from an n-fold to a surface and π : X X be a finite sequence of blowups of points. Then the composition map Φ π : X S is strongly prepared, but not necessarily monomial. 2 Notations We will suppose that is an algebraically closed field of characteristic zero. By a variety we will mean a separated, integral finite type -scheme. A point of a variety will mean a closed point. By a generic point on a variety we will mean a point which satisfies a good condition which holds on an open set of points. Suppose that Z is a variety and p Z is a point. Then m p will denote the maximal ideal of O Z, p. 3

Suppose that P (x) = i=0 c ix i [[x]] is a series. Given e N, P e (x) will denote the polynomial P e (x) = e i=0 c ix i. Given a series f(x,..., x n ) [[x,..., x n ]], ord f will denote the order of f (with ord 0 = ). If x Q, [x] will denote the greatest integer n N such that n x. The greatest common divisor of a,..., a n N will be denoted by (a,..., a n ). Definition 2.. A reduced divisor D on a nonsingular variety X of dimension n is called a simple normal crossing divisor (SNC divisor) if all components of D are nonsingular and the following condition holds. Suppose that p X is a point and D,..., D s are the components of D containing p. Then s n and there exist regular parameters (x,..., x n ) in O X, p such that D,..., D s have at p local equations x = 0,..., x s = 0, respectively. Definition 2.2. A codimension 2 subvarieties C,..., C m of a nonsingular dimension n variety X mae simple normal crossings (SNCs) if C i is nonsingular for all i =,..., m and the following condition holds. Suppose that p X is a ( point ) and C,..., C s are the subvarieties n containing p. Then s and there exist regular parameters 2 (x,..., x n ) in O X, p such that for all i =,..., s x li = x i = 0 are local equations of C i at p and l i < i n. Definition 2.3. Suppose that Φ : X Y is a dominant morphism of nonsingular varieties. Φ is called monomial if for all points p X there exist an étale neighborhood U of p, uniformizing parameters (x,..., x n ) on U, regular parameters y,..., y m in O Y, Φ(p) and a matrix (j ) of nonnegative integers with ran m such that 3 Monomialization y = x a x a n n. y m = x a m x amn n. Definition 3.. Suppose that Φ : X S is a dominant morphism from a nonsingular variety X to a nonsingular variety S with reduced SNC divisors 4

D S on S and E X on X such that Φ (D S ) red = E X. Let sing(φ) be the locus of singular points of Φ. We will say that Φ is quasi prepared (with respect to D S ) if sing(φ) E X. Suppose that Φ : X S is a quasi prepared morphism from a nonsingular n-fold X to a nonsingular surface S. If p E X we will say that p is a, 2,..., n point depending on if p is contained in, 2,..., n components of E X. q D S will be called a or 2 point depending on if q is contained in or 2 components of D S. Regular parameters (u, v) O S, q for q D S are permissible if: ) q is a point and u = 0 is a local equation of D S or 2) q is a 2 point and uv = 0 is a local equation of D S. Definition 3.2. Suppose that Φ : X S is a quasi prepared morphism from a nonsingular n-fold X to a nonsingular surface S. We will say that Φ is strongly prepared at p X (with respect to D S ) if there exist permissible parameters (u, v) at Φ(p) and regular parameters (x,..., x n ) in ÔX, p such that one of the following forms holds: () n p is a point, u = 0 is a local equation of E X and x a )m v = P (x a x a ) + xb x b x + where m > 0, a,..., a > 0 with (a,..., a ) =, b,..., b 0 and P is a series; (2) 2 n p is a point, u = 0 is a local equation of E X and x a )m v = P (x a x a ) + xb x b where( m > 0, a ),..., a > 0 with (a,..., a ) =, b,..., b 0, a... a ran = 2 and P is a series; b... b 5

(3) 2 n p is a point, uv = 0 is a local equation of E X and u = x a x a v = x b 2 2 x b where a 2,..., a, b 2,..., b 0, a, b > 0 and + b i > 0 for all i = 2,...,. In this case the regular parameters (x,..., x n ) in ÔX, p are called - permissible parameters at p for (u, v) and the permissible parameters (u, v) are called prepared. Φ : X S is strongly prepared if it is strongly prepared at every point p X. We will now assume that Φ : X S is a strongly prepared morphism from a nonsingular n-fold X to a nonsingular surface S. Lemma 3.3. Suppose that O X, p R is finite étale and there exist x,..., x n R such that (x,..., x n ) are regular parameters in R q for all primes q R such that q O X, p = m p. Then there exists an étale neighborhood U of p such that (x,..., x n ) are uniformizing parameters on U. Proof. There exists an affine neighborhood V = Spec (A) of p X and a finite étale extension B of A such that B A A mp = R. Set U = Spec (B). Let π : U V be the natural map. There exists an open neighborhood U 2 of π (p) such that (x,..., x n ) are uniformizing parameters on U 2. Let Z = U U 2 and W = π(z). Set U = U π (W ), then U V = V W is finite étale. Thus there exists an étale neighborhood U of p where (x,..., x n ) are uniformizing parameters. Lemma 3.4. Suppose that Φ : X S is strongly prepared at p E X. Then there exist prepared parameters (u, v) for Φ(p) and -permissible parameters (x,..., x n ) for (u, v) at p such that (x,..., x n ) are uniformizing parameters on an étale neighborhood of p and one of the following forms holds: () n p is a point, u = 0 is a local equation of E X and x a )m v = P (x a x a ) + xb x b x + 6 (I)

where m > 0, a,..., a > 0 with (a,..., a ) =, b,..., b 0 and either P 0 or P is a polynomial of order max i { b i } such that if Φ(p) is a point then m ordp ; (2) 2 n p is a point, u = 0 is a local equation of E X and x a )m v = P (x a x a ) + xb x b where( m > 0, a ),..., a > 0 with (a,..., a ) =, b,..., b 0, a... a ran = 2 and either P 0 or P is a polynomial of b... b order max i { b i } such that if Φ(p) is a point then m ordp ; (3) 2 n p is a point, uv = 0 is a local equation of E X and u = x a x a v = x b 2 2 x b (II) (III) where a 2,..., a, b 2,..., b 0, a, b > 0 and + b i > 0 for all i = 2,...,. Proof. Let (u, v) be prepared parameters for Φ(p). Suppose first that there exist regular parameters (x,..., x n ) ÔX, p such that x a )m v = P (x a x a ) + xb x b x +. Then there exist y,..., y O X, p and units α,..., α ÔX, p such that x i = α i y i for all i =. Set γ = (α a α a ) a and R = O X, p [γ]. R is finite étale over O X, p. Let L be the quotient field of R. Set e = max i {[ b i ]} and y + = α b α b x + + P (αa α a ya y a ) P e(α a α a ya y a y b y b 7 ).

Then y + = v Pe(αa αa ya ya ) y b y b of R. Thus y + R q = R. Set ȳ = γy and ȳ + = γ b y +, so that ˆR q L = R q for all maximal ideals q u = (ȳ a y a 2 2 y a )m v = P e (ȳ a y a 2 2 y a ) + ȳb y b 2 2 y b ȳ+. For i = +2 n choose y i O X, p such that y i x i mod m 2 pôx, p. Then ȳ, y 2,..., y, ȳ +, y +2,..., y n R are regular parameters at all maximal ideals of R. Since R is finite étale over O X, p, by Lemma 3.3 there exists an étale neighborhood U of p such that (ȳ, y 2,..., y, ȳ +, y +2,..., y n ) are uniformizing on U. To finish the analysis of this case when Φ(p) is a point and ordp e < we only need to ensure that m ordp in the formula for v. Suppose that P e (t) = i=e i= λ i(x a x a )i. Set and v = v i=[ e m ] i= λ im u i P = P e i=[ e m ] i= λ im (x a x a )im so that regular parameters (u, v) at Φ(p) and regular parameters (ȳ, y 2,..., y, ȳ +, y +2,..., y n ) at p satisfy the conditions of the lemma. Suppose now that there exist regular parameters (x,..., x n ) in ÔX, p such that x a )m v = P (x a x a ) + xb x b, where by permuting x,..., x we can assume that f = a b 2 a 2 b > 0. Then there exist y,..., y O X, p and units α,..., α ÔX, p such that x i = α i y i for all i =,...,. Set γ = (α a α a ) f and R = OX, p [γ]. R is finite étale over O X, p. Let L be the quotient field of R. 8

Set e = max i {[ b i ]} and ω = α b α b + P (αa α a ya y a ) P e(α a α a ya y a y b y b Then ω is a unit and ω = v Pe(αa αa ya ya ) y b y b ) ˆR q L = R q for all maximal ideals q of R. Thus ω R q = R. Let R = R[w f ]. R is finite étale over R and, therefore, it is finite étale over O X, p. Set ȳ = γ b 2 ω a 2 f y and ȳ 2 = γ b ω a f y2, so that u = (ȳ a ȳ a 2 2 y a 3 3 y a )m v = P e (ȳ a ȳ a 2 2 y a 3 3 y a ) + ȳb ȳ b 2 2 y b 3 3 y b. For i = +,..., n choose y i O X, p such that y i x i mod m 2 pôx, p. Then ȳ, ȳ 2, y 3,..., y n R are regular parameters at all maximal ideals of R. Since R is finite étale over O X, p, by Lemma 3.3 there exists an étale neighborhood U of p such that (ȳ, ȳ 2, y 3 y n ) are uniformizing on U. To finish the analysis of this case if Φ(p) is a point we change v to v and P e to P in the same way as we did above, so that regular parameters (u, v) at Φ(p) and regular parameters (ȳ, ȳ 2, y 3,..., y n ) at p satisfy the conditions of the lemma. Finally suppose that there exist regular parameters (x,..., x n ) in ÔX, p such that u = x a x a v = x b 2 2 x b. Then there exist y,..., y O X, p and units α,..., α ÔX, p such that x i = α i y i for all i =,...,. Set γ = (α a α a ) a, ω = (α b 2 2 α b ) b finite étale over O X, p. Set ȳ = γy and ȳ = ωy, so that u = ȳ a y a 2 2 y a v = y b 2 2 y b ȳb. 9 and R = O X, p [γ, ω]. R is.

For i = +,..., n choose y i O X, p such that y i x i mod m 2 pôx, p. Then ȳ, y 2,..., y, ȳ, y +,..., y n R are regular parameters at all maximal ideals of R. Since R is finite étale over O X, p, by Lemma 3.3 there exists an étale neighborhood U of p such that (ȳ, y 2,..., y, ȳ, y +,..., y n ) are uniformizing on U. This completes the proof. Suppose that p E X and (u, v) are permissible parameters for Φ(p). (u, v) will be called strongly prepared at Φ(p) and -permissible parameters (x,..., x n ) for (u, v) at p will be called strongly permissible if they satisfy the conditions of Lemma 3.4 Definition 3.5. Suppose that Φ : X S is strongly prepared with respect to D S. Suppose that p E X. We will say that p is a good point for Φ if there exist permissible parameters (u, v) at Φ(p) and -permissible parameters (x,..., x n ) at p for (u, v) such that one of the following forms holds: (a) n p is a point, u = 0 is a local equation of E X and x a )m v = α(x a x a )t + (x a x a )t x + where m > 0, t 0, a,..., a > 0 with (a,..., a ) = and α ; (b) 2 n p is a point, u = 0 is a local equation of E X and (G.Ia) u = x a x a (G.Ib) v = x b x b x + ( ) a... a where a,..., a > 0, b,..., b 0 and ran = 2; b... b (2) 2 n p is a point, u = 0 is a local equation of E X and u = x a x a v = x b x b (G.II) 0

( ) a... a where a,..., a > 0, b,..., b 0 and ran = 2; b... b (3) 2 n p is a point, uv = 0 is a local equation of E X and u = x a x a v = x b 2 2 x b (G.III) where a 2,..., a, b 2,..., b 0, a, b > 0 and + b i > 0 for all i = 2,...,. p will be called a bad point if p is not a good point. Remar 3.6. If p E X is a good point then following the proof of Lemma 3.4 we can always find strongly prepared parameters (u, v) at Φ(p) and strongly permissible parameters (x,..., x n ) at p for (u, v) such that one of the forms (G.Ia), (G.Ib), (G.II) or (G.III) holds. Lemma 3.7. Suppose that p X is a point and (u,v) are strongly prepared parameters at Φ(p), (x,..., x n ) are strongly permissible parameters at p for (u, v) such that (I) holds and u = x a v = P (x ) + x c x 2, ordp = d. Suppose that (ū, v) are also strongly prepared parameters at Φ(p) and (y,..., y n ) are strongly permissible parameters at p for (ū, v) such that (I) holds and ū = y a 2 v = Q(y ) + y c 2 y 2, ordq = d 2. If Φ(p) is a point then a = a 2, c = c 2 and d = d 2. If Φ(p) is a 2 point then d, d 2 < and c d = c 2 d 2, a +d = a 2 +d 2. Proof. This follows from the discussion before Definition 8.7 in [4].

Suppose that E is a component of E X, p E, f ÔX, p and x = 0 is a local equation of E at p. Then we define ν E (f) = max{n such that x n f}. Suppose that p X is a point and E is the component of E X containing p. Suppose that (u, v) are strongly prepared parameters at Φ(p) such that u = 0 is a local equation of E at p and (x,..., x n ) are strongly permissible parameters for (u, v) at p with u = x a v = P (x ) + x c x 2, ordp = d. Then ν E (v) = d if d < and ν E (v) = c if d =. Thus, in view of Lemma 3.7, c ν E (v) and a + ν E (v) are independent of the choice of strongly prepared parameters (u, v) at Φ(p) and they are also independent of the choice of strongly permissible parameters for (u, v) at p. Definition 3.8. Let p X, E E X, regular parameters (u, v) in O S, Φ(p) and regular parameters (x,..., x n ) in ÔX, p be as above with u = x a v = P (x ) + x c x 2. Define A(Φ, p) = c ν E (v). If A(Φ, p) > 0 define C(Φ, p) = (c ν E (v), a + ν E (v)). Notice that if p X is a point then p is a good point if and only if A(Φ, p) = 0 or, equivalently, ordp = d c. Lemma 3.9. Suppose that p X is a point and E is the component of E X containing p. Then there exists an open neighborhood U of p such that A(Φ, p ) = A(Φ, p) for all p E U and if A(Φ, p) > 0 then C(Φ, p ) = C(Φ, p) for all p E U. Proof. There exist strongly prepared parameters (u, v) at Φ(p) and strongly permissible parameters (x,..., x n ) at p for (u, v) such that u = x a v = P (x ) + x c x 2 and ordp = d. 2

If U is an étale neighborhood of p where (x,..., x n ) are uniformizing parameters then for any p U E there exist α 2,..., α n such that (x, x 2 = x 2 + α 2,..., x n = x n + α n ) are strongly permissible parameters at p for strongly prepared parameters (u, v) at Φ(p ) and v = v if Φ(p ) is a 2 point or a c, v = v + α 2 u c f Φ(p ) is a point and a c. Suppose that the first assumption holds and v = v then at p u = x a v = P (x ) α 2 x c + x c (x 2 + α 2 ) = P (x ) + x c x 2. Thus if d < c then ordp = d and A(Φ, p ) = c d = A(Φ, p) > 0, C(Φ, p ) = (c d, a + d) = C(Φ, p). If d c then ordp c and A(Φ, p ) = 0 = A(Φ, p). Suppose that the second assumption holds and v = v + α 2 u c a then at p u = x a v = P (x ) + x c (x 2 + α 2 ) = P (x ) + x c x 2. Thus A(Φ, p ) = A(Φ, p) and C(Φ, p ) = C(Φ, p) if A(Φ, p) > 0. Now we can define A(Φ, E) = A(Φ, p) for p E a point. If A(Φ, E) > 0 define C(Φ, E) = C(Φ, p). We will call E E X a good component of E X if A(Φ, E) = 0. E will be called a bad component if it is not a good component or, equivalently, if A(Φ, E) > 0. Lemma 3.0. Suppose 2 n. Suppose that p X is a point and E,..., E are the components of E X containing p. Suppose that (u, v) are strongly prepared at Φ(p) and (x,..., x n ) are strongly permissible parameters for (u, v) at p with x i = 0 being a local equation of E i for i =,...,. If or x a )m v = P (x a x a ) + xb x b x + x a )m v = P (x a x a ) + xb x b (I) (II) 3

then A(Φ, E i ) = b i ν Ei (v) for i =,..., and if A(Φ, E i ) > 0 C(Φ, E i ) = (b i ν Ei (v), m + ν Ei (v)). If u = x a x a v = x b 2 2 x b then A(Φ, E i ) = 0 for all i =,...,. (III) Proof. Suppose that p X is a point satisfying (I), (u, v) are strongly prepared parameters at Φ(p) and (x,..., x n ) are strongly permissible parameters for (u, v) at p such that x a )m v = P (x a x a ) + xb x b x + and ordp = d. After possibly permuting x,..., x we can assume that i = and prove only that A(Φ, E ) = b ν E (v) and C(Φ, E ) = (b ν E (v), a m + ν E (v)) if A(Φ, E ) > 0. Suppose that U is an étale neighborhood of p where (x,..., x n ) are uniformizing parameters and p U E is a point. Then there exist α 2,..., α {0} and α +,..., α n such that (x, x 2 = x 2 α 2,..., x n = x n α n ) are regular parameters at p. Set γ = (( x 2 + α 2 ) a2 ( x + α ) a ) a, for i = 2 set f i = a b i b and ω = (( x 2 + α 2 ) f2 ( x + α ) f ) a. Notice that γ a, ω a O U, p are units in O U, p and therefore O U, p [γ, ω] is finite étale over O U, p. Set x = γx and x + = x + ω + α + ω α + (α f 2 2 α f ) a then u = x a m v = P ( x a ) + x b x + + α x b, where α = α + (α f 2 2 α f ) a. Assume first that Φ(p) is a 2 point or a m b, then (u, v) are strongly prepared parameters at Φ(p ). If a d b then ( x,..., x, x +, x +2,..., x n ) are strongly permissible parameters at p for (u, v) and u = x a m v = P ( x ) + x b x + 4

with ordp = a d if a d < b and ordp b if a d = b. Thus A(Φ, E ) = A(Φ, p ) = b ν E (v) in this case and if A(Φ, p ) > 0 or, equivalently, if a d < b then C(Φ, E ) = C(Φ, p ) = (b a d, a m + a d) = (b ν E (v), a m + ν E (v)). If a d > b then set x + = x + + P ( xa ) to get strongly permissible x b parameters ( x,..., x n ) for (u, v) at p, so that u = x a m v = x b x + + α x b and A(Φ, E ) = A(Φ, p ) = 0 = b ν E (v). Assume now that Φ(p) is a point and a m b, then (u, v = v αu b a m ) are strongly prepared parameters at Φ(p ). If a d b then ( x,..., x, x +, x +2,..., x n ) are strongly permissible parameters at p for (u, v) and u = x a m v = P ( x a ) + x b x + = P ( x ) + x b x +. Thus A(Φ, E ) = A(Φ, p ) = b a d = b ν E (v) in this case and if A(Φ, p ) > 0 then C(Φ, E ) = C(Φ, p ) = (b a d, a m + a d) = (b ν E (v), a m + ν E (v)). If a d > b then set x + = x + + P ( xa ) to get strongly permissible x b parameters ( x,..., x n ) for (u, v) at p so that u = x a m v = x b x + and A(Φ, E ) = A(Φ, p ) = 0 = b ν E (v). Suppose that p X is a point satisfying (II), (u, v) are strongly prepared parameters at Φ(p) and (x,..., x n ) are strongly permissible parameters for (u, v) at p such that x a )m v = P (x a x a ) + xb x b and ordp = d. 5

After possibly permuting x,..., x we can assume that i = and prove only that A(Φ, E ) = b ν E (v) and C(Φ, E ) = (b ν E (v), a + ν E (v)) if A(Φ, E ) > 0. Suppose that U is an étale neighborhood of p where (x,..., x n ) are uniformizing parameters and p U E is a point. Then there exist α 2,..., α {0} and α +,..., α n such that (x, x 2 = x 2 α 2,..., x n = x n α n ) are ( regular parameters ) at p. a... a Notice that since ran = 2 there exists j {2,..., } such b... b that a b j a j b 0. So, after possibly permuting x 2,..., x we can assume that j = 2, that is a b 2 a 2 b 0. Set γ = (( x 2 + α 2 ) a2 ( x + α ) a ) a, for i = 2,..., set f i = a b i b, ω = (( x 3 + α 3 ) f3 ( x + α ) f ) a and δ = ( x 2 + α 2 ) f 2 a. Notice that γ a, ω a, δ a O U, p are units in O U, p and therefore O U, p [γ, ωδ] is finite étale over O U, p. Set x = γx and x 2 = δω (α f 2 2 α f ) a then u = x a m v = P ( x a ) + x b x 2 + α x b, where α = (α f 2 2 α f ) a {0}. Now the same analysis as above with x 2 playing the role of x + above shows that A(Φ, E ) = b ν E (v) and C(Φ, E ) = (b ν E (v), a m+ν E (v)) if A(Φ, E ) > 0. Suppose that p X is a point satisfying (III), (u, v) are strongly prepared parameters at Φ(p) and (x,..., x n ) are strongly permissible parameters for (u, v) at p such that u = x a x a v = x a 2 2 x b. Since a, b > 0 and a j + b j > 0 for all j = 2,...,, after possibly permuting x,..., x and u, v we can assume that i = and u = x a x a v = P (x a x a ) + xb x b, where a, b > 0 and P (t) 0. In this notations the proof of the required statement repeats the proof for case (II). 6

Theorem 3.. Suppose that Φ : X S is strongly prepared. Then the locus of bad points in X is a Zarisi closed set of pure codimension, consisting of the union of all bad components of E X. Proof. Let Z be the union of all bad components of E X and let p be a good point on E X, q be a bad point on E X. Then it suffices to show that q Z while p E X Z. Suppose that p is a good point, (u, v) are strongly prepared parameters at Φ(p), (x,..., x n ) are strongly permissible parameters for (u, v) at p such that one of the following forms holds (a) (b) (2) (3) x a )m v = α(x a x a )t + (x a x a )t x + ; u = x a x a v = x b x b x + ; u = x a x a v = x b x b ; u = x a x a v = x b 2 2 x b. In all these cases x = 0,..., x = 0 are local equations of the components of E X containing p. So we can assume that for all i =,..., x i = 0 is a local equation of E i E, a component of E X. By Lemma 3.0 we compute A(Φ, E i ) as follows (a) A(Φ, E i ) = t ν Ei (v) = t t = 0 (b) A(Φ, E i ) = b i ν Ei (v) = b i b i = 0 7

(2) A(Φ, E i ) = b i ν Ei (v) = b i b i = 0 (3) A(Φ, E i ) = 0. Thus all components of E X containing p are good, so p does not lie in Z. Suppose that q is a bad point, (u, v) are strongly prepared parameters at Φ(q), (x,..., x n ) are strongly permissible parameters for (u, v) at q and (I) holds, that is x a )m v = P (x a x a ) + xb x b x +, where P (t) 0. Let x = 0,..., x = 0 be local equations of the components E,..., E E X containing q. By Lemma 3.0 we can { compute A(Φ, E i ) for all i =,..., as follows b i ordp > 0, if ordp < b i ; A(Φ, E i ) = b i ν Ei (v) = 0, otherwise. To prove the statement of the theorem we need to find such j {,..., } that E j is a bad component or, equivalently, A(Φ, E j ) > 0. Assume the contrary. Let ordp b i for all i =,..., then v = x b x b (x + + P (xa If ord P (xa xa ) xa ) x b x b ) and P (x a xa ) x b x b ÔX, q. a xa ) x b x b then set x x b x b + = x + + P (x so that (x,..., x, x +, x +2,..., x n ) are strongly permissible parameters for (u, v) at q and x a )m v = x b x b x +. Thus one of the forms (G.Ia) or (G.Ib) holds for q and, therefore, q is a good point while q was originally chosen to be a bad point. If ord P (xa xa ) = 0 then there exist α {0} and y m q Ô X, q such x b x b that P (xa xa ) = y+α. Set x x b x b + = x + +y so that (x,..., x, x +, x +2,..., x n ) are strongly permissible parameters for (u, v) at q and x a )m v = x b x b ( x + + α). 8

( ) a... a Now if ran < 2 then (G.Ia) holds and, therefore, q is a b... b good point. ( This contradicts ) the choice of q. a... a If ran = 2 then after possibly permuting x b... b,..., x we can assume that f = a b 2 a 2 b 0. Set x = x ( x + + α) a 2 f x 2 = x 2 ( x + + α) a f then ( x, x 2, x 3,..., x, x +, x +2,..., x n ) are -permissible parameters for (u, v) at q and u = ( x a x a 2 2 x a 3 3 x a )m v = x b x b 2 2 x b 3 3 x b. Thus (G.II) holds for q and, therefore, q is a good point while q was originally chosen to be a bad point. This shows that if q is a bad point on E X and (I) holds at q then there exists such a component E of E X containing q that A(Φ, E) > 0. So E Z and, therefore, q Z. Suppose that q is a bad point, (u, v) are strongly prepared parameters at Φ(q), (x,..., x n ) are strongly permissible parameters for (u, v) at q and (II) holds, that is x a )m v = P (x a x a ) + xb x b, where P (t) 0. Let x = 0,..., x = 0 be local equations of the components E,..., E E X containing q. By Lemma 3.0 we can { compute A(Φ, E i ) for all i =,..., as follows b i ordp > 0, if ordp < b i ; A(Φ, E i ) = b i ν Ei (v) = 0, otherwise. To prove the statement of the theorem we need to find such j {,..., } that E j is a bad component or, equivalently, A(Φ, E j ) > 0. Assume the contrary. Let ordp b i for ( all i =,..)., then v = x b x b ( + P (xa xa ) a... a ). Since ran = 2 there exists x b x b b... b l {,..., } such that a l ordp > b l, so P (xa xa ) 9 x b x b m q Ô X, q.

After possibly permuting x,..., x we can assume that f = a b 2 a 2 b 0. Set x = x ( + P (xa x a x b x b x 2 = x 2 ( + P (xa x a x b x b ) ) ) a2 f so that ( x, x 2, x 3,..., x n ) are -permissible parameters for (u, v) at q and ) a f u = ( x a x a 2 2 x a 3 3 x a )m v = x b x a 2 2 x a 3 3 x b. Thus (G.II) holds for q and, therefore, q is a good point while q was originally chosen to be a bad point. This shows that if q is a bad point on E X and (II) holds at q then there exists such a component E of E X containing q that A(Φ, E) > 0. So E Z and, therefore, q Z. Lemma 3.2. Suppose that Φ : X S is strongly prepared, q D S and p Φ (q) is a point on X such that one of the forms (I), (II) or (III) holds at p. Then m q O X, p is not invertible if and only if one of the following holds: (a) n x a )m v = (x a x a )t x + (N.Ia) where m > 0, a,..., a > 0 with (a,..., a ) = and 0 t < m; (b) 2 n x a )m v = (x b x b )x + (N.Ib) where( a,..., a > ) 0 with (a,..., a ) =, b,..., b 0, a... a ran = 2 and min b... b i { b i a i } < m; 20

(c) 2 n x a )m v = P (x a x a ) + xb x b x + (N.Ic) where( a,..., a > ) 0 with (a,..., a ) =, b,..., b 0, a... a ran = 2 and min b... b i { b i a i } < ordp < max i { b i }, min i { b i } < m; (2a) 2 n x a )m v = (x b x b ) (N.IIa) where( a,..., a > ) 0 with (a,..., a ) =, b,..., b 0, a... a ran = 2 and min b... b i { b i a i } < m < max i { b i }; (2b) 2 n x a )m v = P (x a x a ) + xb x b (N.IIb) where( a,..., a > ) 0 with (a,..., a ) =, b,..., b 0, a... a ran = 2 and min b... b i { b i a i } < ordp < max i { b i }, min i { b i } < m; (3) 2 n u = x a x a v = x b 2 2 x b (N.III) where a 2,..., a, b 2,..., b 0, a, b > 0 and + b i > 0 for all i = 2,...,. 2

Proof. ( Suppose that ) (I) holds at p. First consider the case when a... a ran < 2. Then there exist t 0 such that b... b x a )m v = P (x a x a ) + (xa x a )t x +. If d = ordp t then v = (x a x a )d γ, where γ ÔX, p is a unit. So either u is a multiple of v if d m or v is a multiple of u if m d. Thus we may assume that ordp > t. Set x + = x + + P (xa x a ) (x a x a )t O ˆX, p to get strongly permissible parameters (x,..., x, x +, x +2,..., x n ) for (u, v) at p such that x a )m v = (x a x a )t x +. So (u, v)o X, p is not invertible if and only if t < m and we get case (N.Ia) of the lemma. ( ) a... a Suppose now that ran = 2. If P (t) 0 in the formula b... b for v then x a )m v = x b x b x + and v is not a multiple of u if and only if m > b i for some i {,..., }, that is if m > min i { b i }. Thus we meet case (N.Ib) of the lemma. Suppose that P (t) 0 then d = ordp max i { b i }. If d = max i { b i then v = P (x a x a ) + xb x b x + = (x b x b )( P (xa x a + x x b x b + ) ) } where the smallest degree term of x a d b x a d b P (x a xa ) m p Ô X, p since d b i x b x b is x a d b x a d b. for all i =,..., and d b j a j for 22

( ) a... a at least one j {,..., } due to maximality of the ran. So b... b by setting x + = x + + P (xa xa ) we return to the situation when P (t) 0 x b x b in the formula for v. Therefore, we may restrict our considerations to d = ordp < max i { b i }. If d min i { b i } then v = (x a x a )d γ, where γ = P (xa x a ) (x a x a )d + xb a d x b a d x + ÔX, p is a unit. So either v is a multiple of u if m d or u is a multiple of v if d m. On the other hand if ordp > min i { b i } we denote by c i the minimum of d and b i to present v as follows v = x c x c ( P (xa x a x c x c + x b c x b c x + ) = x c x c γ, ) where γ m p O X, p and x i γ for all i =,...,. Then u cannot be a multiple of v and v is a multiple of u if and only if m min { c i } = min i a min{d, b i } = min i i a { b i }. i i Thus, with the assumption m > min i { b i and this is case (N.Ic) of the lemma. } (u, v)o X, p is not invertible Suppose that (II) holds at p. Assume first that P (t) 0 in the formula for v so that x a )m v = x b x b Then v is not a multiple of u if and only if there exist i {, } such that m > b i, that is if m > min i { b i }. The symmetric condition for u not being a multiple of v gives the restriction m < max i { b i }. Thus we meet case (N.IIa) of the lemma. Let P (t) 0 then d = ordp max i { b i }. If d = max i { b i } then v = P (x a x a )+xb x b = (x b x b )( P (xa x a 23 x b x b ) +) = x b x b γ

P (x a xa ) where the smallest degree term of is x a d b x b x b x a d b. So γ ÔX, p is a unit since d b i for all i =,..., and d b j a j for at least one ( ) a... a j {,..., } due to maximality of the ran. b... b After possibly permuting regular coordinates x,..., x n we can assume that f = a b 2 a 2 b 0. Then by setting x = x γ a 2 f x 2 = x 2 γ a f we get strongly permissible parameters ( x, x 2, x 3,..., x n ) for (u, v) at p such that u = ( x a x a 2 2 x a 3 3 x a )m v = x b x b 2 2 x b 3 3 x b. Thus we may assume that ordp < max i { b i }. If d min i { b i } then v = (x a x a )d γ, where γ = P (xa x a ) (x a x a + xb a d x b a d ÔX, p )d is a unit. So either v is a multiple of u if m d or u is a multiple of v if d m. On the other hand if ordp > min i { b i } we denote by c i the minimum of d and b i to present v as follows v = x c x c ( P (xa x a x c x c + x b c x b c ) = x c x c γ, ) where γ m p O X, p and x i γ for all i =,...,. Then u cannot be a multiple of v and v is a multiple of u if and only if m min { c i } = min i a min{d, b i } = min i i a { b i }. i i Thus, with the assumption m > min i { b i and this is case (N.IIb) of the lemma. } (u, v)o X, p is not invertible 24

Suppose that (III) holds at p. Then u = x a x a v = x b 2 x b, with a, b > 0. So m q O X, p is not invertible and this describes case (N.III) of the lemma. Lemma 3.3. Suppose that Φ : X S is strongly prepared. Let π : S S be the blow up of S at a point q D S. Let U be the largest open set of X such that the rational map X S is a morphism Φ : U S on U. Then Φ is strongly prepared with respect to π (D S ), and if all points of U are good for Φ then all points of U are good for Φ also. Proof. This follows from the analysis of the proof of Lemma 3.2. In fact, the conclusions of the Lemma are clear if u v in O X,p. If v u and () or (2) holds in Definition 3.2, we can mae a change of variables in the x i, replacing x i with γ i x i, where γ i is a unit series for i, and maing an appropriate change of x + to get an expression of the form () or (2) of Definition 3.2, with u and v interchanged. If p is a good point, the new expressions of v and u will have the good expressions of (a) or (2) of Definition 3.5. Theorem 3.4. Suppose that Φ : X S is strongly prepared, p X is a point and the rational map Φ from X to the blow up S of q = Φ(p) is a morphism in a neighborhood of p. Then A(Φ, p) A(Φ, p) and if A(Φ, p) = A(Φ, p) > 0 then C(Φ, p) < C(Φ, p). Proof. Let (x,..., x n ) be strongly permissible parameters at p for strongly prepared parameters (u, v) at Φ(p), then u = x a v = P (x ) + x b x 2 with ordp = d. Suppose first that P (x) 0. Then b a since (u, v)o X, p is principal and there exist strongly prepared parameters (u, v ) at Φ (p) such that u = u, v = u v. 25

Thus u = x a v = x b a x 2 and A(Φ, p) = 0 = A(Φ, p). Now suppose that P (x) 0, so d b. If d a then there exist α and strongly prepared parameters (u, v ) at Φ (p) such that Thus u = u, u = x a v = P (x ) x a, v = u (v + α). α + x b a x 2 and A(Φ, p) (b a) (d a) = b d = A(Φ, p), where the equality holds if and only if d > a. In this case C(Φ, p) = (b d, d a + a) = (b d, d) < (b d, d + a) = C(Φ, p). If d < a then there exist strongly prepared parameters (u, v ) at Φ (p) and strongly permissible parameters ( x, x 2, x 3,..., x n ) at p such that and u = x d v = P ( x ) x d u = u v, v = u + x b+a 2d x 2, with ord P = a. Thus A(Φ, p) = (b + a 2d) (a d) = b d = A(Φ, p) and C(Φ, p) = (b d, a d + d) = (b d, a) < (b d, d + a) = C(Φ, p). Suppose that Φ : X S is strongly prepared. We will denote by Z(Φ) the locus of bad points in X. If q D S denote by N q (Φ) the locus of points in X where Φ does not factor through the blowup of q. Then N(Φ) will denote the union of N q (Φ) for all q D S. We will denote by B 2 (X) the set of all 2 points in X. Let B 2 (X) be the Zarisi closure of B 2 (X). We will also say that a codimension 2 subvariety C X is a 2-variety if C = E E 2 for some components E and E 2 of E X. 26

Remar 3.5. B2 (X) is the union of all 2-varieties on X. Lemma 3.6. Suppose that Φ : X S is strongly prepared and q D S is such that Z(Φ) N q (Φ). Then Z(Φ) N q (Φ) is a Zarisi closed set of pure codimension 2, consisting of the union of all 2-varieties in Φ (q) with a generic point in the form of (N.IIb). Suppose that C is a component of Z(Φ) N q (Φ) and π : X X is the blowup of C with exceptional variety E = π (C) red. Then Φ = Φ π is strongly prepared and A(Φ, E) < A(Φ). Proof. Z(Φ) and N q (Φ) are both closed, so to prove the first statement of the theorem it suffices to show that any bad point p N q (Φ) lies on a 2-variety C such that a generic point p C is in the form of (N.IIb) and p Φ (q). Notice also that if p Z(Φ) N q (Φ) then either (N.Ic) or (N.IIb) holds at p. Suppose that p N q (Φ) is a point and (N.Ic) holds at p, (u, v) are strongly prepared parameters at q and (x,..., x n ) are strongly permissible parameters at p for (u, v), then x a )m v = P (x a x a ) + xb x b x +. After possibly permuting x,..., x we can assume that b a < ordp < b 2 a 2 and b a < m. Suppose that E,..., E are the components of E X containing p with local equations x = 0,..., x = 0 respectively. Let U be an étale neighborhood of p where (x,..., x n ) are uniformizing parameters. Set C = E E 2 and fix a 2 point p U C away from the vanishing locus of x +. Then there exist α 3,..., α + {0} and α +2,..., α n such that (x, x 2, x 3 = x 3 α 3, x 4 = x 4 α 4,..., x n = x n α n ) are regular parameters at p. Since f = a b 2 a 2 b 0 we can set γ = (( x 3 + α 3 ) a3 ( x + α ) a ) f and ω = (( x 3 + α 3 ) b3 ( x + α ) b ( x+ + α + )) f. Then γ f, ω f O U, p are units in O U, p and, therefore, O U, p [γ, ω] is finite étale over O U, p. Set x = γ b 2 ω a 2 x and x 2 = γ b ω a x 2 so that ( x,..., x n ) are strongly permissible parameters at p for strongly prepared parameters (u, v) at q and u = ( x a x a 2 2 ) m v = P ( x a x a 2 2 ) + x b x b 2 2. 27

Thus (N.IIb) holds at a generic point p of C and Φ(p ) = q. If π : X X is the blow up of C then Φ = Φ π is strongly prepared above p. If p lies in the intersection of more than 2 components E,..., E of E X then π (p) does not contain any point. Assume that p is a 2 point. If s π (p) is a point then (x, x 2, x 3,..., x n ) are strongly permissible parameters at s where x 2 is defined by x 2 = x ( x 2 + α) for some nonzero α. After setting x = x ( x 2 + α) a 2 f a +a 2 a and x 3 = x 3 ( x 2 + α) +a 2 the following equalities hold x a 2 2 ) m = x (a +a 2 )m v = P (x a x a 2 2 ) + x b x b 2 2 x 3 = P ( x a +a 2 ) + x b +b 2 x 3. If (a +a 2 )ordp (b +b 2 ) then s is a good point, A(Φ, E) = A(Φ, s) = 0 and A(Φ) > 0 since the locus of bad points Z(Φ) is not empty. So, assume that (a + a 2 )ordp < (b + b 2 ). Since b a ordp < 0 A(Φ, E) = A(Φ, s) = b + b 2 (a + a 2 )ordp < b 2 a 2 ordp = = A(Φ, E 2 ) A(Φ). Suppose that p N q (Φ) is a point and (N.IIb) holds at p, (u, v) are strongly prepared parameters at q and (x,..., x n ) are strongly permissible parameters at p for (u, v), then x a )m v = P (x a x a ) + xb x b. After possibly permuting x,..., x we can assume that b a < ordp < b 2 a 2 and b a < m. Suppose that E,..., E are the components of E X containing p with local equations x = 0,..., x = 0 respectively. Let U be an étale neighborhood of p where (x,..., x n ) are uniformizing parameters. Set C = E E 2 and fix a 2 point p U C. Then there exist α 3,..., α {0} and α +,..., α n such that (x, x 2, x 3 = x 3 α 3, x 4 = x 4 α 4,..., x n = x n α n ) are regular parameters at p. Since f = a b 2 a 2 b 0 we can set γ = (( x 3 + α 3 ) a3 ( x + α ) a ) f and ω = (( x 3 + α 3 ) b3 ( x + α ) b ) f. Then γ f, ω f O U, p are units in O U, p and, therefore, O U, p [γ, ω] is finite étale over O U, p. 28

Set x = γ b 2 ω a 2 x and x 2 = γ b ω a x 2 so that ( x,..., x n ) are strongly permissible parameters at p for strongly prepared parameters (u, v) at q and u = ( x a x a 2 2 ) m v = P ( x a x a 2 2 ) + x b x b 2 2. Thus (N.IIb) holds at a generic point p of C and Φ(p) = q. If π : X X is the blow up of C then Φ = Φ π is strongly prepared above p. If p lies in the intersection of more than 2 components E,..., E of E X then π (p) does not contain any point. Assume that p is a 2 point. If s π (p) is a point then (x, x 2, x 3,..., x n ) are strongly permissible parameters at s where x 2 is defined by x 2 = x ( x 2 + α) for some nonzero α. After setting x = x ( x 2 + α) a 2 f a +a 2 a and x 2 = ( x 2 + α) +a 2 α f a +a 2 the following equalities hold x a 2 2 ) m = x (a +a 2 )m v = P (x a x a 2 2 ) + x b x b 2 2 = P ( x a +a 2 ) + x b +b 2 x 2 + α f a +a 2 x b +b 2. If (a + a 2 )ordp (b + b 2 ) then s is a good point and A(Φ, E) = A(Φ, s) = 0 < A(Φ). So, assume that (a + a 2 )ordp < (b + b 2 ). Since b a ordp < 0 A(Φ, E) = A(Φ, s) = b + b 2 (a + a 2 )ordp < b 2 a 2 ordp = = A(Φ, E 2 ) A(Φ). Theorem 3.7. Suppose that Φ : X S is strongly prepared and Z(Φ) N(Φ). Then Z(Φ) N(Φ) is a Zarisi closed set of pure codimension 2, consisting of the union of all 2-varieties with a generic point in the form of (N.IIb). Suppose that C is a component of Z(Φ) N(Φ) and π : X X is the blowup of C with exceptional variety E = π (C) red. Then Φ = Φ π is strongly prepared and A(Φ, E) < A(Φ). Proof. This theorem follows from Lemma 3.6 due to finiteness of the number of 2-varieties in X. 29

Remar 3.8. With the notation of Lemma 3.6, Φ(Z(Φ) N q (Φ)) = {q} and each component of Z(Φ) N q (Φ) is the intersection of a good component E with a bad component E 2. Lemma 3.9. Suppose that p Z(Φ) N q (Φ) is a 2 point and (u, v) are strongly prepared parameters at q, (x,..., x n ) are strongly permissible parameters at p for (u, v) such that (N.IIb) holds and x a 2 2 ) m v = P (x a x a 2 2 ) + x b x b 2 2, where b a < ordp < b 2 a 2 and d = ordp. q is a -point. Suppose that (ū, v) are also strongly prepared parameters at q and (y,..., y n ) are strongly permissible parameters at p for (ū, v) such that (N.IIb) holds and ū = (y a y a 2 2 ) m v = Q(y a y a 2 2 ) + y b y b 2 2, where b < ordq < b a 2 and d = ordq. a 2 Then a = a, a 2 = a 2, b = b, b 2 = b 2, d = d, m = m. Proof. In order to decide whether q is a or 2 point we will compare the varieties given by local equations u = 0 and uv = 0 on X. According to the assumption on d, uv can be presented as uv = x a m+b x a 2m+a 2 d 2 (αx a d b + x b 2 a 2 d P (x a x a 2 2 ) α(x a 2 + x x 2 x a d+ x a 2d+ 2 where 0 α, x a d b, x b 2 a 2 d 2 m p Ô X, p, P (x a xa 2 2 ) (xa xa 2 2 )d x a d+ x a 2 d+ 2 x a 2 2 ) d ), ÔX, p. Thus uv = 0 defines a variety with at least 3 irreducible components at the 2 point p. Therefore uv = 0 cannot be a local equation of D S. So q is a point. This implies that every permissible change of coordinates at q will translate u into αu for some unit series α ÔX, p. Thus (x a x a 2 2 ) m = α(y a y a 2 2 ) m, where α is a unit series. The powers of irreducible factors on the left hand side a m and a 2 m are equal to the powers of irreducible factors on the right hand side a m and 30

a 2m, possibly in reverse order. And since (a, a 2 ) = and (a, a 2) = we can claim that m = m and {a, a 2 } = {a, a 2}. Denote by E and E 2 the components of E X containing p with local equations x = 0 and x 2 = 0 respectively. Then by Lemma 3.0 A(Φ, E ) = 0 and A(Φ, E 2 ) = b 2 a 2 ordp > 0. So E is a good component while E 2 is a bad component. Since ū = 0 is a local equation of E X, y = 0 and y 2 = 0 are local equations of E and E 2, possibly in reverse order. Then by Lemma 3.0 the invariant A of the component of E X with local equation y = 0 is equal to 0. So y = 0 is a local equation of the good component E, while y 2 = 0 is a local equation of E 2. From here and equality of the sets {a, a 2 } and {a, a 2} it follows that a = a and a 2 = a 2. Suppose that U is an étale neighborhood of p where (x,..., x n ) and (y,..., y n ) are uniformizing parameters. Fix a point p U E 2. Following the proof of Lemma 3.0 we can find strongly permissible parameters ( x,..., x n ) and strongly permissible parameters (ȳ,..., ȳ n ) at p such that and u = x a 2m 2 v = P ( x 2 ) + x b 2 2 x, with ord P = a 2 d ū = ȳ a 2 m 2 v = Q(ȳ 2 ) + ȳ b 2 2 ȳ, with ord Q = a 2d. So by Lemma 3.7 b 2 = b 2, a 2 d = a 2d, and therefore, d = d. To show that b = b we fix a point p U E. Then following the proof of Lemma 3.0 we find strongly permissible parameters ( x,..., x n ) and strongly permissible parameters (ȳ,..., ȳ n ) at p such that and So by Lemma 3.7 b = b. u = x a m v = α x b + x b x 2, with α ū = ȳ a m v = α 2 ȳ b + ȳ b ȳ, with α 2. 3

If α, β are real numbers, define S(α, β) = max{(α, β), (β, α)} where the maximum is in the Lexicographic ordering. Definition 3.20. Suppose that Φ : X S is strongly prepared and p E X is a 2 point such that (II) holds at p. Define { S( b a ordp, b 2 a 2 ordp ), if p N(Φ) Z(Φ); σ(p) = 0, otherwise. If C is a 2-variety in E X containing a 2 point p in the form of (II), set σ(c) = σ(p). Set σ(c) = 0, otherwise. Finally, define σ(φ) = max{σ(c) C E X is a 2-variety}. Remar 3.2. In view of Lemmas 3.6 and 3.9, at every 2 point p N(Φ) Z(Φ), where (II) holds, b a ordp and b 2 a 2 ordp are independent of the choice of strongly prepared parameters (u, v) at Φ(p) and they are also independent of the choice of strongly permissible parameters for (u, v) at p. So, σ(p) is well defined at every 2 point p E X in the form of (II). To justify the definition of σ(c) for a 2-variety C we will prove the following Lemma 3.22. Suppose that p E X is a 2 point in the form of (N.IIb) and C is a 2-variety containing p. Then there exists an open neighborhood U of p such that σ(p) = σ(p ) for all p U C. Proof. There exist strongly prepared parameters (u, v) at Φ(p) and strongly permissible parameters (x,..., x n ) at p such that x a 2 2 ) m ( ) v = P (x a x a 2 2 ) + x b x b 2 2 and b a < d = ordp < b 2 a 2, b a < m. Let U be an étale neighborhood of p where (x,..., x n ) are uniformizing parameters. Since x = x 2 = 0 are local equations of C, for any p U C there exist α 3,..., α n such that (x, x 2, x 3 = x 3 + α 3,..., x n = x n + α n ) are strongly permissible parameters at p for strongly prepared parameters (u, v) at Φ(p ). Then the same equations ( ) hold at p and, therefore, σ(p ) = S( b a ordp, b 2 a 2 ordp ) = σ(p). 32

Theorem 3.23. Suppose that Φ : X S is strongly prepared. Then there exists a sequence of blowups of 2-varieties X X such that the induced map Φ : X S is strongly prepared, A(Φ, E) < A(Φ ) = A(Φ) if E is an exceptional component of E X for X X and Z(Φ ) N(Φ ) =. Proof. Z(Φ) N(Φ) = if and only if σ(φ) = 0. Suppose that σ(φ) > 0 and C Z(Φ) N(Φ) is a 2-variety such that σ(c) = σ(φ). Let π : X X be the blowup of C. Then by Theorem 3.7 Φ = Φ π is strongly prepared and A(Φ, E) < A(Φ), so we will show that at every 2 point s π (C) in the form of (II) σ(s) < σ(φ). Suppose that p C is a point and (N.Ic) holds at p, (u, v) are strongly prepared parameters at Φ(p) and (x,..., x n ) are strongly permissible parameters at p for (u, v) such that x a )m v = P (x a x a ) + xb x b x + and x = x 2 = 0 are local equations of C. Then there will not be any 2 point in the form of (II) in π (p). Suppose that p C is a point and (N.IIb) holds at p, (u, v) are strongly prepared parameters at Φ(p) and (x,..., x n ) are strongly permissible parameters at p for (u, v) such that x a )m v = P (x a x a ) + xb x b and x = x 2 = 0 are local equations of C. Then after possibly permuting x and x 2 we can assume that b a < ordp < b 2 a 2 and b a < m. Suppose that E,..., E are the components of E X containing p with local equations x = 0,..., x = 0 respectively. If p lies in the intersection of more than 3 components E,..., E of E X there will not be any 2 point in π (p). Assume first that p is a 2 point with x a 2 2 ) m v = P (x a x a 2 2 ) + x b x b 2 2 and d = ordp. Then σ(φ) = σ(c) = σ(p) = S(a d b, b 2 a 2 d). 33

Suppose that a 2 point s π (p) has -permissible parameters (x, x 2, x 3,..., x n ) such that x 2 = x x 2, then +a 2 x a 2 2 ) m v = P (x a +a 2 x a 2 2 ) + x b +b 2 x b 2 2. Since d < b 2 a 2, following the proof of Lemma 3.4 we can find strongly permissible parameters (y,..., y n ) at s such that u = (y a +a 2 y a 2 2 ) m v = P (y a +a 2 y a 2 2 ) + y b +b 2 y b 2 2. Then σ(s) > 0 if and only if (u, v)o X, s is not invertible and (N.IIb) holds at p. Let it be the case, then d > b +b 2 a +a 2 and (a + a 2 )d (b + b 2 ) < a d b since a 2 d b 2 < 0. Thus σ(s) = S((a + a 2 )d (b + b 2 ), b 2 a 2 d) < S(a d b, b 2 a 2 d) = σ(φ). Suppose that a 2 point s π (p) has -permissible parameters ( x, x 2, x 3,..., x n ) such that x = x x 2 then If d b +b 2 a +a 2 that u = ( x a x a +a 2 2 ) m v = P ( x a x a +a 2 2 ) + x b x b +b 2 2. there exist strongly permissible parameters (y,..., y n ) at s such u = (y a y a +a 2 2 ) m v = y b y b +b 2 2. Thus σ(s) = 0 in this case. Assume that d < b +b 2 a +a 2, then following the proof of Lemma 3.4 we can find strongly permissible parameters (y,..., y n ) at s such that u = (y a y a +a 2 2 ) m v = P (y a y a +a 2 2 ) + y b y b +b 2 2. Suppose that σ(s) > 0, that is (N.IIb) holds at s. Then since b a d < 0, (b + b 2 ) (a + a 2 )d < b 2 a 2 d and σ(s) = S(a d b, (b + b 2 ) (a + a 2 )d) < S(a d b, b 2 a 2 d) = σ(φ). 34

Assume now that p is a 3 point with x a 2 2 x a 3 3 ) m v = P (x a x a 2 2 x a 3 3 ) + x b x b 2 2 x b 3 3 and d = ordp. Suppose that s π (p) is a 2 point, then s has regular parameters (x, x 2, x 3,..., x n ) defined by x 2 = x ( x 2 + α) for some nonzero α and +a 2 ( x 2 + α) a 2 x a 3 3 ) m v = P (x a +a 2 ( x 2 + α) a 2 x a 3 3 ) + x b +b 2 ( x 2 + α) b 2 x b 3 3. ( ) a + a If ran 2 a 3 < 2 then σ(s) = 0 since (II) cannot hold at s. So, b + b 2 b 3 ( ) a + a consider the case when ran 2 a 3 = 2. b + b 2 b 3 Set h = (a + a 2 )b 3 a 3 (b + b 2 ) and x = x ( x 2 + α) a 2 b 3 a 3 b 2 h x 3 = x 3 ( x 2 + α) a b 2 a 2 b h to get -permissible parameters ( x, x 2, x 3, x 4,..., x n ) at s with If d max{ b +b 2 at s such that a +a 2, b 3 u = ( x a +a 2 x a 3 3 ) m v = P ( x a +a 2 x a 3 3 ) + x b +b 2 x b 3 3. a 3 } there exist strongly permissible parameters (y,..., y n ) u = (y a +a 2 y a 3 3 ) m v = y b +b 2 y b 3 3. Thus σ(s) = 0 in this case. Assume that d < max{ b +b 2 a +a 2, b 3 a 3 }, then following the proof of Lemma 3.4 we can find strongly permissible parameters (y,..., y n ) at s such that u = (y a +a 2 y a 3 3 ) m v = P (y a +a 2 y a 3 3 ) + y b +b 2 y b 3 3. 35

Suppose that σ(s) > 0, so (N.IIb) holds at s and min{ b +b 2 a +a 2, b 3 a 3 } < d < max{ b +b 2 a +a 2, b 3 a 3 }. If b +b 2 a +a 2 < d < b 3 a 3 then according to the proof of Lemma 3.6 the 2-variety E E 3 lies in Z(Φ) N(Φ) and σ(e E 3 ) = S(a d b, b 3 a 3 d). Thus since a 2 d b 2 < 0, (a + a 2 )d (b + b 2 ) < a d b and σ(s) = S((a + a 2 )d (b + b 2 ), b 3 a 3 d) < S(a d b, b 3 a 3 d) = = σ(e E 3 ) σ(φ). If b 3 a 3 < d < b +b 2 a +a 2 then notice that σ(s) > 0 implies that b 3 a 3 < m. So according to the proof of Lemma 3.6 the 2-variety E 2 E 3 lies in Z(Φ) N(Φ) and σ(e 2 E 3 ) = S(a 3 d b 3, b 2 a 2 d). Thus since b a d < 0, (b + b 2 ) (a + a 2 )d < b 2 a 2 d and σ(s) = S(a 3 d b 3, (b + b 2 ) (a + a 2 )d) < S(a 3 d b 3, b 2 a 2 d) = = σ(e 2 E 3 ) σ(φ). By Theorem 3.7, induction on the number of 2-varieties C X with σ(c) = σ(φ) and induction on σ(φ) we achieve the conclusions of the theorem. Theorem 3.24. Suppose that Φ : X S is strongly prepared and q S. Suppose also that N(Φ) does not contain any bad point. If N q (Φ) then N q (Φ) is a pure codimension 2 subscheme which maes SNCs with B 2 (X). Suppose that C is a component of N q (Φ) and π : X X is the blowup of C, E = π (C) red and Φ = Φ π. Then Φ is strongly prepared, Z(Φ ) N(Φ ) = and A(Φ, E) = 0. Proof. Suppose that p N q (Φ) is a point, (u, v) are strongly prepared parameters at q and (x,..., x n ) are strongly permissible parameters for (u, v) at p. Let U be an étale neighborhood of p where (x,..., x n ) are uniformizing parameters. Denote by E,..., E the components of E X containing p with local equations x = 0,..., x = 0, respectively. The assumption that N(Φ) does not contain bad points implies that there is no point of the form (N.Ic) or (N.IIb) in N q (Φ). Suppose that (N.Ia) holds at p : x a )m v = (x a x a )t x +. 36