Toroidalization of Locally Toroidal Morphisms from N-folds to Surfaces

Toroidliztion of Loclly Toroidl Morphisms from N-folds to Surfces Krishn Hnumnthu 1. Introduction Fix n lgebriclly closed field k of chrcteristic 0. A vriety is n open subset of n irreducible proper k-scheme. A simple norml crossing (SNC) divisor on nonsingulr vriety is divisor D on X, ll of whose irreducible components re nonsingulr nd whenever r irreducible components Z 1,..., Z r of D meet t point p, then locl equtions x 1,..., x r of Z i form prt of regulr system of prmeters in O X,p. If D is SNC divisor nd point p D belongs to exctly k components of D, then we sy tht p is point. A toroidl structure on nonsingulr vriety X is SNC divisor D X. The divisor D X specifies toric chrt (V p, σ p ) t every closed point p X where p V p X is n open neighborhood nd σ p : V p X p is n étle morphism to toric vriety X p such tht under σ p the idel of D X t p corresponds to the idel of the complement of the torus in X p. The ide of toroidl structure is fundmentl to lgebric geometry. It is developed in the clssic book Toroidl Embeddings I [10] by G. Kempf, F. Knudsen, D. Mumford nd B. Sint-Dont. Definition 1.1. ([10], [1]) Suppose tht D X nd D Y re toroidl structures on X nd Y respectively. Let p X be closed point. A dominnt morphism f : X Y is toroidl t p (with respect to the toroidl structures D X nd D Y ) if the germ of f t p is formlly isomorphic to toric morphism between the toric chrts t p nd f(p). f is toroidl if it is toroidl t ll closed points in X. 1

A nonsingulr subvriety V of X is possible center for D X if V D X nd V intersects D X trnsverslly. Tht is, V mkes SNCs with D X, s defined before Lemm 2.3. The blowup π : X 1 X of possible center is clled possible blowup. D X1 = π 1 (D X ) is then toroidl structure on X 1. Let Sing(f) be the set of points p in X where f is not smooth. It is closed set. The following toroidliztion conjecture is the strongest possible generl structure theorem for morphisms of vrieties. Conjecture 1.2. Suppose tht f : X Y is dominnt morphism of nonsingulr vrieties. Suppose lso tht there is SNC divisor D Y on Y such tht D X = f 1 (D Y ) is SNC divisor on X which contins the singulr locus, Sing(f), of the mp f. Then there exists commuttive digrm of morphisms f 1 X 1 Y 1 X π 1 f Y π where π, π 1 re possible blowups for the preimges of D Y nd D X respectively, such tht f 1 is toroidl with respect to D Y1 = π 1 (D Y ) nd D X1 = π 1 1 (D X ) A slightly weker version of the conjecture is stted in the pper [2] of D. Abrmovich, K. Kru, K. Mtsuki, nd J. Wlodrczyk. When Y is curve, this conjecture follows esily from embedded resolution of hypersurfce singulrities, s shown in the introduction of [5]. The cse when X nd Y re surfces hs been known before (see Corollry 6.2.3 [2], [3], [7]). The cse when X hs dimension 3 is completely resolved by Dle Cutkosky in [5] nd [6]. A specil cse of dim(x) rbitrry nd dim(y ) = 2 is done in [8]. For detiled history nd pplictions of this conjecture, see [6]. A relted, but weker question sked by Dle Cutkosky is the following Question 1.4. To stte the question we need the following definition. Definition 1.3. Let f : X Y be dominnt morphism of nonsingulr vrieties. Suppose tht the following re true. 2

1. There exist open coverings {U 1,..., U m } nd {V 1,..., V m } of X nd Y respectively such tht the morphism f restricted to U i mps into V i for ll i = 1,..., m. 2. There exist simple norml crossings divisors D i nd E i in U i nd V i respectively such tht f 1 (E i ) U i = D i nd Sing(f Ui ) D i for ll i = 1,..., m. 3. The restriction of f to U i, f Ui : U i V i, is toroidl with respect to D i nd E i for ll i = 1,..., m. Then we sy tht f is loclly toroidl with respect to the open coverings U i nd V i nd SNC divisors D i nd E i. For the reminder when we sy f is loclly toroidl, it is to be understood tht f is loclly toroidl with respect to the open coverings U i nd V i nd SNC divisors D i nd E i s in the definition. We will usully not mention U i, V i, D i nd E i. We hve the following. Question 1.4. Suppose tht f : X Y is loclly toroidl. Does there exist commuttive digrm of morphisms f 1 X 1 Y 1 X π 1 f Y π where π, π 1 re blowups of nonsingulr vrieties such tht there exist SNC divisors E, D on Y 1 nd X 1 respectively such tht Sing(f 1 ) D, f 1 1 (E) = D nd f 1 is toroidl with respect to E nd D? The im of this pper is to give positive nswer to this question when Y is surfce nd X is rbitrry. The result is proved in Theorem 4.2. Brief outline of the proof: The core results (Theorems 4.1 nd 4.2) re proved in section 4. Sections 2 nd 3 consist of preprtory mteril. Let f : X Y be loclly toroidl morphism with the nottion s in definition 1.3. The essentil observtion is this: if there is SNC divisor E 3

on Y such tht E i E for ll i, then f is toroidl with respect to E nd f 1 (E). A proof of this observtion is contined in the proof of Theorem 4.2. The min tsk, then, is to construct the divisor E. This is not hrd: consider the divisor E = Ē1 +... + E m where Ēi is the Zriski closure of E i in Y. By embedded resolution of singulrities, there exists finite sequence of blowups of points π : Y 1 Y such tht π 1 (E ) is SNC divisor on Y 1. The problem now reduces to constructing sequence of blowups π 1 : X 1 X such tht there is loclly toroidl morphism f 1 : X 1 Y 1. This is done in Theorem 4.1. Sections 2 nd 3 prepre the ground for Theorem 4.1. Given the sequence of blowups π : Y 1 Y s bove, there exist principliztion lgorithms which give sequence of blowups π 1 : X 1 X so tht there exists morphism f 1 : X 1 Y 1. The min difficulty we fce is tht such morphism f 1 my not be loclly toroidl. So blnket ppel to existing principlizing lgorithms cn not be mde. In sections 2 nd 3, we construct specific lgorithm tht works in our sitution. Section 2 dels with the blowups tht preserve the locl toroidl structure. We cll these permissible blowups (definition 2.4). The min result of section 2 is Lemm 2.5 which nlyzes the effect of permissible sequence of blowups. In section 3, we define invrints on nonprincipl locus of the morphism f. These invrints re positive integers nd we prescribe permissible sequences of blowups under which these invrints drop (Theorems 3.3 nd 3.4). Finlly we chieve principliztion in Theorem 3.6. 2. Permissible Blowups Let f : X Y be loclly toroidl morphism from nonsingulr n-fold X to nonsingulr surfce Y with respect to open coverings {U 1,..., U m } nd {V 1,..., V m } of X nd Y respectively nd SNC divisors D i nd E i in U i nd V i respectively. Then we hve the following Lemm 2.1. Let p D i. Then there exist regulr prmeters x 1,..., x n in Ô X,p nd regulr prmeters u, v in O Y,q such tht one of the following forms holds: 1 k n 1 : u = 0 is locl eqution of E i, x 1 = 0 is locl eqution of D i nd u = x 1, v = x k+1, (1) 4

where,..., > 0. 1 k n 1 : uv = 0 is locl eqution for E i, x 1 = 0 is locl eqution of D i nd u = (x 1 ) m, v = (x 1 ) t (α + x k+1 ), (2) where,...,, m, t > 0 nd α K {0}. 2 k n : uv = 0 is locl eqution of E i, x 1 = 0 is locl eqution of D i nd u = x 1, v = x 1 b k, (3) where[,...,,,..., ] b k 0, i + b i > 0 for ll i nd 1.. rnk k = 2... b k Proof. This follows from Lemm 4.2 in [8]. Definition 2.2. Suppose tht D is SNC divisor on vriety X, nd V is nonsingulr subvriety of X. We sy tht V mkes SNCs with D t p X if there exist regulr prmeters x 1,..., x n in O X,p nd e, r n such tht x 1...x e = 0 is locl eqution of D t p nd x σ(1) =... = x σ(r) = 0 is locl eqution of V t p for some injection σ : {1,..., r} {1,..., n}. We sy tht V mkes SNCs with D if V mkes SNCs with D t ll points p X. Let q Y nd let m q be the mximl idel of O Y,q. Define W q = {p X m q O X,p is not principl}. Note tht the closed subset W q f 1 (q) nd tht m q O X,p is principl if nd only if m q Ô X,p is principl. Lemm 2.3. For ll q Y, W q is union of nonsingulr codimension 2 subvrieties of X, which mke SNCs with ech divisor D i on U i. Proof. Let us fix q Y nd denote W = W q. Let I W be the reduced idel shef of W in X, nd let I q be the reduced idel shef of q in Y. Since the conditions tht W is nonsingulr nd hs codimension 2 in X re both locl properties, we need only check tht for ll p W, I W,p is n intersection of height 2 prime idels which re regulr. Since X is nonsingulr, I q O X = O X ( F )I where F is n effective Crtier divisor on X nd I is n idel shef such tht the support of O X /I hs 5

codimension t lest 2 on X. We hve W = supp(o X /I). The idel shef of W is I W = I. Let p W. We hve tht p U i for some 1 i m. Suppose first tht q / E i. Then f is smooth t p becuse it is loclly toroidl. This mens tht there re regulr prmeters u, v t q which form prt of regulr sequence t p. So we hve regulr prmeters x 1,..., x n in O X,p such tht u = x 1, v = x 2. I q O X,p = (u, v)o X,p = (x 1, x 2 )O X,p. It follows tht I W,p = (x 1, x 2 )O X,p. This gives us the lemm. Suppose now tht q E i. Since p W q, there exist regulr prmeters x 1,..., x n in ÔX,p nd u, v in O Y,q such tht one of the forms (1) or (3) holds. Suppose tht (1) holds. Since D j is SNC divisor, there exist regulr prmeters y 1,..., y n in O X,p nd some e such tht y 1...y e = 0 is locl eqution of D j. Since x 1 = 0 is locl eqution for D j in ÔX,p, there exists unit series δ ÔX,p such tht y 1...y e = δx 1. Since the x i nd y i re irreducible in ÔX,p, it follows tht e = k, nd there exist unit series δ i ÔX,p such tht x i = δ i y i for 1 i k, fter possibly reindexing the y i. Note tht y 1,..., y k, x k+1, y k+2,..., y n is regulr system of prmeters in Ô X,p, fter possibly permuting y k+1,..., y n. So the idel (y 1,..., y k, x k+1, y k+2,..., y n )ÔX,p is the mximl idel of ÔX,p. Since x k+1 = v O X,p, y 1,..., y k, x k+1, y k+2,..., y n generte n idel J in O X,p. Since ÔX,p is fithfully flt over O X,p, nd JÔX,p is mximl, it follows tht J is the mximl idel of O X,p. Hence y 1,..., y k, x k+1, y k+2,..., y n is regulr system of prmeters in O X,p. Rewriting (1), we hve u = y 1...y k k δ, where δ is unit in Ô X,p. Since δ u = y 1...yk, δ QF(O X,p ) ÔX,p, where QF(O X,p ) is the quotient field of O X,p. By Lemm 2.1 in [4], it follows tht δ O X,p. Since δ is unit in ÔX,p, it is unit in O X,p. We hve s required. I W,p = I q O X,p = (u, v)o X,p = (y 1...yk, xk+1 ) = (y 1, x k+1 ) (y 2, x k+1 )... (y k, x k+1 ), 6

We rgue similrly when (3) holds t p. Let Z be nonsingulr codimension 2 subvriety of X such tht Z W q for some q. Let π 1 : X 1 X be the blowup of Z. Denote by (W 1 ) q the set {p X 1 m q Ô X1,p is not invertible}. Given ny sequence of blowups X n X n 1... X 1 X, we define (W i ) q for ech X i s bove. Definition 2.4. Let q Y. A sequence of blowups X k X k 1... X 1 X is clled permissible sequence with respect to q if for ll i, ech blowup X i+1 X i is centered t nonsingulr codimension 2 subvriety Z of X i such tht Z (W i ) q. We will often write simply permissible sequence without mentioning q if there is no scope for confusion. Lemm 2.5. Let f : X Y be loclly toroidl morphism. Let π 1 : X 1 X be permissible sequence with respect to q Y. I Suppose tht 1 i m nd p (f π 1 ) 1 (q) π 1 1 (U i ) nd q E i. Then I.A nd I.B s below hold. I.A. There exist regulr prmeters x 1,..., x n such tht one of the following forms holds: in ÔX 1,p nd (u, v) in O Y,q 1 k n 1: u = 0 is locl eqution of E i, x 1 = 0 is locl eqution of π 1 1 (D i ) nd where b i i. u = x 1, v = x 1 b k x k+1, (4) 1 k n 1: u = 0 is locl eqution of E i, x 1 x k+1 = 0 is locl eqution of π 1 1 (D i ) nd u = x 1 x k+1 +1, v = x 1 b k x k+1 b k+1, (5) where b i i for i = 1,..., k nd b k+1 < +1. 7

1 k n 1: u = 0 is locl eqution of E i, x 1 = 0 is locl eqution of π 1 1 (D i ) nd u = x 1, v = x 1 b k (x k+1 + α), (6) where b i i for ll i nd 0 α K. 1 k n 1: uv = 0 is locl eqution for E i, x 1 = 0 is locl eqution of π 1 1 (D i ) nd u = (x 1 ) m, v = (x 1 ) t (α + x k+1 ), (7) where,...,, m, t > 0 nd α K {0}. 2 k n: uv = 0 is locl eqution of E i, x 1 = 0 is locl eqution of π 1 1 (D i ) nd u = x 1 1 b k, v = x 1 b 1 k, (8) [ ] 1.. where,...,,,..., b k 0, i +b i > 0 for ll i nd rnk k =.. b k 2. I.B. Suppose tht p 1 (W 1 ) q. There exist regulr prmeters x 1,..., x n Ô X1,p nd (u, v) in O Y,q such tht one of the following forms holds: 1 k n 1: u = 0 is locl eqution of E i, x 1 = 0 is locl eqution of π 1 1 (D i ) nd u = x 1, v = x 1 b k x k+1, (9) where b i i nd b i < i for some i. Moreover, the locl equtions of (W 1 ) q re x i = x k+1 = 0 where b i < i. 2 k n: uv = 0 is locl eqution of E i, x 1 = 0 is locl eqution of π 1 1 (D i ) nd u = x 1, v = x 1 b k, (10) where,...,,,..., b k 0, [ i + b i > 0 for ] ll i, u does not divide v, 1.. v does not divide u, nd rnk k = 2. Moreover, the locl.. b k equtions of (W 1 ) q re x i = x j = 0 where ( i b i )(b j j ) > 0. 8 in

II Suppose tht 1 i m nd p (f π 1 ) 1 (q) π 1 1 (U i ) nd q / E i. Then II.A nd II.B s below hold. II.A There exist regulr prmeters x 1,..., x n such tht one of the following forms holds: in ÔX 1,p nd (u, v) in O Y,q u = x 1, v = x 2 (11) u = x 1, v = x 1 (x 2 + α) for some α K. (12) u = x 1 x 2, v = x 2. (13) II.B Suppose tht p 1 (W 1 ) q. There exist regulr prmeters x 1,..., x n Ô X1,p nd (u, v) in O Y,q such tht the following form holds: in The locl equtions of (W 1 ) q re x 1 = x 2 = 0. u = x 1, v = x 2. (14) III (W 1 ) q is union of nonsingulr codimension 2 subvrieties of X 1. Proof. I We prove this prt by induction on the number of blowups in the sequence π 1 : X 1 X. In X the conclusions hold becuse of Lemm 2.3 nd f is loclly toroidl. Suppose tht the conclusions of the lemm hold fter ny sequence of l permissible blowups where l 0. Let π 1 : X 1 X be permissible sequence (with respect to q) of l blowups. Let π 2 : X 2 X 1 be the blowup of nonsingulr codimension 2 subvriety Z of X 1 such tht Z (W 1 ) q. Let p π 2 1 (π 1 1 (U i )) (f π 1 π 2 ) 1 (q) for some 1 i m. If p 1 = π 2 (p) / Z then π 2 is n isomorphism t p nd we hve nothing to prove. Suppose then tht p 1 π 1 1 (U i ) Z π 1 1 (U i ) (W 1 ) q. Then by induction hypothesis (I.B) p 1 hs the form (9) or (10). Suppose first tht it hs the form (9). 9

Then the locl equtions of Z t p 1 re x i = x k+1 = 0 for some 1 i k. Note tht b i < i. As in the proof of Lemm 2.3, there exist regulr prmeters y 1,..., y k, x k+1, y k+2,..., y n in O X1,p 1 nd unit series δ i ÔX 1,p 1 such tht y i = δ i x i for 1 i k. Then O X2,p hs one of the following two forms: () O X2,p = O X1,p 1 [ x k+1 y i ] (yi, x k+1 α) y i for some α K, or (b) O X2,p = O X1,p 1 [ y i x k+1 ] y (xk+1, i ) x k+1 In cse(), set ȳ k+1 = x k+1 y i α. Then y 1,..., y k, ȳ k+1, y k+2,..., y n re regulr prmeters in O X2,p nd so ÔX 2,p = k[[y 1,..., y k, ȳ k+1, y k+2,..., y n ]]. Let c 0 be the constnt term of the unit series δ i. Then evluting δ i in the locl ring O X2,p we get, δ i (y 1,..., y k, x k+1, y k+2,..., y n ) = δ i (y 1,..., y k, y i (ȳ k+1 + α), y k+1,..., y n ) = c + 1 y 1 +... + k y k + k+2 y k+2 +... + n y n for some i O X2,p. Set ᾱ = cα. Note tht x k+1 x x i ᾱ = δ k+1 i y k cα = δ i (ȳ k+1 + α) cα = δ i ȳ k+1 + (δ i c)α. Since y 1,..., y k, ȳ k+1, y k+2,..., y n re regulr prmeters in ÔX 2,p the bove clcultions imply tht x 1,..., x k, x k+1 x i ᾱ, y k+2,..., y n re regulr prmeters in ÔX 2,p. Set x k+1 = x k+1 x k ᾱ. We get u = x 1 k, v = b x1 1 b... x i +1 b i k( xk+1 + α). This is the form (6) if α 0 nd form (4) if α = 0. In cse (b), set ȳ k+1 = y i x k+1. Then y 1,..., y k, ȳ k+1, y k+2,..., y n re regulr prmeters in O X2,p nd so ÔX 2,p = k[[y 1,..., y k, ȳ k+1, y k+2,..., y n ]]. x Then x 1,..., x k, i x k+1, y k+2,..., y n re regulr prmeters in ÔX 2,p. Set x i = x i x k+1. u = x 1... x i i xk+1 i b k, v = x 1... x i +1 b i xk+1 k. This is the form (5). By the bove nlysis, when p 1 = π 2 (p) hs form (9), if p (W 2 ) q, then it lso hs to be of the form (9). 10

Suppose now tht p 1 hs the form (10). Then the locl equtions of Z t p 1 re x i = x j = 0 for some 1 i, j k. Then s in the bove nlysis there exist regulr prmeters y 1,...,..., y n in O X1,p 1 nd unit series δ i ÔX 1,p 1 such tht y i = δ i x i for 1 i k. Then O X2,p hs one of the following two forms: () O X2,p = O X1,p 1 [ y i y j ] for some α K, or (yj, y i α) y j (b) O X2,p = O X1,p 1 [ y j y i ] (yi, y j y j ) Arguing s bove in cse () we obtin regulr prmeters x 1,..., x i,..., x n in ÔX 2,p so tht u = x 1...( x i + α) i...x j i + j, v = x 1...( x i + α) b i...x j b i +b j b k. This is the form (8) if α = 0. If α 0, we obtin either the form (8) or the form (7) ccording s rnk of [ ] 1.. i + j.. j 1 j+1.. is = 2 or < 2... b i + b j.. b j 1 b j+1.. b k Agin rguing s bove in cse (b) we obtin regulr prmeters x 1,..., x j,..., x n in ÔX 2,p so tht u = x 1...x i i + j... x j j, v = x 1...x i b i +b j... x j b j b k. This is the form (8). By the bove nlysis, when p 1 = π 2 (p) hs the form (10), if p (W 2 ) q, then it lso hs to be of the form (10). This completes the proof of I.A for X 2. Now I.B is cler s the forms (9) nd (10) re just the forms (4) nd (8) from I.A. II We prove this prt by induction on the number of blowups in the sequence π 1 : X 1 X. Since q / E i nd f is loclly toroidl, f is smooth t ny point p 1 f 1 (q). This mens tht the regulr prmeters t q form prt of regulr sequence t p. So we hve regulr prmeters x 1,..., x n in ÔX,p 1 nd u, v in O Y,q such tht u = x 1, v = x 2. This is the form (11). Thus the conclusions hold in X. Suppose tht the conclusions of the lemm hold fter ny sequence of l permissible blowups where l 0. 11

Let π 1 : X 1 X be permissible sequence (with respect to q) of l blowups. Let π 2 : X 2 X 1 be the blowup of nonsingulr codimension 2 subvriety Z of X 1 such tht Z (W 1 ) q. Let p π 2 1 (π 1 1 (U i )) (f π 1 π 2 ) 1 (q) for some 1 i m. If p 1 = π 2 (p) / Z then π 2 is n isomorphism t p nd we hve nothing to prove. Suppose then tht p 1 π 1 1 (U i ) Z π 1 1 (U i ) (W 1 ) q. Then by induction hypothesis (II.B) p 1 hs the form (14). Then the locl equtions of Z t p 1 re x 1 = x 2 = 0. There exist regulr prmeters x 1, x 2 in ÔX 2,p such tht one of the following forms holds: x 1 = x 1, x 2 = x 1 ( x 2 + α) for some α K or x 1 = x 1 x 2, x 2 = x 2. These two cses give the forms (12) nd (13). Now II.B is cler s the form (14) is just the form (11) from II.A. III Since {π 1 1 (U i )} for 1 i m is n open cover of X 1 nd π 1 1 (U i ) (W 1 ) q is union of nonsingulr codimension 2 subvrieties of X 1 for ll i by I nd II, (W 1 ) q is union of nonsingulr codimension 2 subvrieties of X 1. 3. Principliztion Let f : X Y be loclly toroidl morphism from nonsingulr n-fold X to nonsingulr surfce Y with respect to open coverings {U 1,..., U m } nd {V 1,..., V m } of X nd Y respectively nd SNC divisors D i nd E i in U i nd V i respectively. In this section we fix n i between 1 nd m nd q Y. Let π 1 : X 1 X be permissible sequence with respect to q. Our im is to construct permissible sequence π 2 : X 2 X 1 such tht π 2 π 1 : X 2 X is permissible sequence nd π 2 1 (π 1 1 (U i )) (W 2 ) q is empty. First suppose tht q / E i. If p π 1 1 (U i ), then by Lemm 2.5 one of the forms (11), (12) or (13) holds t p. Theorem 3.1. Let π 1 : X 1 X be permissible sequence with respect to q Y. Let i be such tht q / E i. Then there exists permissible sequence π 2 : X 2 X 1 with respect to q such tht π 2 1 (π 1 1 (U i )) (W 2 ) q is empty. Proof. If π 1 1 (U i ) (W 2 ) q is empty, then there is nothing to prove. So suppose tht π 1 1 (U i ) (W 2 ) q. By Lemm 2.3, it is union of codimension 2 subvrieties of π 1 1 (U i ). 12

Let Z π 1 1 (U i ) (W 1 ) q be subvriety of π 1 1 (U i ) of codimension 2. Let π 2 : X 2 X 1 be the blowup of the Zriski closure Z of Z in X 1. Let Z 1 π 2 1 (Z) be codimension 2 subvriety of π 2 1 (π 1 1 (U i )) such tht Z 1 π 2 1 (π 1 1 (U i )) (W 2 ) q. By the proof of Lemm 2.5 it follows tht Z 1 (W 2 ) q =. The theorem now follows by induction on the number of codimension 2 subvrieties Z in π 1 1 (U i ) (W 1 ) q. Now we suppose tht q E i. Remrk 3.2. Suppose tht π 1 : X 1 X is permissible sequence with respect to some q E i. Let π 2 : X 2 X 1 be permissible blowup with respect to q. Let p 1 π 2 1 (π 1 1 (U i )) (W 2 ) q. Then clerly p = π 2 (p 1 ) π 1 1 (U i ) (W 1 ) q. Suppose tht p 1 is point. Then the nlysis in the proof of Lemm 2.5 shows tht p lso is point. Suppose tht p 1 is 2 point where the form (10) holds. Then the nlysis in the proof of Lemm 2.5 shows tht p is 2 or 3 point where the from (10) holds. Suppose tht π 1 : X 1 X is permissible sequence with respect to q E i. Let p π 1 1 (U i ) (W 1 ) q be point. By Lemm 2.5, there exist regulr prmeters x 1,..., x n in ÔX 1,p nd u, v in O Y,q such tht u = x 1, v = x 1 b x 2 where > b. Define Ω i (p) = b > 0. Let Z π 1 1 (U i ) (W 1 ) q be codimension 2 subvriety of π 1 1 (U i ). Define Ω i (Z) = Ω i (p) if there exists point p Z. This is well defined becuse Ω i (p) = Ω i (p ) for ny two points p, p Z. If Z contins no 1 points, we define Ω i (Z) = 0. Finlly define Ω i (f π 1 ) = mx{ω i (Z) Z π 1 1 (U i ) (W 1 ) q is n irreducible subvriety of π 1 1 (U i ) of codimension 2} Theorem 3.3. Let π 1 : X 1 X be permissible sequence with respect to q E i. There exists permissible sequence π 2 : X 2 X 1 with respect to q such tht Ω i (f π 1 π 2 ) = 0. 13

Proof. Suppose tht Ω i (f π 1 ) > 0. Let Z π 1 1 (U i ) (W 1 ) q be subvriety of π 1 1 (U i ) of codimension 2 such tht Ω i (f π 1 ) = Ω i (Z). Let π 2 : X 2 X 1 be the blowup of the Zriski closure Z of Z in X 1. Let Z 1 π 2 1 (Z) be codimension 2 subvriety of π 2 1 (π 1 1 (U i )) such tht Z 1 π 2 1 (π 1 1 (U i )) (W 2 ) q. We clim tht Ω i (Z 1 ) < Ω i (Z). If there re no 1 points of Z 1 then we hve nothing to prove. Otherwise, let p 1 Z 1 be point. Then π 1 (p 1 ) = p is point of Z by Remrk 3.2. There re regulr prmeters x 1,..., x n in ÔX 1,p nd u, v in O Y,q such tht u = x 1, v = x 1 b x 2. There exist regulr prmeters x 1, x 2,..., x n in ÔX 2,p 1 such tht x 2 = x 1 (x 2 + α). u = x 1, v = x 1 b+1 (x 2 + α). Since p 1 (W 2 ) q, α = 0. Ω i (Z 1 ) = Ω i (p 1 ) = b 1 < b = Ω i (Z). The theorem now follows by induction on the number of codimension 2 subvrieties Z in π 1 1 (U i ) (W 1 ) q such tht Ω i (f π 1 ) = Ω i (Z) nd induction on Ω i (f π 1 ). Let π 1 : X 1 X be permissible sequence with respect to q E i. Let Z π 1 1 (U i )) (W 1 ) q be codimension 2 subvriety of π 1 1 (U i ). Let p Z be 2 point where the form (10) holds. There exist regulr prmeters x 1,..., x n in ÔX 1,p nd u, v in O Y,q such tht u = x 1 x 2 2 nd v = x 1 x 2 b 2. Define ω i (p) = ( )(b 2 2 ). Then since p (W 1 )q, ω i (p) > 0. Now define ω i (Z) = ω i (p) if p Z is 2 point where the form (10) holds. If there re no 2 points of the form (10) in Z define ω i (Z) = 0. Then ω i (Z) is well-defined. Finlly define ω i (f π 1 ) = mx{ω i (Z) Z π 1 1 (U i ) (W 1 ) q is n irreducible subvriety of π 1 1 (U i ) of codimension 2} Theorem 3.4. Let π 1 : X 1 X be permissible sequence with respect to q E i. Suppose tht Ω i (f π 1 ) = 0. There exists permissible sequence π 2 : X 2 X 1 with respect to q such tht Ω i (f π 1 π 2 ) = 0 nd ω i (f π 1 π 2 ) = 0. Proof. Since Ω i (f π 1 ) = 0, there re no 1 points in π 1 1 (U i ) (W 1 ) q. Let X 2 X 1 be ny permissible blowup. Then by Remrk 3.2 it follows tht π 2 1 (π 1 1 (U i )) (W 2 ) q hs no 1 points. Hence Ω i (f π 1 π 2 ) = 0. Suppose tht ω i (f π 1 ) > 0. Let Z π 1 1 (U i ) (W 1 ) q be codimension 2 irreducible subvriety of π 1 1 (U i ) such tht ω i (f π 1 ) = ω i (Z). 14

Let π 2 : X 2 X 1 be the blowup of the Zriski closure Z of Z in X 1. Let Z 1 π 1 2 (Z) be codimension 2 subvriety of π 1 2 (π 1 1 (U i )) such tht Z 1 π 1 2 (π 1 1 (U i )) (W 2 ) q. We prove tht ω i (Z 1 ) < ω i (Z) = ω i (f π 1 ). If there re no 2 points of the form (10) in Z 1 then ω i (Z 1 ) = 0 nd we hve nothing to prove. Otherwise let p 1 Z 1 be 2 point of the form (10). By Remrk 3.2, p = π 2 (p 1 ) Z is 2 or 3 point of form (10). Suppose tht p Z is 2 point. There exist regulr prmeters x 1,..., x n in ÔX 1,p nd u, v in O Y,q such tht u = x 1 x 2 b 2 nd v = x 1 x 2 2. Also the locl equtions of Z re x 1 = x 2 = 0. Then there exist regulr prmeters x 1, x 2, x 3..., x n in ÔX 2,p 1 such tht x 2 = x 1 x 2 nd u = x 1 +2 x 2 b 2 nd v = x 1 +b2 x 2 2. ω i (Z 1 ) = ω i (p 1 ) = ( + 2 b 2 )(b 2 2 ) = ( )(b 2 2 ) + ( 2 b 2 )(b 2 2 ) < ( )(b 2 2 ) = ω i (p) = ω i (Z) = ω i (f π 1 ). Suppose tht p Z is 3 point. There exist regulr prmeters x 1,..., x n in Ô X1,p nd u, v in O Y,q such tht u = x 1 x 2 2 x 3 b 3 nd v = x 1 x 2 b 2 x 3 3. After permuting x 1, x 2, x 3 if necessry, we cn suppose tht the locl equtions of Z re x 2 = x 3 = 0. Then there exist regulr prmeters x 1, x 2, x 3..., x n in ÔX 2,p 1 such tht x 3 = x 2 ( x 3 +α) nd u = x 1 x 2 + 3 2 ( x 3 + α) 3 b nd v = x 1 x 2 +b 3 2 ( x 3 + α) b 3. Since p 1 is 2 point, we hve α 0 nd (b 2 + b 3 ) ( 2 + 3 ) 0. After n pproprite chnge of vribles x 1, x 2 we obtin regulr prmeters x 1, x 2, x 3, x 4,..., x n in ÔX 2,p 1. 1 u = x 1 x 2 + 3 b1 b 2 nd v = x 1 x 2 +b 3 2. Since the locl equtions of Z π 1 1 (U i ) (W 1 ) q re x 2 = x 3 = 0, b 2 2 nd b 3 3 hve different signs. So hs the sme sign s exctly one of b 2 2 or b 3 3. Without loss of generlity suppose tht ( )(b 2 2 ) > 0 nd ( )(b 3 3 ) < 0. Let Z be the codimension 2 vriety whose locl equtions re x 1 = x 2 = 0 defined in n ppropritely smll neighborhood in π 1 1 (U i ). Then the closure Z of Z in π 1 1 (U i ) is n irreducible codimension 2 subvriety contined in π 1 1 (U i ) (W 1 ) q. ω i (Z 1 ) = ω i (p 1 ) = ( )(b 2 + b 3 2 3 ) = ( )(b 2 2 ) + ( )(b 3 3 ) < ( )(b 2 2 ) = ω i ( Z ) ω i (f π 1 ). 15

The theorem now follows by induction on the number of codimension 2 subvrieties Z in π 1 1 (U i ) (W 1 ) q such tht ω i (f π 1 ) = ω i (Z) nd induction on ω i (f π 1 ). Remrk 3.5. Let π 1 : X 1 X be permissible sequence with respect to q. Let i be such tht 1 i m. If q E i, then it follows from Theorems 3.3 nd 3.4 tht there exists permissible sequence π 2 : X 2 X 1 with respect to q such tht Ω i (f π 1 π 2 ) = 0 nd ω i (f π 1 π 2 ) = 0. Theorem 3.6. Let f : X Y be loclly toroidl morphism between nonsingulr n-fold X nd nonsingulr surfce Y. Let q Y. Then there exists permissible sequence π 1 : X 1 X with respect to q such tht (W 1 ) q is empty. Proof. First we pply Theorem 3.1 nd Remrk 3.5 to X nd i = 1. Suppose tht q / E 1. Then by Theorem 3.1, there exists permissible sequence π 1 : X 1 X with respect to q such tht π 1 1 (U 1 ) (W 1 ) q =. Now suppose tht q E 1. It follows from Remrk 3.5 tht there exists permissible sequence π 1 : X 1 X with respect to q such tht Ω 1 (f π 1 ) = 0 nd ω 1 (f π 1 ) = 0. So there re no 1 points or 2 points of the form (10) in π 1 1 (U 1 ) (W 1 ) q. But if Z π 1 1 (U 1 ) (W 1 ) q is ny codimension 2 irreducible subvriety of π 1 1 (U i ), then generic point of Z must either be point or 2 point of the form (10). It follows then tht π 1 1 (U 1 ) (W 1 ) q is empty. Now we pply Theorem 3.1 nd Remrk 3.5 to the permissible sequence π 1 : X 1 X nd i = 2. If q / E 2, then by Theorem 3.1, there exists permissible sequence π 2 : X 2 X 1 such tht π 2 1 (π 1 1 (U 2 )) (W 2 ) q =. If q E 2, then s bove there exists permissible sequence π 2 : X 2 X 1 such tht π 2 1 (π 1 1 (U 2 )) (W 2 ) q is empty. Notice tht we lso hve π 2 1 (π 1 1 (U 1 )) (W 2 ) q =. Repeting the rgument for i = 3, 4,..., m we obtin the desired permissible sequence. 4. Toroidliztion Theorem 4.1. Let f : X Y be loclly toroidl morphism from nonsingulr n-fold X to nonsingulr surfce Y with respect to open coverings 16

{U 1,..., U m } nd {V 1,..., V m } of X nd Y respectively nd SNC divisors D i nd E i in U i nd V i respectively. Let π : Y 1 Y be the blowup of point q Y. Then there exists permissible sequence π 1 : X 1 X such tht there is loclly toroidl morphism f 1 : X 1 Y 1 such tht π f 1 = f π 1. Proof. By Theorem 3.6 there is permissible sequence π 1 : X 1 X such tht there exists morphism f 1 : X 1 Y 1 nd π f 1 = f π 1. Let p X 1. Suppose tht p π 1 1 (U i ) for some i such tht 1 i m. If π 1 (p) / f 1 (q) then we hve nothing to prove. So we ssume tht π 1 (p) f 1 (q). Suppose first tht q / E i. Then by Lemm 2.5 one of the forms (12) or (13) holds t p. So there exist regulr prmeters x 1,..., x n in ÔX 1,p nd u, v in O Y,q such tht u = x 1, v = x 1 (x 2 + α) for some α K, or u = x 1 y 1, v = x 2. Let f 1 (p) = q 1. There exist regulr prmeters u 1, v 1 O Y1,q 1 such tht u = u 1, v = u 1 (v 1 + α) or u = u 1 v 1, v = v 1 ccording s the form (12) or the form (13) holds. In either cse, we hve u 1 = x 1, v 1 = x 2, nd f 1 is smooth t p. Now suppose tht q E i. By Lemm 2.5 there exist regulr prmeters x 1,..., x n in ÔX 1,p nd u, v in O Y,q such tht one of the forms (4), (5), (6), (7), or (8) of Lemm 2.5 holds. Suppose first tht the form (4) holds. Then since m q Ô X1,p is invertible, there exist regulr prmeters x 1,..., x n in ÔX 1,p nd u, v in O Y,q such tht u = x 1 k, v = x1 1 xk+1 k for some 1 k n 1. Further x 1 = 0 is locl eqution of π 1 1 (D i ) nd u = 0 is locl eqution for E i. Let f 1 (p) = q 1. There exist regulr prmeters (u 1, v 1 ) in O Y1,q 1 such tht u = u 1 nd v = u 1 v 1. Hence the locl eqution of π 1 (E i ) t q 1 is u 1 = 0. u 1 = x 1, v 1 = x k+1. This is the form (1). Suppose now tht the form (5) holds t p for f π 1. There exist regulr prmeters x 1,..., x n in ÔX 1,p nd u, v in O Y,q nd 1 k n 1 such tht 17

u = 0 is locl eqution of E i, x 1 x k+1 = 0 is locl eqution of π 1 1 (D i ) nd u = x 1 x k+1 +1, v = x 1 b k x k+1 b k+1, where b i i for i = 1,..., k nd b k+1 < +1. Let f 1 (p) = q 1. There exist regulr prmeters u 1, v 1 in O Y1,q 1 such tht u = u 1 v 1 nd v = v 1. Hence the locl eqution of π 1 (E i ) t q 1 is u 1 v 1 = 0. u 1 = x 1 b k x k+1 +1 b k+1, v 1 = x 1 b k x k+1 b k+1. This is the form (3). Note tht the rnk condition follows from the dominnce of the mp f 1. Suppose now tht the form (6) holds. There exist regulr prmeters x 1,..., x n in ÔX 1,p nd u, v in O Y,q nd 1 k n 1 such tht u = 0 is locl eqution of E i, x 1 = 0 is locl eqution of π 1 1 (D i ) nd u = x 1, v = x 1 b k (x k+1 + α), where b i i for ll i nd 0 α K. Let f 1 (p) = q 1. There exist regulr prmeters u 1, v 1 in O Y1,q 1 such tht u = u 1 v 1 nd v = v 1. Hence the locl eqution of π 1 (E i ) t q 1 is u 1 v 1 = 0. u 1 = x 1 1 b k k (x k+1 + α) 1 b, v 1 = x 1 b 1 k (x k+1 + α). [ ] 1 b If rnk 1.. b k = 2 then there exist regulr prmeters.. b k x 1,..., x n in ÔX 1,p such tht u 1 = x 1... x k b k k, b v1 = x 1... x k k. This is the form (3). [ ] 1 b If rnk 1.. b k < 2 then there exist regulr prmeters.. b k x 1,..., x n in ÔX 1,p such tht u 1 = ( x 1... x k k) m, v = ( x 1... x k k) t (x k+1 +β), with β 0. This is the form (2). Suppose tht the form (7) holds. There exist regulr prmeters x 1,..., x n in ÔX 1,p nd u, v in O Y,q nd 1 k n 1 such tht uv = 0 is locl eqution for E i, x 1 = 0 is locl eqution of π 1 1 (D i ) nd u = (x 1 ) m, v = (x 1 ) t (α + x k+1 ), where,...,, m, t > 0 nd α K {0}. 18

Suppose tht m t. There exist regulr prmeters u 1, v 1 in O Y1,q 1 such tht u = u 1 nd v = u 1 (v 1 + β) for some β K. u 1 = (x 1 ) m, v 1 = (x 1 ) t m (α + x k+1 ) β. If m < t then β = 0. So u 1 v 1 = 0 is locl eqution of π 1 (E i ) nd we hve the form (2). If m = t then α = β 0 nd u 1 is locl eqution of π 1 (E i ). In this cse we hve the form (1). Suppose tht m > t. Then there exist regulr prmeters u 1, v 1 in O Y1,q 1 such tht u = u 1 v 1 nd v = v 1. u 1 = (x 1 ) m t (α + x k+1 ) 1, v 1 = (x 1 ) t (α + x k+1 ). We obtin the form (2). Finlly suppose tht the form (8) holds. There exist regulr prmeters x 1,..., x n in ÔX 1,p nd u, v in O Y,q nd 2 k n such tht uv = 0 is locl eqution of E i nd x 1 = 0 is [ locl eqution ] of π 1 1 (D i ) nd u = x 1 k, v = b x1 1 b k, 1.. where rnk k = 2... b k We hve either i b i for ll i or i b i for ll i. Without loss of generlity, suppose tht i b i for ll i. Let f 1 (p) = q 1. There exist regulr prmeters u 1, v 1 in O Y1,q 1 such tht u = u 1 nd v = u 1 v 1. Hence the locl eqution of π 1 (E i ) t q 1 is u 1 v 1 = 0. u 1 = x 1 1 b k, v 1 = x 1 b 1 k. [ ] Further, rnk 1.. = 2. This is the form (1)... b k Now we re redy to prove our min theorem. Theorem 4.2. Suppose tht f : X Y is loclly toroidl morphism between vriety X nd surfce Y. Then there exists commuttive digrm of morphisms f 1 X 1 Y 1 π 1 f X Y where π, π 1 re blowups of nonsingulr vrieties such tht there exist SNC divisors E, D on Y 1 nd X 1 respectively such tht Sing(f 1 ) D, f 1 1 (E) = D nd f 1 is toroidl with respect to E nd D. π 19

Proof. Let E = Ē1 +... + E m where Ēi is the Zriski closure of E i in Y. There exists finite sequence of blowups of points π : Y 1 Y such tht π 1 (E ) is SNC divisor on Y 1. By Theorem 4.1, there exists sequence of blowups π 1 : X 1 X such tht there is loclly toroidl morphism f 1 : X 1 Y 1 with f π 1 = π f 1. Let E = π 1 (E ) nd D = f 1 1 (E). We now verify tht E nd D re SNC divisors on Y 1 nd X 1 respectively nd tht f 1 : X 1 Y 1 is toroidl with respect to D nd E. Let p X 1 nd let q = f 1 (p). Suppose tht p / D, so tht q / E. There exists i such tht 1 i m nd p π 1 1 (U i ). Then q / E = π 1 (E ) q / π 1 (E i ). So p / f 1 1 (π 1 (E i )) = π 1 1 (D i ). Then f 1 is smooth t p becuse f 1 π1 1 (U i ) is toroidl. Thus Sing(f 1 ) D. Suppose now tht p D. Let p π 1 1 (U i ) for some i between 1 nd m. If q / π 1 (E i ) then f 1 is smooth t p nd then D = f 1 1 (E) is SNC divisor t p. We ssume then tht q π 1 (E i ). Cse 1 q E is point. q is necessrily point of π 1 (E i ). Then π 1 (E i ) nd E re equl in neighborhood of q. Hence π 1 1 (D i ) nd D re equl in neighborhood of p. Since π 1 1 (D i ) is SNC divisor in neighborhood of p, D is SNC divisor in neighborhood of p. Since f 1 π1 1 (U i ) is toroidl there exist regulr prmeters u, v in O Y1,q nd regulr prmeters x 1,..., x n in ÔX 1,p such tht the the form (1) holds t p with respect to E nd D. Cse 2 q E is 2 point. q is either point or 2 point of π 1 (E i ). Cse 2() q is point of π 1 (E i ). There exists regulr prmeters u, v in O Y1,q nd regulr prmeters x 1,..., x n in ÔX 1,p such tht the form (1) holds t p. There exists ṽ O Y1,q such tht u, ṽ re regulr prmeters in O Y1,q, uṽ = 0 is locl eqution for E t q, u = 0 is locl eqution of π 1 (E i ) t q, nd for some β K with β 0. ṽ = αu + βv + higher degree terms in u nd v, 20

Since π 1 1 (D i ) is SNC divisor in neighborhood of p, there exist regulr prmeters x 1,..., x n in O X1,p such tht x 1... x k = 0 is locl eqution of π 1 1 (D i ) t p. Since x 1 = 0 is lso locl eqution of π 1 1 (D i ) t p, there exist units δ 1,..., δ k ÔX 1,p such tht, fter possibly permuting the x j, x j = δ j x j for 1 j k. ṽ = αu + βv + higher degree terms in u nd v = αx 1 + βx k+1 + higher degree terms in u nd v = αδ 1 k 1...δ k x 1... x k + βx k+1 + higher degree terms in u nd v Let m be the mximl idel of O X1,p nd let ˆm = môx 1,p be the mximl idel of Ô X1,p. Since β 0, x 1,..., x k, ṽ re linerly independent in ˆm/ ˆm 2 = m/m 2, so tht they extend to system of regulr prmeters in O X1,p. Sy x 1,..., x k, ṽ, x k+2,..., x n. uṽ = x 1... x k ṽ = 0 is locl eqution of D t p, so D is SNC divisor in neighborhood of p, nd u, ṽ give the form (3) with respect to the forml prmeters x 1,..., x k, ṽ, x k+2,..., x n. Cse 2(b) q is 2 point of π 1 (E i ). Then π 1 (E i ) nd E re equl in neighborhood of q. Hence π 1 1 (D i ) nd D re equl in neighborhood of p. Since π 1 1 (D i ) is SNC divisor in neighborhood of p, D is SNC divisor in neighborhood of p. Since f 1 π1 1 (U i ) is toroidl there exist regulr prmeters u, v in O Y1,q nd regulr prmeters x 1,..., x n in ÔX 1,p such tht the one of the forms (2) or (3) holds t p with respect to E nd D. Acknowledgment: I m sincerely grteful to my dvisor Dle Cutkosky for his continued support nd help with this work. References [1] Abrmovich D., Kru K., Wek semistble reduction in chrcteristic 0, Invent. Mth. 139 (2000), 241-273. 21

[2] Abrmovich D., Kru K., Mtsuki K., Wlodrczyk J., Torifiction nd fctoriztion of birtionl mps, JAMS 15 (2002), 531-572. [3] Akbulut S., King H., Topology of lgebric sets, MSRI publictions 25, Springer-Verlg, Berlin. [4] Cutkosky S.D., Locl monomiliztion nd fctoriztion of morphisms, Asterisque 260, Societe Mthemtique de Frnce, 1999. [5] Cutkosky S.D., Monomiliztion of morphisms from 3-folds to Surfces, Lecture Notes in Mthemtics, 1786, Springer Verlg 2002. [6] Cutkosky S.D., Toroidliztion of dominnt morphisms of 3-folds, Memoirs of the Americn Mthemticl Society, Volume 190, Number 890, Amer. Mth. Soc., 2007. [7] Cutkosky S.D., Piltnt O., Monomil resolutions of morphisms of lgebric surfces, Communictions in Algebr 28 (2002), 5935-5959. [8] Cutkosky S.D., Kscheyev, O. Monomiliztion of strongly prepred morphisms from nonsingulr n-folds to surfces, J. Algebr 275 (2004), 275-320. [9] Dnilov V.I., The geometry of toric vrieties, Russin Mth, Surveys 33:2 (1978), 85-134. [10] Kempf, G., Knudsen, F., Mumford, D. nd Sint-Dont, B., Toroidl embeddings I, Lecture Notes in Mthemtics 339, Springer-Verlg, Berlin, Heidelberg, New York, 1973. 22