Resolution and alteration with ample exceptional divisor
aa r X i v : . [ m a t h . AG ] F e b RESOLUTION AND ALTERATION WITHAMPLE EXCEPTIONAL DIVISOR
J ´ANOS KOLL ´AR AND JAKUB WITASZEK
Abstract.
In this short note we explain how to construct resolutions or regu-lar alterations admitting an ample exceptional divisor, assuming the existenceof projective resolutions or regular alterations. In particular, this implies theexistence of such resolutions for arithmetic three-dimensional singularities.
It is frequently advantageous to have resolutions or alterations that have an am-ple exceptional divisor. While Hironaka-type methods automatically produce sucha resolution, neither the resolution of 3-dimensional schemes [CP19] nor alterations[dJ96] yield ample exceptional divisors right away. The aim of this note is to outlinea simple trick that does ensure the existence of ample exceptional divisors.Let X be an integral scheme. A proper, birational morphism π : Y → X is a resolution if Y is regular, and a log resolution if, in addition, the exceptional locusEx( π ) is a simple normal crossing divisor. A proper, dominant, generically finitemorphism π : Y → X is an alteration. It is called regular if Y is regular, and Galois with group G = Aut( Y /X ) if
Y /G → X is generically purely inseparable.We let Ex( π ) ⊂ Y denote the smallest closed subset such that π is quasi-finite on Y \ Ex( π ). Theorem 1.
Let X be a Noetherian, normal scheme. Assume that projectiveresolutions (resp. log resolutions) exist for every scheme X ′ → X that is projectiveand birational over X .Then X has a projective resolution (resp. log resolution) g : R ( X ) → X by ascheme R ( X ) , such that Ex( g ) supports a g -ample divisor. Theorem 2.
Let X be a Noetherian, normal scheme. Assume that regular, pro-jective, Galois alterations exist for every scheme X ′ → X that is projective andgenerically purely inseparable over X .Then X has a regular, projective, Galois alteration g : A ( X ) → X by a scheme A ( X ) , such that Ex( g ) supports a g -ample divisor. Note that Theorems 1–2 are also valid for algebraic spaces and stacks; see Re-mark 11 for details.
Corollary 3.
Let X be a normal, integral, quasi-excellent scheme (or algebraicspace) of dimension at most three, that is separated and of finite type over an affinequasi-excellent scheme S . Then X admits a projective log resolution g : R ( X ) → X by a scheme R ( X ) , such that Ex( g ) supports a g -ample divisor. Corollary 4.
Let X be a Noetherian, normal, integral scheme (or algebraic space),that is separated and of finite type over an excellent scheme S with dim S ≤ . Then X admits a regular, projective, Galois alteration g : A ( X ) → X by a scheme A ( X ) ,such that Ex( g ) supports a g -ample divisor. Remark 5.
It is clear from the proof that one can find g : R ( X ) → X and g : A ( X ) → X with other useful properties. For example, we can choose R ( X )(resp. A ( X )) to dominate any finite number of resolutions (resp. alterations).Also, if Z i ⊂ X are finitely many closed subschemes, and embedded resolutions(resp. regular, Galois alterations) exist over X , then we can choose R ( X ) (resp. A ( X )) to be an embedded resolution (resp. regular, Galois alteration) for the Z i .The log version of alterations does not seem to be treated in the literature.To fix our notation, recall that a normal scheme X is Q -factorial if, for everygenerically invertible sheaf L , there is an m > L [ m ] (the reflexive hullof L ⊗ m ) is invertible.We start with three lemmas; the first two are well known. Lemma 6.
Let X be a Noetherian, normal, Q -factorial scheme, π : X ′ → X a projective, birational morphism with X ′ normal. Then there is a π -ample, π -exceptional divisor E on X ′ .Proof. Let H be a π -ample line bundle on X ′ . Choose m > π ∗ H ) [ m ] isinvertible. Then H m ⊗ π ∗ (cid:0) ( π ∗ H ) [ − m ] (cid:1) is π -ample and trivial on X ′ \ Ex( π ). Thusit is linearly equivalent to a π -exceptional divisor E . (cid:3) Lemma 7.
Let X be a Noetherian, normal scheme, π : X → X a projective,generically purely inseparable morphism, and H a line bundle on X . Set U := X \ Ex( π ) .Then there is a coherent, generically invertible sheaf L on X and q > , suchthat, π ∗ L | U ∼ = H q | U .Proof. Consider the Stein factorization X ρ ′ −→ X ′ ρ −→ X of π . The images of U give U ′ ⊂ X ′ and U ⊂ X . So ρ ′∗ H is a line bundle on U ′ . Since U ′ → U is finite andpurely inseparable, it factors through a power of Frobenius; cf. [Sta15, Tag 0CNF].Hence there is a line bundle L U on U such that ρ ∗ L U ∼ = ρ ′∗ H q | U ′ , where we cantake q = deg ρ . We can then extend L U to a coherent sheaf L on X . (cid:3) Lemma 8.
Let X be a Noetherian, normal scheme and π : X → X a projective,generically purely inseparable morphism. Assume that X is Q -factorial and let H be a π -ample line bundle on X . Let L be a coherent, generically invertible sheafon X as in Lemma 7. Set L := H om X ( L , O X ) and π : X := Proj X P m ≥ L ⊗ m → X. Let π : X → X be a projective, generically purely inseparable morphism thatdominates both X and X . Then there is a π -ample, π -exceptional divisor E on X .Proof. Let τ i : X → X i be the natural maps, H := O X (1), and X τ ′ −→ X ′ τ −→ X the Stein factorization of τ . Since X is Q -factorial and τ is finite and purelyinseparable (and so, as above, it is an isomorphism or it factors through a power ofFrobenius), X ′ is also Q -factorial.By Lemma 6 there is a τ ′ -ample, τ ′ -exceptional divisor E on X . Then τ ∗ H m ( E )is π -ample for m ≫ H is π -nef, its pull-back τ ∗ H is π -nef. Therefore τ ∗ H m ⊗ τ ∗ H qm ( E )is π -ample as well, where q is as in Lemma 7. ESOLUTION AND ALTERATION 3
Set U := X \ Ex( π ); its images give open subschemes U ⊂ X and U i ⊂ X i .Then τ ∗ H m ⊗ τ ∗ H qm ( E ) | U ∼ = π ∗ (cid:0) L m | U ⊗ L m | U (cid:1) ∼ = O U . This gives a rational section of τ ∗ H m ⊗ τ ∗ H qm ( E ) whose divisor is π -ample and π -exceptional. (cid:3) (Proof of Theorem 1) . Start with a projective (log) resolution π : X → X andconstruct π : X → X as in Lemma 8. Let X ⊂ X × X X be the irreduciblecomponent that dominates X , and X → X a projective (log) resolution. ByLemma 8, π : X → X has a π -ample, π -axceptional divisor. (cid:3) (Proof of Theorem 2) . Start with a regular, projective, Galois alteration ¯ π :¯ X → X . Let π : X → X be its quotient by the Galois group of k ( ¯ X /X ). Notethat X is Q -factorial.Construct π : X → X as in Lemma 8. Let X ⊂ X × X X be the irreduciblecomponent that dominates X , and ¯ X → X a regular , projective, Galois alter-ation. Let X → X be its quotient by the Galois group of k ( ¯ X /X ). By Lemma8, π : X → X has a π -ample, π -axceptional divisor. Its pull-back to ¯ X is a¯ π -ample, ¯ π -exceptional divisor, where ¯ π : ¯ X → X is the natural morphism. (cid:3) Remark 11.
Theorems 1–2 are valid for every integral, Noetherian algebraic space(resp. stack) X with R ( X ) or A ( X ) being an algebraic space (resp. stack), assum-ing the appropriate representable resolutions or regular alterations by algebraicspaces (resp. stacks) exist for every algebraic space (resp. stack) X ′ admitting arepresentable projective birational (resp. generically purely inseparable) morphismto X . As for algebraic spaces, we note that all of the above constructions can beperformed in the category of algebraic spaces and their validity may be verified´etale locally. As for algebraic stacks, we note that every algebraic stack admitsa presentation as a quotient of an algebraic space by a smooth groupoid [Sta15,Tag 04T3], and that quotients of algebraic spaces by smooth groupoids always ex-ist [Sta15, Tag 04TK]. We can then conclude as each step in our constructions isequivariant with respect to a chosen presentation.If X is an algebraic space and the appropriate resolutions or regular alterationsof all algebraic spaces admiting representable, projective, birational or genericallypurely inseparable morphisms to X exist as schemes, then we can assume that R ( X ) or A ( X ) is a scheme.Here, a representable morphism of quasi-compact quasi-separated algebraic spaces(resp. algebraic stacks) is projective if it is proper and there exists a relatively ampleinvertible sheaf (cf. [R15, Definition 8.5 and Theorem 8.6]). (Proof of Corollary 3) . When X is a scheme, the assumptions of Theorem 1are valid for integral affine quasi-excellent schemes of dimension at most three by[CP19], see [BMP +
20, Theorem 2.5 and 2.7].If X is an algebraic space, then by Chow’s lemma [Sta15, Tag 088U] we canfind a projective birational morphism h : Y → X such that the scheme Y is quasi-projective over S . M. Temkin extended [CP19] to give a projective resolution forsuch a scheme Y ; the proof will be contained in the revised version of [BMP + X . By Remark 11 we can obtain R ( X ) as ascheme. J´ANOS KOLL´AR AND JAKUB WITASZEK (Proof of Corollary 4) . When X is a scheme, the assumptions of Theorem 2 arevalid for all integral schemes that are separated and of finite type over an excellentscheme S with dim S ≤ X is an algebraic space, then a regular, projective, Galois alteration of X (and of all algebraic spaces admitting a projective generically purely inseparablemorphism to X ) exists by Chow’s lemma as in the proof of Corollary 3, and so wecan conclude by Remark 11 to get A ( X ), which is a scheme. Remark 14.
The above proofs of Corollaries 3–4 do not immediately apply to alge-braic stacks. Indeed, Chow’s lemma for algebraic stacks only ensures the existenceof a proper surjective cover by a quasi-projective scheme. This cover need not bebirational. On the other hand, one could try to construct a resolution equivariantlywith respect to a presentation, but we do not know whether the algorithms for theexistence of resolutions and regular alterations from [CP19] and [dJ96] can be runequivariantly (in contrast to the characteristic zero case). For Deligne-Mumfordstacks of finite type over a Noetherian scheme, the proper surjective cover fromChow’s lemma may be assumed to be generically ´etale [LMB00, Corollaire 16.6.1].In particular, they admit regular alterations (and so also regular, Galois alterations)and Corollary 4 holds for them.
Acknowledgments.
We thank M. Temkin for very helpful e-mails on alterationsand B. Bhatt, L. Ma, Z. Patakfalvi, K. Schwede, K. Tucker, and J. Waldron forvaluable conversations. Partial financial support to JK was provided by the NSFunder grant number DMS-1901855.
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