Frobenius-stable lattices in rigid cohomology of curves

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FROBENIUS-STABLE LATTICES IN

RIGID COHOMOLOGY OF CURVES

vorgelegt von

Diplom-Mathematiker und Diplom-Informatiker

MORITZ MINZLAFF

Karlsruhe

Von der Fakult¨at II – Mathematik und Naturwissenschaften der Technischen Universit¨at Berlin

zur Erlangung des akademischen Grades Doktor der Naturwissenschaften

Dr. rer. nat. genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. John Sullivan, Ph. D. Berichter: Prof. Dr. rer. nat. Florian Heß

Berichter: Prof. Dr. rer. nat. Remke Kloosterman Tag der wissenschaftlichen Aussprache: 8. M¨arz 2013

Berlin 2013 D83

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Document History

Version Remarks Date

1.1 Accepted version; minor revisions 2013-04-06

1.0 Submitted version; Section 2.2 and Section 2.3 finished, Proposition 3.13 fixed, minor revisions

2012-12-18 0.9 Complete and proofread draft; except for Section 2.2

and Section 2.3

2012-05-31

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Ich bedanke mich bei Florian Heß f¨ur den spannenden Themenvorschlag und bei Remke Kloosterman f¨ur die

”vor Ort“-Betreuung seit Florian Hess nicht mehr in Berlin war. Desweiteren bedanke ich mich bei der Berlin Mathematical School sowie meinen Eltern Angelika und Volker Minzlaff für die finanzielle Unterstützung während der Anfertigung dieser Doktorarbeit. Anja Bewersdorff, Tanja Fagel, Do-minique Schneider, Mariusz Szmerlo und Nadja Wisniewski vom BMS One-Stop Office möchte ich dafür danken, dass jegliche administrativen Gänge so angenehm waren, wie man es sich nur wünschen kann. Viel Dank gilt auch John Sullivan für die kurzfristige Übernahme des Vorsitzes im Promotionsausschuss.

Bei Claus Diem, Felix Fontein, Frank Herrlich und Gabriela Weitze-Schmithüsen, Kiran Kedlaya, Thorsten Lagemann, Alan Lauder und Jan Tuitman, George Walker und Stefan Wewers bedanke ich mich für Einladungen zu Seminarvorträgen und die Zeit verschiedene Aspekte meiner Arbeit mit mir zu diskutieren. Ganz besonders danke ich Nils Bruin dafür, daß er mir einen mehrmonatigen Besuch bei ihm an der Simon Fraser University in British Columbia ermöglicht hat.

Für das Korrekturlesen und für Verbesserungsvorschläge bedanke ich mich bei Fabian Januszewski, Stefan Keil, Thorsten Lagemann, Fabian Müller, Christopher Özbek und Osmanbey Uzunkol. Christina Neuhaus schließlich gebührt ein ganz großer Dank für die moralische Unterstützung auf der Zielgeraden dieser Arbeit.

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CHAPTER 1 INTRODUCTION

It is probably fair to say that a major part of mathematics is ultimately motivated by the desire to solve equations or to study their sets of solutions. The present thesis is no exception. Its guiding problem is that of efficiently counting the number of solutions to bivariate polynomial equations

f (x, y) = 0

over finite fields. Formulated from the viewpoint of algebraic geometry, the problem is that of efficiently counting the number of rational points on plane curves. More generally, let X be any curve over a finite field. If N1, N2, N3, . . . are the numbers of

rational points of X over the extensions of the finite field of degrees 1, 2, 3, . . ., then the zeta function of X is defined as

Z_X(t) = exp ∞ X r=1 Nr r t r ! .

It is uniquely determined by N1, . . . , Nmwhenever m is large enough. For example,

if X is smooth, proper, geometrically integral, then one may take m equal to the genus [Sti09, Cor. V.1.17]. Conversely, the zeta function is a rational function in t and if αi are the reciprocal roots of the numerator and βj the reciprocal roots of the

denominator, then one recovers the number of points as Nr=Pjβ r j− P iα r i. A more

detailed account of these statements is given by Wan [Wan08]. The aim of this thesis is to improve the state of the art of computing zeta functions.

The zeta function has a cohomological interpretation: Let X be a smooth, geomet-rically integral curve over the finite field k. Its rigid cohomlogy H•_rig(X) consists of finite dimensional vector spaces over the fraction field K of the ring of Witt vec-tors W (k) [Ber97, Thm. 3.1]. The Frobenius morphism F on X induces linear maps

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on those spaces and Z_X(t) = det id − Ft | H 1 rig(X) detid − Ft | H0 rig(X) detid − Ft | H2 rig(X)

whenever X is proper. If X is affine, then the relation is

Z_X(t) = det id − #k · F −1_{t | H}1 rig(X) detid − #k · F−1_{t | H}0 rig(X) detid − #k · F−1_{t | H}2 rig(X) .

The only interesting map in the above determinants is the one on the first cohomology group. Indeed, in both cases the zeroeth cohomology group is 1-dimensional and the Frobenius acts as the identity. If the curve is proper, then the second cohomology group is also 1-dimensional and the Frobenius action is multiplication by #k. In case the curve is affine, the second cohomology group vanishes. To see this, use, e.g. the comparison theorems with crystalline cohomology and with Monsky-Washnitzer cohomology [Ber97, Prop. 1.9 & 1.10] and the properties of the respective cohomolo-gies [Ber74, MW68, vdP86].

1.1. State of the art

The task of computing the Frobenius action on H1

rig(X) has received a lot of attention

over the past few years. First came Kedlaya’s algorithm for hyperelliptic curves in odd characteristic [Ked01] which has been generalised several times to superelliptic curves [GG01], Ca,b-curves [DV06a], and nondegenerate curves [CDV06].

Further-more there are algorithms for hyperelliptic curves in even characteristic [DV06b] and smooth plane curves (as special case of smooth hypersurfaces) [AKR11].

More recent are algorithms that add a deformation step: The given curve is first embedded in a family of curves with a particular “nice” fibre. The Frobenius action on the cohomology of this fibre is then computed and finally transformed to give the action on the cohomology of the original curve. By now this has been applied to most classes of curves mentioned above [Ger07, Hub07, Hub08, CHV08, Tui11]. Related to the approach with a deformation step is the approach that uses a fibration on X: Computing the Frobenius action for a given variety is reduced to the same prob-lem for a smooth hyperplane section. This method was introduced by Lauder [Lau06] and improved by Walker [Wal09]. In the case of curves, Walker focuses on smooth plane curves and on Ca,b-curves.

Typically the runtimes of these algorithms are polynomial in the genus g of the curve and the degree n of k, but linear or worse in the characteristic p. In particular, it

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1.1. STATE OF THE ART 3

is exponential in log(p). For example, the time complexities of the algorithms for hyperelliptic, superelliptic, and Ca,b-curves (without deformation or fibration) lie in

e

O(p1n3g5).

To contrast, if X is proper, then O(log(p)ng2_{) bits suffice to determine X [Hes,}

Thm. 56] and to write down the determinant of 1−Ft on H1_rig(X) [Sti09, Thm. V.1.15]. Using deformation can yield a lower exponent for n as demonstrated in case of hyper-elliptic curves. Regarding p, Harvey describes an algorithm for hyperhyper-elliptic curves in odd characteristic which improves the dependence on the characteristic to eO(p0.5₎

at the cost of higher exponents for the other parameters [Har07].

Fixing the characteristic, the most general class of curves for which a practical al-gorithm exists arguably is the class of nondegenerate curves. If the cardinality of k is large enough, then this class includes all Ca,b-curves and thus all superelliptic and

hyperelliptic curves. Nonetheless, nondegenerancy is quite special: The moduli space of nondegenerate genus g curves has dimension 2g + 1 whereas the moduli space of all genus g curves has dimension 3g − 3 [CV09].(1)

Further algorithms. — If the only goal is to compute the zeta function and not the Frobenius action on rigid cohomology, there are further methods available. Most notably are those using “Dwork cohomology”, ´etale (or `-adic) cohomlogy, or the canonical lift of the Jacobian. Currently the most general class of curves these meth-ods can practically handle are ordinary hyperelliptic curves in even and superelliptic curves in any characteristic.

Lauder and Wan proposed an algorithm based on Dwork cohomology for arbitrary affine (not necessarily smooth) varieties. It runs in polynomial time when the char-acteristic is kept fixed [LW08], but is deemed not very practical [LW04, p. 332]. Related algorithms for superelliptic curves [Lau03], Artin-Schreier curves [LW04], and smooth hypersurfaces with a deformation step [Lau04] have time complexity comparable to those using rigid cohomology. In particular, they share the bad be-haviour with respect to log(p).

Schoof described an algorithm for elliptic curves using ´etale cohomology [Sch85]. Apart from several optimisations, there is an adaption to genus 2 [GH00] and even a generalisation to arbitrary genus [Pil90], but the latter runs doubly-exponential in the genus and is considered impractical [The04, Sec. F.3.b]

(1)_{The only exceptions are genus 7 where the dimension is 16 and genera less than 4 where 3g − 3 is}

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Canonical lifts of Jacobians are used in Satoh’s and Mestre’s algorithms for ordinary elliptic curves in odd respectively even characteristic [Sat00, Mes00]. The most gen-eral variants of these algorithms are for ordinary hyperelliptic curves [CL07, LL06]. More in-depth accounts of the various approaches to compute zeta functions can be found in a book by Cohen et al. [CFA05] and a survey by Chambert-Loir [CL08].

1.2. This work

The previous section indicates that when one wants to compute the Frobenius action on rigid cohomology with current methods, one has to restrict to special classes of curves such as the class of nondegenerate curves and to “small” characteristic. This thesis makes three main contributions to weaken these restrictions.

Lifting birational equivalence classes of curves. — A central ingredient to com-puting zeta functions with rigid cohomology is a comparison theorem with de Rham cohomology: Let X be a smooth, proper, geometrically integral curve and X be a proper lifting. If Z is a reduced strict relative normal crossing divisor on X, then there is a natural isomorphism

H•_dR(XK\ ZK) ∼

−→ H•rig(X \ Z) (1.1)

between the rigid cohomology of the affine curve X \ Z over k and the de Rham cohomology of the affine curve XK\ ZK over K (see (3.2)). For the types of curves

mentioned in the previous section a proper lifting is easily found. Indeed, each of those curves is a smooth hypersurface in some ambient surface which is obviously the reduction of some proper, integral scheme over the ring of Witt vectors: For smooth plane curves one may take the projective plane as the ambient scheme, for hyper-elliptic and Ca,b-curves a weighted projective plane, and for nondegenerate curves a

toric surface. One may identify a hypersurface with an element of the homogeneous coordinate ring. Under this identification, lifting the hypersurface amounts to picking a lift of the element in the homogeneous coordinate ring of the lifted surface. The first contribution of this thesis is a treatment of the case when such a nice embedding is not known. It suffices to lift the birational equivalence class of the curve and we present an algorithm that approximates a lifting of the class of any curve (Chapter 2).

An explicit description of Frobenius-stable lattices. — Much like lifting a root modulo p of a polynomial over the integers in general only yields an element of the p-adic integers, it seems that one cannot expect a proper lifting of X to be defined over the integers or a finite extension of them. Rather, one has to consider the ring of Witt vectors W (k). A Frobenius action on lattices over W (k) is therefore of interest for two reasons: When one is only able to write down an approximation of

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1.3. NOTATIONS AND CONVENTIONS 5

a lifting over Witt vectors WN(k) of finite length, one cannot write down the generic

fibre of the exact lifting as needed in (1.1). In this case, a Frobenius-stable lattice over W (k) naturally yields a WN(k)-module on which to approximate the Frobenius

action. Another reason is that even in the case when one does obtain an exact lifting, if the chosen basis of H1

rig(X \ Z) does not span a Frobenius-stable lattice, then the

matrix representing the Frobenius will not have coefficients in W (k). Since we can only compute with Witt vectors of finite length, this causes a loss of precision. The second contribution of this thesis is therefore an explicit description of a Frobenius-stable lattice in the vector space H1

rig(X \ Z) for any choice of X and Z (Chapter 3).

Computing the Frobenius action in larger characteristic. — The third con-tribution of this thesis is an extension of Harvey’s algorithm from hyperelliptic curves in odd characteristic to more general “Kummer coverings” of the rational line. It computes the Frobenius action in eO(p0.5_{)-time (Chapter 4). Like Harvey’s algorithm,}

the runtime is polynomial in the degree of k and the genus.

Finally, the effectiveness of all algorithms in this thesis is demonstrated using imple-mentations in Sage [S+_{12] and Magma [BCP97] (Chapter 5).}

Original contributions of this thesis are usually enclosed in numbered environments such as propositions, lemmas, and others. Conversely, to our best knowledge such numbered environment contain original contribution. Exceptions to this are marked as such. The main theorems are numbered with capital letters.

1.3. Notations and conventions

We denote by N the monoid of finite cardinals 0, 1, 2, . . . and by Z the ring of integers. Throughout this thesis, k is a finite field of characteristic p. The ring of Witt vectors over k is denoted by W (k), its fraction field by K, its valuation by v, and the ring of Witt vectors of length N by WN(k) = W (k)/pNW (k). Liu’s book “Algebraic

Geometry and Arithmetic Curves” [Liu06] will be our default reference for algebraic geometry.

Objects over k. — Objects over k usually come with an overline · in their notation, e.g. we write X for a scheme over k. We also sometimes use this notation to denote the base change of an object over W (k) to k, e.g. given a W (k)-scheme X, we might write X for the special fibre Xk.

Frobenius morphisms. — Let X be a scheme over k of finite type. The morphism Fp: X → X given by the identity on the topological space of X and p-th powering on

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is not a k-morphism. In general, let n be the degree of k. Then we define the (k-linear or relative) Frobenius morphism F as the n-th iterate of the absolute Frobenius, i.e. F = Fn

p. This is the Frobenius that was already mentioned earlier in this introduction.

Curves. — Let S be a locally Noetherian scheme. A curve over S is a flat, separated scheme of finite type over S whose fibres are geometrically reduced and equidimen-sional of dimension 1. Curves are proper unless otherwise mentioned. A curve over S is plane if it is a hypersurface in P2

S. In general, we say that a curve X has a certain

property whenever X has that property as a scheme. The arithmetic genus of a curve X over a field is pa(X)

df

= 1 − χ(OX). If the

normali-sation eX of X is smooth, then the genus g(X) of X is the geometric genus of eX, i.e. g(X)df= h0_(Ω

e X).

Deformations. — Our main references for deformation theory and its terminology are Hartshorne’s “Deformation Theory” [Har10] and Sernesi’s “Deformations of Al-gebraic Schemes” [Ser06]. For example, the deformation of a scheme X over k to a local Noetherian ring A with residue field k is a scheme X that is flat and of finite type over A together with an isomorphism of its special fibre with X. We use the notion of a deformation and a lifting interchangeably, in particular when we think of lifting objects from characteristic p to characteristic 0.

Abbreviations. — Given a sheaf F on a scheme X, we will usually write Hi(F ) for Hi(X,F ). Similarly, if X is understood to be a scheme over another scheme S, we will write ΩX for ΩX/Sand HidR(X) instead of H

i

dR(X/S). Further abbreviations like

these may be found throughout the thesis and hopefully will not lead to confusion. A bold letter usually denotes a tuple or sequence, e.g. x might be a sequence x1, . . . , xr

of variables or α might be a tuple (α1, . . . , αr) ∈ Nrof finite cardinals. In the latter

case, we also have |α| = α1+ . . . + αr.

Complexity. — In addition to the usual Oh notation to measure the asymptotic be-haviour of functions, e.g. n2_{+ n ∈ O(n}2_{), we also use the soft Oh notation [vzGG03,}

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CHAPTER 2 LIFTING BIRATIONAL EQUIVALENCE CLASSES OF

CURVES

This chapter addresses the issue of computing a proper lifting to W (k) of a smooth curve X over k up to birational equivalence. Our motivation to study this problem arose in the context of describing the p-adic cohomology of open affine subschemes of X (Chapter 3). Deformation theory tells us that such a proper lifting always exists [Har10, Thm. 22.1] and our main result will be to turn this existence statement into an effective algorithm.

Finding a lifting is easy when the curve is embedded in projective space as com-plete intersection: Pick any homogeneous liftings of its defining equations [Har10, Thm. 9.4]. But while every smooth curve over k can be embedded already in P3

` for

some finite extension ` of k, in general it cannot be embedded as smooth complete intersection in P3_`. It is conjectured that in general an embedding in P3_` does not lift [EH99, Conj. C].

One idea to circumvent this problem is to embed curves as complete intersections in more general ambient schemes. This is done by most algorithms that compute zeta functions with p-adic cohomology. However, this approach also seems to have limitations: For example, the step from hyperelliptic curves (hypersurfaces in weighted projective planes) to nondegenerate curves (hypersurfaces in toric surfaces) only gives a modest increase in the dimension of the moduli space. Beyond toric surfaces, there is (to us) no clear candidate for ambient schemes that are suitable from a computational point of view.

Therefore, we take a different approach: We restrict to the simplest ambient scheme, the projective plane, but allow singularities in a controlled fashion. In particular, we allow nodes so that X is always birational to such a curve [Har97, Thm IV.3.10] over some finite extension of k. The idea is to lift the plane curve along with its singularities.

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For this chapter, we use a more general definition for k and K: Let Λ be a complete discrete valuation ring with maximal ideal m, fraction field K, and perfect residue field k. Set ΛN

df

= Λ/mN_{Λ and denote by S the}

spectrum of Λ.

This chapter is organised as follows: In Section 2.1, we formally define liftings of birational equivalence classes of curves. We will see that a proper lifting X of a representative may fail to lift the class. The solution is an adaption of Teissier’s δ-invariant criterion to our situation: A proper lifting X lifts the class if and only if the δ-invariants of its fibres coincide (Proposition 2.5). For plane curves whose singularities are all ordinary, e.g. nodes, the δ-invariant is completely determined by the multiplicities of the singularities and the degrees of their underlying points. An “equimultiple proper lifting” of a plane ordinary curve therefore solves the problem of

lifting its birational equivalence class (Proposition 2.11).

The definition of an “equimultiple” local Hilbert functor (Definition 2.15) in Section 2.2 frames the problem of computing such liftings in the context of formal deformation theory. Finally in Section 2.3, we explain how the infinitesimal lifting property of smooth schemes can be used to compute equimultiple proper liftings of plane curves over k to ΛN (Algorithm 1). We use the formal deformation theory introduced in the

previous section to show the algorithm is successful whenever the plane curve has only nodes and not too many ordinary singularities of higher multiplicity (Corollary 2.27). Most results of this chapter are classical knowledge when one takes Λ as the ring of formal power series over the complex numbers. We are not aware, however, of any references for the general case.

Further notation and terminology. — Let Set be the category of sets. The category of Artinian local Λ-algebras with residue field k is denoted by Ar. We write mAfor the the maximal ideal of a local ring A. The category of complete Noetherian

local Λ-algebras A such that A/mi

Ais in Ar, all i, will be denoted by cAr. A surjection

A0 _{A in Ar is a small extension (of A by I) if the kernel I is annihilated by m}A0.

The kernel has thus the structure of a k-vector space. We write Ex(A, I) for the set of small extensions of A by I. One can give Ex(A, −) the structure of a covariant functor. In case of a surjection I J of finitely generated k-vector spaces with kernel C, this functor maps the extension A0_{A to A}0_{/C A.}

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2.1. TEISSIER’S CRITERION FOR LIFTING CLASSES 9

2.1. Teissier’s criterion for lifting classes

The (birational equivalence) class of a curve over a field contains a unique-up-to-isomorphism normal curve.(1) _{Consequently, we make}

Definition 2.1. — A lifting of a class [X] of a curve X over k to Λ is a class of curves over Λ that contains a proper lifting of the normal curve of [X].

While a proper lifting Y of a normal curve over k is a normal curve over Λ, there may be several nonisomorphic normal curves in its class. Nonetheless, the generic fibres of all normal curves in [Y ] are isomorphic and thus define the same class [YK] over K.

The existence of a unique representative in [Y ] is not relevant to our goals; we refer the interested reader to Liu’s book [Liu06, Chp. 9 & 10].

Example 2.2. — Let X be a curve over k. By definition, a proper lifting of the nor-mal curve in [X] also lifts the class. However, given any representative of [X], a proper lifting does not need to lift the class. Take, for example, the geometrically integral genus 1 curve X defined by y2= x3(x−1)(x−2), where the characteristic of k does not divide 6. Assume that K is perfect. The equation y2_{= x(x − m)(x + m)(x − 1)(x − 2),}

m ∈ m nonzero, defines a proper lifting X to Λ. Let Y be a proper lifting of the nor-mal curve in [X]. Then Y is smooth since it is a proper lifting of a nornor-mal, hence smooth, curve over k. So the genera of Yk and YK must coincide and be equal to 1.

But the genus of XK is 2, so Y and X cannot be birational.

The general case. — Let X be a curve over Λ. We want to know when [X] lifts [Xk]. The above example suggests that the genera of the fibres of X should be equal.

As it turns out, that is the exact criterion except that when Xk is not irreducible we

replace the genera by the arithmetic genera of the normalisations of the fibres. In this subsection, we follow a paper by Diaz and Harris which treats curves over the complex numbers [DH88, Sec. 2].

Lemma 2.3. — Let X be a curve over Λ. Its normalisation π : Y → X is a finite morphism of curves over S.

Proof. — Since X is of finite type over Λ, X is excellent [Gro65, Sch. 7.8.3]. So its normalisation π is finite [Gro65, Prop. 7.8.6]. Moreover, the structure map Y → X → S is flat: A reduced scheme over Λ is flat if and only if the generic point of each irreducible component maps to the generic point of S [Gro66, Prop 14.5.6]. Since X is flat over Λ and since the normalisation induces a bijection between generic points of Y and X, Y is flat over Λ.

(1)_{In contrast to the definition in Liu’s book [Liu06, Def. 4.1.1], we do not assume that a normal}

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Hence, Y is a proper lifting of Yk and it remains to show that Yk is a curve. Since

Y → X is the normalisation, the normalisation of the special fibre Xk must

fac-tor through Yk → Xk and Yk must be equidimensional of dimension 1. To show

that Yk is reduced (this implies Yk geometrically reduced as k is perfect), consider a

point P ∈ Yk. Its local ring is reduced if and only if the zero ideal is radical. Let

0 = ∩p∈Ass(OYk,P)q(p) be a reduced primary decomposition, i.e. q(p) is p-primary.

Since Y is normal and of dimension 2, it is Cohen-Macaulay [Liu06, Cor. 8.2.22] and so is Yk. Hence Ass(OYk,P) is the set of minimal primes [Liu06, Prop. 8.2.15]. Let Z

be the irreducible component of Yk passing through P and corresponding to p. Let

ξ be its generic point. The image π(ξ) is a generic point of Xk. Since Xk must be

regular along an open subset of π(Z), Xk must be regular at π(ξ). Hence, X must be

regular at π(ξ). Thus,OY,ξ andOX,π(ξ) are isomorphic since π is the normalisation.

In particular,OYk,ξis isomorphic toOXk,π(ξ)and thus reduced. So q(p) equals p. This

argument applies to all minimal primes p ofOYk,P and so its zero ideal is radical.

Let X be a curve over Λ and s ∈ S. Let π(s) : fXs→ Xs be the normalisation of the

fibre over s. Denote bySXs the quotient sheaf of π(s)∗OXfs byOXs, i.e. there is the

short exact sequence

0 →OXs→ π(s)∗OXfs →SXs → 0. (2.1)

One defines the δ-invariant of Xsas

δ(Xs) df

= h0(SXs).

The δ-invariant of a closed point P of Xsis δP(Xs) df

= length_O

Xs,P(SXs,P)

= [k (P ) : k (s)]−1dimk (k)SXs,P. SinceSXs is a skyscraper sheaf, there is an equality

δ(Xs) =

X

P ∈X0 s

[k (P ) : k (s)]δP(Xs).

The sequence (2.1) gives an identity χ(π(s)∗OX_fs) = χ(OXs) + χ(SXs) [Liu06,

Lem. 7.3.16]. Since χ(π(s)∗O_X_f

s) and χ(OXfs) are equal, one arrives at

pa(Xs) = pa( fXs) + δ(Xs). (2.2)

Lemma 2.4. — Let X be a curve over Λ. Let fXk → Xk and gXK → XK be the

normalisations of its fibres. Then pa( fXk) is at most pa(gXK).

Proof. — Let Y → X be the normalisation of X. Then gXK equals YK and the

nor-malisation fXk → Xk factors through Yk → Xk. Since Y is flat over Λ by Lemma 2.3,

the arithmetic genera of its fibres coincide. Hence, the genus formula (2.2) gives pa(gXK) = pa(YK) = pa(Yk) = pa( fXk) + δ(Yk). This proves the claim.

Proposition 2.5. — Let X be a curve over Λ. Let π : Y → X, π(k) : fXk → Xk, and

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(1) The δ-invariants δ(Xk) and δ(XK) coincide,

(2) The arithmetic genera pa( fXk) and pa(gXK) coincide,

(3) the fibres of π are π(k) and π(K), and (4) the class [X] lifts the class [Xk].

Proof. — The genus formula (2.2) implies that (1) and (2) are equivalent. Now fix a very ample sheaf OX(1) on X and let s ∈ S. As in the lead-up to

the genus formula, tensoring (2.1) with OXs(n) = OX(n)|Xs yields equalities

χ(π(s)∗O_X_f_s(n)) = χ(OXs(n)) + χ(SXs(n)) for all n ∈ N. In terms of Hilbert

polynomials this means P (π(s)∗OX_fs) = P (OXs) + P (SXs). Note that SXs is a

skyscraper sheaf, so χ(SXs(n)) equals δ(SXs) for all n ∈ N. Since X is flat over Λ,

P (OXs) is independent of s. Therefore (1) holds if and only if the Hilbert polynomial

P (π(s)∗O_X_f_s) is independent of s. In this formulation, the equivalence of (1) and (3)

was proven by Chiang-Hsieh and Lipman [CHL06, Cor. 3.4.2]. (In the context of complex analytic spaces the result goes back to Teissier [Tei80].)

To finish, we will show that (3) implies (4) implies (2). Assume (3) holds. By Lemma 2.3, Y is a proper lifting of Yk = fXk. Hence, [X] lifts [Xk] and (4) holds.

Assuming (4), one can now argue as in Example 2.2: If [X] contains a proper lifting Z of fXk, then Z is normal. In particular, Zsis isomorphic to fXsfor both points s of S.

Since Z is flat over Λ, the arithmetic genera of its fibres agree, i.e. (2) is satisfied. Definition 2.6. — A curve over Λ is equinormalisable if it satisfies any of the equiv-alent properties in Proposition 2.5.

Example 2.7. — In the later sections of this chapter, we will use the δ-invariant to lift classes of curves. Hurwitz’s theorem gives another approach to finding a lifting: Let Y be a curve over Λ with normal, geometrically integral fibres. Moreover, let φ : Y → P1

Sbe a finite morphism and s be a point of S. Hurwitz relates the arithmetic

genus of Ys to the degree a(s) of φs and the degree of the ramification divisor (or

different) Rφs of φs:

2pa(Ys) = 2 − 2a(s) + deg Rφs.

Now consider the geometrically integral curve X ⊂ P2

kdefined by y

a _{= h(x) where the}

characteristic of k does not divide a and h(x) is a polynomial over k. For simplicity, assume that a and deg h are coprime. Let Y be the normalisation of X. The function x defines a finite morphism φ : Y → P1

k of degree a. Let h =

Q

ih ri

i be the factorisation

into irreducibles and sibe the greatest common divisor of a and ri, each i. The degree

of R_φisP

i(a − si) deg hi[Sti09, Sec. 3.7]. So for each hi, choose a degree-preserving

lifting hi∈ Λ[x]. Each hiis necessarily irreducible. The equation ya=Qih ri

i defines

a proper lifting X of X to Λ. Let Y be its normalisation. The rational function x defines a finite, flat morphism φ : Y → P1

Λ of degree a whose special fibre is φ. For

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generic fibre φK is completely analogous to above, i.e. its degree isPi(a − si) deg hi.

By Hurwitz, the arithmetic genera of Y and YK agree and since YK is a normalisation

of XK, [X] lifts [X].

Remark 2.8. — Except potentially for “special” cases, all geometrically integral curves X in Pr

k have an equinormalisable proper lifting in PrΛ: The normalisation

π : Y → X corresponds to a closed immersion φ : Y → Pr_k and thus to a base-point free linear system on Y . SetL = φ∗OPr

k(1). The linear system is given by a subspace

V of H0₍_{L ). Choose a proper lifting Y of Y to Λ and an invertible sheaf L on Y that}

restricts toL . If L is acyclic, then so is L and the natural map H0₍_{L ) → H}0₍_{L ) is}

surjective. The inverse image V of V under this map defines a base-point free linear system. Indeed, let P ∈ Y . ThenL_P⊗ k equals L_P. Since L is generated by V at P , L is generated by V at P by Nakayama. Now the set Z of points at which L is not generated by V is closed. As Y is proper over Λ, Z must be empty (oth-erwise it would intersect Y ). Hence, V and L define a morphism φ: Y → Pr_Λ. By construction φ lifts φ and induces a finite morphism π : Y → X = φ(Y ) that lifts π. If ξ denotes the generic point of Y , then π(ξ) is the generic point of X andOY,ξis a

finite extension of OX,π(ξ). In particular, the genus of XK is at most YK. This gives

g(XK) ≤ g(YK) = g(Yk) = g(Xk), so by Lemma 2.4, the genera of the fibres of X

coincide, i.e. X is equinormalisable. Example 2.9. — Let X ⊂ P2

k be a geometrically integral curve of degree d and

genus g. With the notation of the above remark, the sheafL is acyclic whenever d is at least 2g − 1. The genus formula (2.2) for X is (d − 1)(d − 2)/2 = g + δ(X). So d is at least 2g − 1 if and only if δ(X) is at least 2g2− 6g + 3. In other words, any sufficiently singular geometrically integral plane curve over k admits an equinormalisable proper lifting in P2

Λ.

The case of plane ordinary curves. — The δ-invariant of a plane curve whose singularities are all ordinary and whose underlying points are rational is completely de-termined by the multiplicities of its singularities. Similar to Wahl’s [Wah74b, Sec. 1] and Markwig’s [Mar07, Paper V] definitions over the complex numbers, we will now define equimultiple proper liftings of a plane curve X over k. When all singularities of X are ordinary and have residue field k, such liftings are equinormalisable. Let F be a field and U ⊆ P2

F be open. Let X ⊆ U be a curve and P be an F -rational

point of P2

F with maximal ideal mP. The multiplicity of X at P is the largest integer

m such that mm_P contains the ideal OU(−X)P. The point P lies on X if and only

if the multiplicity is at least 1. If the multiplicity of X at P is exactly 1, then the local ringOX,P is regular. If it is greater than 1, then P is called a multiple point or

singularity of X. To generalise these notions to curves over an algebra A of cAr, note that the F -rational point P corresponds to an F -section of U . So now let U ⊆ P2_Λ be

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open and σ be an A-section of U . Then σ : Spec A → U is a closed immersion with ideal sheafIσ. We define the multiplicity mσ(X) of a curve X ⊆ UA along σ as

mσ(X) df = max{n ∈ N |OP2 A(−X) ⊆I n σ}.

From now on we assume that the underlying point of a singularity is rational.

Definition 2.10. — Let A be an object of cAr, U ⊆ P2_Λ be open, and σ ∈ U (A). A plane curve X over A is equimultiple along σ (of multiplicity m) if mσk(Xk) = mσ(X)

(= m). The curve X is called equimultiple if there are sections σ1, . . . , σe∈ U (A) such

that: (1) X is equimultiple along σi for each i = 1, . . . , e, and (2) σ1, . . . , σe induce

the k-sections that correspond to the multiple points of Xk.

Let x0, x1, x2 be coordinates of P2Λ over Λ. The image of an A-section σ of P2Λ

is contained in D(xi) for some i = 0, 1, 2, say for i = 0. Then its ideal sheaf Iσ

is trivial outside D(x0) and on D(x0) it is given by an ideal of A[x1, x2] of the form

(x1−λ1, x2−λ2) with λ1, λ2∈ A. Now let X be a plane curve over A and F (x0, x1, x2)

be a defining homogeneous polynomial. Then X has multiplicity m along σ if and only if the homogeneous parts of degrees 0, 1, . . . , m−1 of F (1, x1+λ1, x2+λ2) vanish

and the the degree m part does not vanish. Now let A be equal to k. Assume that σ is the section corresponding to a singularity P of X. We say that P is an ordinary singularity if the degree m part of F (1, x1+ λ1, x2+ λ2) factors into m distinct linear

forms over k.(2) If in addition m equals 2, then P is a node of X. We call X an ordinary curve when all its singularities are ordinary and a nodal curve when all its singularities are nodes. The δ-invariant of a singularity P of multiplicity mP satisfies

δP(X) ≥ mP(mP− 1)/2. (2.3)

The inequality is an equality if and only if P is ordinary [Hir57, Thm. 1].

Proposition 2.11. — Let X be a plane curve over Λ with ordinary special fibre. If X is equimultiple, then it is equinormalisable.

Proof. — Lemma 2.4 and the genus formula (2.2) imply that δ(XK) is at most δ(Xk).

Let us use the notation from Definition 2.10 and let mi be the multiplicity of X along

σi. Each section σigives rise to a section σi,K∈ P2K(K) such that XK has multiplicity

mi along σi,K. In other words, each σi corresponds to a singularity of multiplicity

mi on XK. Moreover, pairwise distinct σi give pairwise distinct singularities. From

the formula (2.3) for the δ-invariant of a singularity, we deduce that δ(XK) is at least

(2)_{The definition of ordinary singularities given here is more restrictive than the general one. In}

particular, ordinarity can be defined when the multiple points are not rational. For our purposes, the definition given here suffices.

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δ(Xk). Hence, the δ-invariants of the fibres of X must agree and X is equinormalisable.

Example 2.12. — The converse to the proposition does not hold: A trivial example is the union X of the three lines defined by x, y, and x + y + mz in the projective plane over Λ, m ∈ m nonzero. Its generic fibre has three singularities, each a node, at [0 : 0 : 1], [m : 0 : 1], and [0 : m : 1]. Therefore, δ(XK) is 3. However, its special fibre

has a unique singularity, an ordinary multiple point of multiplicity 3, at [0 : 0 : 1]. So δ(Xk) also equals 3. Hence, X is equinormalisable but not equimultiple.

2.2. The equimultiple local Hilbert functor

For this section, we fix a plane curve X of degree d over k and a finite set Σ ⊆ P2

k(k) of sections. Let m = {mσ}σ be the multiplicities of X

along the σ ∈ Σ.

Due to Proposition 2.5 and Proposition 2.11, we are interested in equimultiple proper liftings of X to Λ. The goal of this section is to introduce and understand the following natural diagram of functors. The first half of this section introduces the necessary terminology and the functors on the top square. The bottom functors are the topic of the second half.

H_X // Pd(d+3)/2_Λ H_(X,Σ) // 99 99 Pd(d+3)/2_Λ ×Q σ∈ΣP 2 Λ 99 99 E?_X OO E_(X,Σ)? // OO 88 88 E(d,m) ? OO (2.4)

The three functors on the right-hand side are (the functors of sections of) projective schemes over S. The left-hand square is made up of prorepresentable functors of Artin rings. They are the “local counterparts” at X and Σ to the functors on the right. The functors on the top square are smooth. If A is an object of Ar and Σ is the set of sections corresponding to the multiple points of X, then E_X(A) is the set of equimultiple proper liftings of X to A.

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2.2. THE EQUIMULTIPLE LOCAL HILBERT FUNCTOR 15

Formal deformation theory. — A functor of Artin rings is a covariant functor F : Ar → Set, #F (k) = 1.

For example, if R is an object of cAr, then HomΛ(R, −) is a functor of Artin rings.

A functor of Artin rings that is isomorphic to some HomΛ(R, −) is said to be

prorep-resentable or prorepresented by R. The functor F of Artin rings is called smooth if F (A0) → F (A) is surjective whenever A0 → A is. If F is prorepresented by R, then F is smooth if and only if R is a power series ring over Λ [Sch68, Prop. 2.5]. Any functor F of Artin rings can be extended to a functor bF on cAr by

b

F (A)df= lim_←−F (A/mNA).

Let k[ε] be the ring of dual number of k, i.e. ε2 _{equals 0. The tangent space of F is}

defined as

t(F )df= F (k[ε]).

Given surjections A0_{A and A}00_{A in Ar, there is the natural map}

F (A0×AA00) → F (A0) ×F (A)F (A00). (2.5)

Consider the following “Schlessinger’s conditions”: (H1) (2.5) is surjective whenever A00_{A is small,} (H2) (2.5) is bijective whenever A = k and A00_{= k[ε],}

(H3) t(F ) is a k-vector space of finite dimension, and (H4) (2.5) is bijective whenever A00_{A is small.}

If F satisfies (H2), then the tangent space t(F ) has a natural structure as k-vector space [Sch68, Lem. 2.10]. Moreover, let A0_{A be a small extension with kernel I.} There is a natural action

F (A0) × (t(F ) ⊗ I) → F (A0) ×F (A)F (A0)

of t(F ) ⊗ I on the nonempty fibres of F (A0) → F (A). If F satisfies (H1) and (H2), then the action is transitive. If F satisfies (H2) and (H4), then the action is transitive and free [Sch68, (2.15)]. Finally, a functor of Artin rings is prorepresentable if and only if it satisfies (H2), (H3), and (H4) [Sch68, Thm. 2.11].

Example 2.13. — Let Z be a closed subscheme of Pr_k. The local Hilbert functor H_Z: Ar → Set sends an algebra A to the set

H_Z(A)df=Z ⊆ Pr

A| Z is a deformation of Z to A .

The functor H_Z is prorepresentable [Har10, Thm. 17.1]. LetN_Z/Pr k

be the normal sheaf of the closed immersion Z ,→ Pr

k. The tangent space of the local Hilbert functor

is t(H_Z) ∼= H0(N_Z/Pr

k) [Har10, Thm. 2.4]. A particular case is that of a k-rational

point P of Pr_k. Its local Hilbert functor is smooth and can be identified with a subfunctor of Pr

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When σ is a k-section of Pr_Λand P is the corresponding k-rational point, we write Hσfor HP. Similarly, we may identify a plane curve X of degree

d over an object A of Ar with an A-section of Pd(d+3)/2_Λ . In particular, the local Hilbert functor H_X of a plane curve over k is a subfunctor of some Pd(d+3)/2_Λ .

Remark 2.14. — Let {Fi}i∈I be a finite family of functors of Artin rings. Then

their product F =Q

i∈IFiin the category of functors Ar → Set is a functor of Artin

rings. Let (H*) be any of Schlessinger’s conditions or smoothness. If each Fi satisfies

(H*), then so does F .

By the above remark and the previous example, the functor H_(X,Σ)df= H_X×Y

σ∈Σ

Hσ

is prorepresentable and smooth.

Equimultiple Hilbert functors. — We are now able to consider equimultiple proper liftings of plane curves in the framework just introduced. Following Wahl again [Wah74b, Sec. 1], we make

Definition 2.15. — The equimultiple local Hilbert functor of (X, Σ) sends an alge-bra A of Ar to

E_(X,Σ)(A)df=(X, Σ) ∈ H_(X,Σ)(A) | For all σ ∈ Σ, X is equimultiple along σ . Remark 2.16. — Let E_X be the image of E_(X,Σ) in H_X. Assume that the sections in Σ correspond to the multiple points of X. Then for any object A of cAr, bE_X(A) is the set of equimultiple proper liftings of X to A.

The equimultiple local Hilbert functor is prorepresentable and – at least in some cases – smooth. To see this, we introduce a scheme that represents a “global equimultiple Hilbert functor”. Let σ be a lifting of σ ∈ Σ to an object A of cAr. Let (x0, x1, x2)

be coordinates of P2

Λ over Λ. If the image of σ is contained in D(xi), then so is

the image of σ. Hence, the ideal sheaf of σ is trivial outside D(xi) and on D(xi) it

corresponds to an ideal generated by xj− tj, j ∈ {0, 1, 2} \ {i}, for some tj ∈ A. As

noted following Definition 2.10, the multiplicity of a plane curve V (F ) over A is given by the least degree among the nonvanishing homogeneous parts of F (1, x1+t1, x2+t2)

(or F (x0+ t0, 1, x2+ t2) or F (x0+ t0, x1+ t1, 1) depending on i). This motivates the

following notation and definition:

Let x = (x0, x1, x2) be fixed coordinates for P2Λ. For each section σ ∈ Σ,

pick a fixed xσ ∈ x such that the image of σ is contained in D(xσ). Let

ˆ

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Definition 2.17. — Pick coordinates λ = (λα)α∈N3_,|α|=d and t_σ= (t_σ,0, t_σ,1, t_σ,2),

σ ∈ Σ, for the multiprojective space Pd(d+3)/2_Λ ×Q

σ∈ΣP 2 Λ. Set F = P αλαx α_.

For each σ ∈ Σ, let fσ ∈ Λ[ ˆxσ][λ, tσ] be the polynomial F (1, x1+ tσ,1, x2+ tσ,2) or

F (x0+ tσ,0, 1, x2+ tσ,2) or F (x0+ tσ,0, x1+ tσ,1, 1) (choose the polynomial that sets

xσ to 1). For each σ ∈ Σ, β ∈ N2, and |β| ≤ mσ− 1, let fσ,β ∈ Λ[λ, tσ] be the

coefficient of ˆxβ_σ in fσ. We set E(d,m) df = V ({fσ,β}σ,β) ⊆ P d(d+3)/2 Λ × Y σ∈Σ P2_Λ.

Even though the notation suggests that the scheme E(d,m)depends only on d and m,

in general it also depends on how the k-rational points specified by Σ are spread out in the projective plane: Unless k is infinite, there may not be coordinates x for P2

Λsuch

that a single D(xi) contains all points specified by Σ. For an example over the field of

2 elements, consider the union of the lines V (x0), V (x1), V (x2) and V (x0+ x1+ x2)

and let Σ be the set of sections corresponding to the multiple points.

Remark 2.18. — Let A be an object of Ar. By the discussion preceding Defini-tion 2.17,

E_(X,Σ)(A) = H_(X,Σ)(A) ∩ E(d,m)(A),

where the intersection is taken inPd(d+3)/2_Λ ×Q

σ∈ΣP 2 Λ

(A).

We now turn to prorepresentability and smoothness of E_(X,Σ) and E_X.

Proposition 2.19. — The functor E_(X,Σ) is prorepresented by the completion bO of the local ring O = O_E

(d,m),(X,Σ).

Proof. — We need to show that the functors HomΛ( bO, −) and E_(X,Σ)are isomorphic.

So let A be an object of Ar and let N be its length. Write RN = R/mNR for any

Λ-algebra R. On the one hand, there is a natural bijection between HomΛ( bO, A) and

HomΛN( bON, A). Moreover, bON andON are naturally isomorphic.

On the other hand, by definition an element of E_(X,Σ)(A) may be interpreted as an A-section of E(d,m) that lifts the unique element of E(X,Σ)(k). In other words, the

elements of E_(X,Σ)(A) correspond bijectively to local morphismsO ON → A of

Λ-algebras. These in turn correspond bijectively to the elements of HomΛN(ON, A).

Lemma 2.20. — If Σ contains at most one section, then E_(X,Σ) is smooth.

Proof. — If Σ is empty, then E_X,Σ equals H_X and we already mentioned that this functor is smooth in Example 2.13. So let Σ = {σ}. Let A0 _{A be a surjection} in Ar and let (X, Σ) ∈ E_(X,Σ)(A). We need to show that there exists an element

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(X0, Σ0) ∈ E_(X,Σ)(A0) that maps to (X, Σ). We may choose coordinates (x, y, z) for P2

Λ such that the ideal sheaf of σ is the ideal generated by x and y in A[x, y]. Let

Pd

i=0Fi(x, y)zi be a defining polynomial of X and Fi(x, y) be homogeneous part of

degree d − i. Let m be the unique member of m, i.e. the multiplicity of X along σ. Since (X, Σ) is a section of E(d,m), the polynomials F0, F1, . . . , Fm−1 vanish. For

i = m, . . . , d, pick a homogeneous lifting F_i0 of Fi to A0. Let X0 be the plane curve

over A0 defined by Pd

i=mF 0

izi and σ0 be the A0-section of P2Λ that corresponds to

the ideal generated by x and y in A0[x, y]. Then (X0, Σ0) is the desired element of E_(X,{σ})(A0).

In general, the question of smoothness seems more complicated. Since E_(X,Σ) maps onto E_Xby definition, the latter functor is smooth whenever the first is. The converse holds when all points corresponding to the sections in Σ are ordinary singularities on X. In fact, we have

Lemma 2.21. — Assume that the point corresponding to the section σ ∈ Σ is an ordinary singularity on X. Then the natural map E_(X,Σ) → E_(X,Σ\{σ}) is injective. In particular, when all sections in Σ correspond to ordinary singularities on X, then the natural surjection E_(X,Σ)_E_X is an isomorphism.

For Λ the power series ring over the complex numbers and X an algebroid curve, the lemma was proven by Wahl [Wah74b, Prop. 1.9]. We follow his proof.

Proof. — Since E_(X,Σ) and E_(X,Σ\{σ}) are both prorepresentable by the previous proposition, it suffices to show that the map on tangent spaces t(E_(X,Σ)) → t(E_(X,Σ\{σ})) is injective [Wah74a, Prop. 1.1.4]. (Wahl’s proof is given in a more restrictive setting than needed here. However, his proof can be used verbatim also in our context.) Let (X, Σ) and (X, Σ0) be elements of t(E_(X,Σ)) that map to the same element of t(E_(X,Σ\{σ})), e.g. Σ = Θ ∪ {σ} and Σ0= Θ ∪ {σ0} and σ, σ0 _{are liftings of σ. Let m}

be the multiplicity of X along σ. We may choose coordinates (x, y, z) of P2_Λ such that σ and σ0 _{correspond to the ideals (x, y) and (x − εa, y − εb) of k[ε][x, y] respectively.}

We may also assume that k is algebraically closed. Then in addition we can choose the coordinates so that X is defined by a polynomial F (x, y, z) such that the degree m part of f (x, y) = F (x, y, 1) is equal to yh(x, y).

We may assume that a and b are elements of k and thus must show that both equal 0. So let F + εG be a defining polynomial of X in k[ε][x, y, z] and write g(x, y) = G(x, y, 1). Since (X, Σ0) is an element of t(E_(X,Σ)), we calculate

f (x+εa, y+εb)+εg(x+εa, y+εb) = f (x, y)+εa∂f

∂x(x, y)+εb ∂f

∂y(x, y)+εg(x, y) ∈ (x, y)

m_.

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But we also know that (X, Σ) is an element of t(E_(X,Σ)), so f (x, y) + εg(x, y) also belongs to (x, y)m_{. Hence,} a∂f ∂x(x, y) + b ∂f ∂y(x, y) ∈ (x, y) m_.

For the degree m part of f (x, y) written as yh(x, y), this means ay∂h

∂x(x, y) + bh(x, y) + by ∂h

∂y(x, y) = 0.

Therefore, y divides bh(x, y). Since σ corresponds to an ordinary singularity, the degree m part of f splits into pairwise distinct linear factors. Hence, y does not divide h(x, y), but b and so b is equal to 0. This in turn implies that a(∂h/∂x)(x, y) vanishes. So either a equals 0 as desired or h is a p-th power. (Use that y does not divide h(x, y).) The latter cannot happen since σ corresponds to an ordinary singularity.

Remark 2.22. — With the notions from the proof, we can identify the tangent space of E_Xwhen Σ contains a unique section: Indeed, letI denote the ideal sheaf in O_P2 k

defined by the ideal (f , ∂f /∂x, ∂f /∂y) + (x, y)m_{⊆ k[x, y]. Recall that there is a short}

exact sequence

0 →OP2

k →I (X) → IX(X) → 0, (2.7)

whereI (X) is the tensor product I ⊗ OP2

k(X). Equation (2.6) implies

t(E_X) ∼= H0(I_X(X)). (2.8)

In general, define an ideal sheaf for each section in Σ as above and let I be their sum. Then an isomorphism as in (2.8) still exists. Note also that this identification is independent of the types of singularities that the elements of Σ correspond to. Definition 2.23. — The ideal sheaf I defined in the remark above is called the equimultiple ideal sheaf for X and Σ. If Σ is the set of all sections of X along which X has multiplicity greater than 1, then we also say thatI is the equimultiple ideal sheaf of X.

Proposition 2.24. — Let I be the equimultiple ideal sheaf. The functor E_X is smooth if H1(I_X(X)) vanishes.

For curves over complex surfaces (in fact, characteristic 0 seems to suffice) Greuel and Lossen show that E_X is smooth already when Im H1₍_{I (X)) → H}1₍_I

X(X))

vanishes [GL01, Prop. 1.3]. In case that no sections are considered, i.e. Σ is the empty set, their statement is the “Severi-Kodeira-Spencer” theorem. A proof is in Mumford’s “Lectures on Curves on an Algebraic Surface” [Mum66, Lec. 23]. We follow this proof.

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Proof. — To begin, let us fix some notation: Let {Ui}i∈Ibe a covering of P2k by open

affines such that each Ui contains the image of at most one section σ ∈ Σ. Let Σi

be the subset of Σ that consists of this unique section or is empty when Ui does not

contain the image of any section. Set Uij = Ui∩ Uj and Uij` = Ui∩ Uj∩ U` for

i, j, ` ∈ I. Note that P2

Ahas the same topological space as P 2

k for any object A of Ar.

Now let A0_{A be a surjection in Ar. We need to show that any element of E}_X(A) lifts to an element of E_X(A0). It suffices to do this when A0_{A is a small extension} whose kernel is generated by a single element η as k-vector space. (Since any surjection in Ar is the composite of finitely many such extensions.) So let X ∈ E_X(A) be given. For each i ∈ I, let E(i)

X be the image of E(X,Σi) in HX. By Lemma 2.20 and the

fact that E_(X,Σ

i) → E

(i)

X is surjective, E (i)

X is smooth. Moreover, there are natural

injections E_X ,→ E(i)

X. Hence, for each i ∈ I, there exists a lifting X 0 i∈ E

(i) X (A

0_{) of X.}

There exists a lifting X0 ∈ E_(X,Σ)(A0) of X if and only if there exist liftings X_i0 ∈ E_X(i)(A0) of X that glue along the Uij. Assume that X = V (F ) for some

homogeneous polynomial F (x, y, z) and for each i ∈ I, let X_i0= V (F_i0), i ∈ I, be any lifting of X in E(i)

X (A

0_{). We may assume that each F}0

i is a lifting of F to A0. Write

f_i0 for the element induced inOP2

A0(Ui) by F

0

i and fi for the image of fi0 inOP2 A(Ui).

Then

f_i0= u0_ijf_j0+ ηh0_ij (2.9)

over Uij for some unit u0ij ∈ OP2 A0(Uij)

∗ _{and h}0

ij ∈ OP2

A0(Uij). We must show that

the f_i0 and u0_ij can be chosen so that the h0_ij vanish. First, we calculate

η(h0_ij+ u0_ijh0_j`) = f_i0− u0_ijf_j0+ u0_ij(f_j0− u0_j`f_`0) = fi0− u0iju0j`f`0

= ηh0_i`+ (u0_i`− u0

iju0j`)f`0.

Given an element a0 or a ofOP2

A0(Uij), let a denote its image inOP 2 k(Uij). From the above we get hij+ uijhj`= hi`+ _u0 i`− u 0 iju0j` η f_`. Moreover, f_i= uijfj. Hence, hij f_i + hj` f_j = hj` f_i + 1 − u0_iju0_j`u0_i`−1 η ! .

Since X_i0 and X_j0 are equimultiple along the unique sections (if any) of Uiand Uj, the

sections hij lie inI (Uij). Therefore, {hij/fi}i,jrepresents an element of H1(IX(X))

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2.3. COMPUTING EQUIMULTIPLE PROPER LIFTINGS 21

Hence, there exist elements gi ∈ I (Ui) such that hij/fi equals gj/fj− gi/fi as

section ofI_X(X)(Uij). Pick any liftings gi0∈OP2

A0(Ui) of the gi. Then f

0

i+ ηg0iis an

equimultiple(!) lifting of fi and

f_i0+ ηg0_i= (u0_ij+ η(hij+ gi0− g 0 ju 0 ij)/f 0 j)(f 0 j+ ηg 0 j).

Since uij is a unit, so is u0ij+ η(hij+ g0i− g0ju0ij)/fj0 and the fi0+ ηg0i glue along the

Uij as desired.

2.3. Computing equimultiple proper liftings

We are now ready to describe an algorithm that computes equimultiple proper liftings of a given plane curve X over k to WN(k). It is an application of formal smoothness

of E(d,m) in neighbourhoods of smooth points.

Since E_(X,Σ) is prorepresented by the completion of O_E

(d,m),(X,Σ), the functor is

smooth if and only if the completion is a power series ring over Λ. This in turn happens if and only if E(d,m) is smooth at (X, Z) [Bou06, Prop VIII.5.1 & Thm. VIII.5.2].

Now let A be an algebra of Ar. We may identify an element of E_(X,Σ)(A) with an A-section of E(d,m). If E(d,m) is smooth at (X, Z), then it is formally smooth in

a neighbourhood of (X, Z) [Gro67, Thm. 17.5.1]. Hence, for any small extension A0 _{A in Ar, there exists an A}0-section of E(d,m) that lifts the A-section [Gro67,

Def. 17.1.1]. This is also sometimes called the infinitesimal lifting property. Proposition 2.25 (Effective infinitesimal lifting property)

Let V be a scheme of finite type over Λ, σ ∈ V (k), and Im σ be a smooth point of V . Given an integer N ≥ 1 and a presentation ΛN[x]/(f ) of an open affine neighbourhood

U of Im σ, Algorithm 1 computes a section σ ∈ U (ΛN) whose special fibre is σ.

Proof. — The case N equal to 1 is trivial. Assume that N is at least 2. Let x = (x1, . . . , xr), f = (f1, . . . , fs), t be a generator of the maximal ideal of Λ, and

n be an integer between 2 and N . A section in U (Λn) corresponds to an r-tuple of

elements of Λn, i.e. an element of the free module Λrn. We need to show that if a

represents a lifting of σ to U (Λn−1) at the beginning of the loop in Algorithm 1, then

a represents a lifting to U (Λn) at the end of the loop.

To find a lifting of a to Λrnthat represents a section in U (Λn) the algorithm proceeds

as follows: First, it picks any lifting a ∈ Λr

n, i.e. a section in Ar(Λn). This section

restricts to an element of U (ΛN) if and only if o = f (a) vanishes. Now any other

lifting of a to Λr

n is of the form a0= a + tN −1∆, ∆ ∈ ΛrN. This element a0 defines a

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Algorithm 1: Applies the infinitesimal lifting property to lift a k-section σ of an affine scheme U ⊆ Ar

ΛN to a ΛN-section if Im σ is a smooth point.

J ← Jacobian(U ) 1 a ← represent in kr_(σ) 2 for n = 2 to N do 3 // _{Choose lifting to A}r_(Λ n) a ← lift(Λn, a) 4

// _{Compute obstruction for lifting to lie in U (Λ}n)

o ← f (a)

5

// _{Modify lifting so that obstruction vanishes} ∆ ← solve(J (a), −o/tn−1₎

6

a ← a + tn−1_{· lift(Λ} n, ∆) 7

// _{Now a represents a lifting of σ to U (Λ}N)

return a

8

if

tN −1J (a) · ∆ = −o,

where J denotes the Jacobian matrix of U . Regard this as an equation in ∆. Both sides are multiples of tN −1_{, so the solutions to the equation are in one-to-one}

corre-spondence to the elements tN −1_{∆ where ∆ is any lifting to Λ}r

n of a solution ∆ to the

equation

J (a) · ∆ = −o/tN −1_. _(2.10)

Since U is formally smooth in a neighbourhood of Im σ, a lifting of a to Λr n that

defines a section in U (Λn) has to exist, i.e. (2.10) has a solution ∆. The algorithm

can now pick any lifting ∆ of ∆ and replace the initial choice of a by a + tN −1∆. By the above, this a represents the desired lifting of σ.

Remark 2.26. — Li describes this algorithm in his thesis “Anwendung deformations-theoretischer Methoden zur Liftung des Frobeniusmorphismus” [Li08] when U is an affine plane curve. In this case, J (a) is a matrix in k1×2_{. Li argues that a solution}

to (2.10) always exists when U is smooth at Im σ since smoothness is equivalent to J (a) being of full rank. In the higher dimensional case of course, J (a) does not need to be of full rank if U is smooth at Im σ. A priori it is not clear that (2.10) has a solution and this is why we argue with formal smoothness.

Theorem A. — Let X be an ordinary, plane curve over k with irreducible compo-nents Xi of degrees di, i = 1, . . . , s, Σ ⊆ P2k(k) be the set of sections that correspond

to the multiple points of X, and I be the equimultiple ideal sheaf of X and Σ. As-sume that the curves Xiare geometrically irreducible. The space H1(IX(X)) vanishes

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2.3. COMPUTING EQUIMULTIPLE PROPER LIFTINGS 23 whenever 3di> X mσ(X)>2 mσ(Xi), for all i.

Before we spent the remainder of this chapter on the proof of this theorem, let us first discuss the result. There is

Corollary 2.27. — In the situation of Theorem A, E(d,m) is smooth at (X, Σ) and

Algorithm 1 (applied to a suitable affine patch of E(d,m)) computes a lifting of (X, Σ)

to ΛN.

Proof. — Since H1₍_I

X(X)) vanishes, EX is smooth (Proposition 2.24). As X is

or-dinary, E_(X,Σ)is smooth as well (Lemma 2.21). So by the arguments at the beginning of this section, the scheme Ed,mis smooth at (X, Σ). So Proposition 2.25 shows that

the algorithm computes a lifting of (X, Σ) to ΛN.

Remark 2.28. — Beerenwinkel also considered the problem of effectively lifting curves. In his Diplomarbeit [Bee99], he derives the same lifting algorithm as ours but with two restrictions: All multiple points of the given plane curve must be nodes and only at most b(d + 1)/2c nodes are allowed (where d is as usual the degree of the curve). This is due to the fact that Beerenwinkel relies on a Hensel-lifting lemma instead of formal deformation theory. Therefore, he needs that the Jacobian of Ed,m

at (X, Σ) has full rank. However, when there are more than b(d + 1)/2c nodes, the rank might no longer be full. (As example consider three nodes on a degree 4 curve that lie on a line).

The first restriction – that all multiple points are nodes – is a mild one from a the-oretical point of view. However, in the context of lifting a smooth curve in practice, it is of course of interest to allow more general (ideally, any) plane images. A more serious restriction is the one on the number of nodes. If there are at most b(d + 1)/2c such points, then the genus formula (2.2) and formula (2.3) force

d2− 4d + 3 ≤ 2g. (2.11)

Hence, the Brill-Noether number ρ(g, 2, d) = g − 3(g − d + 2) is negative whenever the genus is at least 7. In particular, in general it is not clear whether smooth geometrically integral curves of large genus can be mapped birationally to a plane curve so that (2.11) is satisfied. Hence, it is not clear whether Beerenwinkel’s result is sufficient to lift arbitrary curves.

Let us now turn to the proof of Theorem A. We will use the notion of isomorphism defect : LetF and G be coherent sheaves on a curve Y over k and assume that both have the same rank on each irreducible component of Y . In the following, a_e· denotes the reduction of a sheaf modulo torsion. Let Z be a closed subset of Y and denote

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by ·_Z the restriction of a sheaf to Z. The local isomorphism defect ofF in G at a (closed) point P of Z is

isod_Z,P(F , G )df= minlength_O

Z,P

Coker( fF_Z,P −→ eG_Z,P),

where the minimum is over all injective local homomorphisms induced by local homo-morphismsF_P →G_P. The local isomorphism defect is nonzero at only finitely many points of Z and one defines

isod_Z(F , G )df= X

P ∈Z

[k(P ) : k]isod_Z,P(F , G )

as the isomorphism defect ofF in G along Z. When Z is the whole curve, then we drop the Z in the indices. A key lemma for the proof of Theorem A is

Lemma 2.29. — Let Y be a curve on a smooth surface(3) _{over k, Y}

1, . . . , Ys its

irreducible components, ω_Y its canonical sheaf, and F be a coherent O_Y-module of rank 1 on each irreducible component of Y . Then H1(F ) vanishes if for all i = 1, . . . , s

χ( fF_Y

i) > χ(ωY ,Yi) − isodYi(F , ωY).

Proof. — Greuel and Karras’ proof over the complex numbers [GK89, Prop. 5.2] works over any field (the fact that k is perfect is not needed): Assume that H1(F ) does not vanish. Since Y is a curve on a smooth surface, it is a local complete intersection and the canonical sheaf of Y is (isomorphic to) its dualising sheaf [Liu06, Thm. 6.4.32], so there exists a nonzero morphism ϕ :F → ω_Y [Liu06, Rem. 6.4.20]. Since ω_Y is torsion-free, the image of ϕ must have rank 1 on at least one irreducible component Yi of Y . Let ϕ_ei: fFYi → ωeY ,Yi be the map induced by ϕ. Note that

e ω_{Y ,Y}

i equals ωY ,Yi as the latter sheaf is already torsion-free. By the above, ϕei is injective and thus

χ( fF_Y

i) = χ(ωY ,Yi) − χ(Cokerϕei) ≤ χ(ωY ,Yi) − isodYi(F , ωY).

We introduce one final notion that we will use in the following proof: Given a plane curve Y over k with equimultiple ideal sheafI and a closed point P on Y , set

τ_Pem(Y )= lengthdf _O Y ,P(OY ,P IY ,P) and τem(Y )df= h0(O_{Y ,P}I_{Y ,P}) =X P [k(P ) : k]τ_Pem(Y ).

(3)_{A surface over k is a geometrically reduced, separated k-scheme of finite type all of whose}

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2.3. COMPUTING EQUIMULTIPLE PROPER LIFTINGS 25

Proof of Theorem A. — We follow the proof of Greuel and Lossen who consider the case that Λ is the ring of formal power series over the complex numbers [GL96, Cor. 5.1]. To begin, note that the sheafI_X(X) vanishes outside X, so we may regard it asO_X-module when computing its cohomology [Har97, Lem. III.2.10]. Our goal is to apply Lemma 2.29. To this end, note that there is an exact sequence

0 →I_X(X) →O_X(X) →Q → 0

which we may take to defineQ. By construction, the stalk of Q at a closed point P of X is

QP ∼=OX,P

_I

X,P,

Hence,Q is torsion and vanishes outside the singular locus of X. Let us indicate with an index i the restriction of a sheaf on X to Xi. Then Qi is also torsion and (with

the_e·-notation introduced before Lemma 2.29) there is an exact sequence

0 → fI_X(X)i→OX(X)i →Qi→ 0 (2.12)

for all i = 1, . . . , s. SinceO_X(X) is an invertible sheaf, so is O_X(X)i and we deduce

from the exact sequences that I_X(X) is a coherent O_X-module of rank 1 on each irreducible component of X. Let ω_X be the canonical sheaf of X. We apply Lemma 2.29 and find that H1(I_X(X)) vanishes if

χ( fI_X(X)i) > χ(ωX,i) − isodXi(IX(X), ωX) (2.13)

for all i = 1, . . . , s. Let us first consider χ(ω_X,i). Let ω_P2

k be the canonical sheaf

of P2

k. The adjunction formula tells us that ωX,i equals (OP2

k(X) ⊗ ωP2k)|Xi [Liu06,

Thm. 9.1.37]. This in turn equals (OP2

k(Xi)⊗ωP 2 k)|Xi⊗OP2k(X 0_)| Xiwith X = X 0 +Xi. By adjunction again, (O_P2

k(Xi) ⊗ ωP2k)|Xi is isomorphic to the canonical sheaf ωXi

of Xi. For the Euler-Poincar´e characteristics this means

χ(ω_X,i) = χ(ω_X i) + χ(OP2k(X 0_)| Xi) − χ(OXi). Now, χ(OP2 k(X 0_)|

Xi) − χ(OXi) is just the intersection number X

0

· Xi of X 0

and Xi [Liu06, Thm. 9.1.12]. Furthermore, χ(ωXi) equals −χ(OXi) by duality [Liu06,

Rem. 6.4.21] and therefore is equal to pa(Xi) − 1. Putting everything together, (2.13)

is equivalent to

χ( fI_X(X)i) > pa(Xi) − 1 + X 0

· Xi− isodXi(IX(X), ωX). (2.14)

Let us now turn to χ( fI_X(X)i). By(2.12) and since the support ofQiis 0-dimensional,

χ( fI_X(X)i) = χ(OX(X)i) − χ(Qi)

= χ(O_X(X)i) − h0(Qi).

Now, χ(O_X(X)i) equals 1 − pa(Xi) + X · Xi [Liu06, Thm. 9.1.12]. Plugging this into

(2.14) gives

X · Xi− 2pa(Xi) + 2 > X 0

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and by adjunction

X · Xi− (W + Xi) · Xi> X 0

· Xi+ h0(Qi) − isodXi(IX(X),OX),

where W is a canonical divisor of P2

k. This simplifies to

− W · Xi> h0(Qi) − isodXi(IX(X), ωX). (2.15)

Let H be a line on P2

k. Then one may take W as −3H. So B´ezout tells us that the

above inequality is equivalent to

3di> h0(Qi) − isodXi(IX(X), ωX).

We are left to show thatP

m_P(X)>2mP(Xi) is at least h0(Qi) − isodXi(IX(X), ωX).

By Lemma 2.30, the latter equals X m_P(X)>0 τ_Pem(Xi) + i_P(Xi, X 0 ) − isod_X i,P(IX(X), ωX).

By Lemma 2.33 and Corollary 2.31, this simplifies to X m_P(X)>0 τ_Pem(Xi) − δP(Xi) + length (I_X,Pcond(I_X,P)) ⊗O_X i,P ,

where cond(·) denotes the conductor (see below). Clearly, the individual summands are each 0 whenever P is a regular point of X, i.e. whenever m_P(X) equals 1. So, together with Lemma 2.34, the above simplifies once more to

X m_P(X)>1 min{0, m_P(Xi) − 2} + length (I_X,Pcond(IX,P)) ⊗OXi,P .

The theorem is proven with Lemma 2.35.

Lemmas for the Proof. — This subsection collects several lemmas that were used in the proof of Theorem A. To this end, we assume that k is algebraically closed (this is not strictly needed for every lemma). Let Y be a plane, ordinary curve over k, P be a closed point of Y , m be the multiplicity of Y at P , I the equimultiple ideal sheaf of Y , and ω_Y be the canonical sheaf of Y . In the application to the proof, Y will play the role of X or Xi. Let x and y be local coordinates P2k at P . We may

and do assume that Y is defined at P by f ∈ k[x, y] and that x does not divide the degree m part of f .

Lemma 2.30. — In the situation of the proof Theorem A, let P be a closed point of X. Then

length(Q_i,P) = τ_Pem(Xi) + i_P(Xi, X 0

).

Proof. — If P is not a multiple point of X, then the three cardinals in each of the two equations are each equal to 0. So let P be a multiple point of X. By our earlier assumption in Section 2.1, the degree of P is 1, so we find defining elements

Frobenius-stable lattices in rigid cohomology of curves