A birational journey: From plane curve singularities to the Cremona group over perfect fields

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From plane curve singularities to the Cremona group over perfect fields

Inauguraldissertation zur

Erlangung der W¨urde eines Doktors der Philosophie vorgelegt der

Philosophisch-Naturwissenschaftlichen Fakult¨at der Universit¨at Basel

von

Julia Noemi Schneider

2021

Originaldokument gespeichert auf dem Dokumentenserver der Universit¨at Basel

edoc.unibas.ch

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auf Antrag von

Prof. Dr. Jérémy Blanc, Prof. Dr. Philipp Habegger, Prof. Dr. Serge Cantat und Prof. Dr. Stefan Schröer.

Basel, den 17. November 2020

Prof. Dr. Martin Spiess

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Acknowledgements

I am immensly grateful to all the people that have been accompanying me on the journey of my PhD. You are all responsible for making the years of my PhD a wonderful time that I have been enjoying thoroughly. As my defense took place virtually and I did not get the chance to give a speech, I would like to take this opportunity to thank each and every one of the almost one hundred (!) people that were watching online from their offices, from home, from Switzerland or from France, Italy, Spain, the UK. All my colleagues, friends, all the Schneiders and the Saladins, my students, and all the people that were by chance in the same room with someone watching and taking an interest in the turtle.

Most of all, I thank Jérémy Blanc for his supervision and the enormous amount of time he spent mentoring me, for the ideas that he shared with me and the research questions he proposed. I also thank Philipp Habegger for being my second adivsor, and Serge Cantat as well as Stefan Schröer for accepting to be the experts for my defense.

I want to thank my doctoral siblings Susanna Zimmermann, Christian Urech, Mattias Hemmig, Aline Steiner, Pascal Fong, Sokratis Zikas, Anna Bot and our doctoral half brother Philipp Mekler, as well as the other members of our group, Enrica Floris, Immanuel van Santen, Anne Lonjou, Egor Yasinski and Pierre- Marie Poloni. In particular, I want to thank Susanna for the many discussions we had and will continue to have, I thank Christian, Mattias and Aline for the many fun moments and discussions during the first part of my PhD, and Pascal, Sokratis and Egor for making the second half of my PhD great again with Carbonara, Pizza, Tonic and mathematics.

Moreover, I want to thank all the other members of the department that I was lucky enough to get to know, among them Gabriel, Linda, Francesco, Fabrizio, Stefan, Jung-Kyu, Marta, Gerold, Harry, Richard, Robert, Myrto, as well as Christine and Annette, and the crew from the 2nd floor, Leon, Pascal, Ramon, and Yannik. A special thanks goes out to the secretary Barbara Fridez

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Lastly, I thank my family for the constant support. I thank my parents for being proud of the texts with many symbols and fancy words that I have produced. I thank my brother Andreas for many discussions and for making me the aunt of my beautiful niece. Also, I congratulate my parents for having

Rpmy parentsq “1,

where RpPq denotes the ratio of the number of children of the parentsP with a PhD to the total number of children ofP.

Finally: Merci Kevin!

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Introduction

The Cremona group Cr_npkq of rank n over the field k is the group of field automorphisms overkof the field of rational functionskpx1, . . . , xnq. This group can be identified with the group of birational transformations of the affine space Aⁿk or the projective space Pⁿk. We take the last viewpoint on the Cremona group and write Cr_npkq “ Bir_kpPⁿq. A birational map of the projective space Pⁿ over a fieldkis a rational map f: Pⁿ99KPⁿ given by

px₀:¨ ¨ ¨:x_nq ÞÑ pf₀pxq:¨ ¨ ¨:f_npxqq,

where f₀, . . . , f_n Pkrx0, . . . , x_ns are homogeneous polynomials of the same degree without common factors, such that there is an inverse map of the same shape.

Some of the driving questions to study the Cremona groups are: How can it be described in terms of generators and relations? Does it have proper normal subgroups? In general, the answers depend on the rank and on the field.

As all birational maps of the projective line are automorphisms, the Cremona group of rank 1 is PGL2pkq. The plane Cremona group (that is, the Cremona group of rank 2) contains more interesting birational maps, for example the standard quadratic transformation, which is the map given by px : y : zq ÞÑ pyz : xz : xyq. Over algebraically closed fields such as the complex numbers the plane Cremona group is well understood. The classical result of Noether and Castelnuovo [Noe70,Cas01] states that Cr₂pkq is generated by the group

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formation. After Gizatullin described a complete set of relations [Giz82], more presentations of the Cremona group with respect to generators and relations followed [Isk85,Wri92,Bla12,Zim16b,UZ19]. Already in 1899, Kantor gave a list of sixteen types that generate the Cremona group over the field of rational numbers [Kan99]. In the 1990’s, the Russian school around Iskovskikh have studied this question for perfect fields that are not necessarily algebraically closed, and they provided a list of generators [Isk91] and a list of relations [IKT93]. How- ever, since they studied generators and relations in the Cremona group itself, the list is very long and hard to handle. Instead of only considering elements of the group itself, the Sarkisov program allows to decompose a birational map between Mori fiber spaces (simple fibrations) into Sarkisov links (simple birational maps). Iskovskikh described all Sarkisov links for n “ 2 and perfect fields [Isk96]. Over C, the Sarkisov program is established in any dimension:

For n “ 3 it was proved in [Cor95], and in any dimension in [HM13] using the Minimal Model Program. So Sarkisov links form a generating set of the groupoid that consists of birational maps between Mori fiber spaces. In arbi- trary dimension over C, elementary relations have been described in [BLZ19]

(see also [Kal13]), and in dimension 2 over perfect fields in [LZ19].

The question whether the Cremona groups contain non-trivial normal subgroups was already asked by Enriques in 1895, who guessed the answer is no [Enr95]. In 2013, Cantat and Lamy showed that the answer is yes for the plane Cremona group over algebraically closed fields, using the action of the Cremona group on an infinite dimensional hyperbolic space and methods from geometric group theory [CL13]. Lonjou adapted this result for all fields in dimension 2 [Lon16]. So the plane Cremona groups are not simple groups.

Zimmermann has proved that the abelianisation of the real Cremona group is isomorphic to an uncountable sum ofZ{2Z[Zim18]. Then, Lamy and Zimmer- mann used the approach via Sarkisov links for many perfect fields that are not algebraically closed: They constructed a surjective group homomorphism from the plane Cremona group to a free product ofZ{2Z, implying non-perfectness of the group. As a consequence, non-simplicity of the Cremona group is recov- ered [LZ19]. As they need the existence of Bertini involutions, their method requires the existence of a Galois orbit of size 8. Note that as a consequence of the Theorem by Noether and Castelnuovo, over algebraically closed fields all group homomorphisms from Cr₂pkqto an abelian group are trivial, that is, the plane Cremona group is perfect. With the idea in mind that the plane Cremona group over perfect fields “behaves like” high dimensional Cremona groups over

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the complex numbers (Galois orbits of points vs. covering gonality), Blanc, Lamy and Zimmermann constructed a surjective group homomorphism from Cr_npkqto a free product of direct sums ofZ{2Z, whereně3 andkis any sub- field of the complex numbers [BLZ19], proving that the Cremona group in high dimensions is not a simple group. The strategy of [BLZ19] fails in dimension 2:

The constructed group homomorphism is trivial in this case.

This thesis travels through different topics in algebraic geometry, where the common denominator are birational maps between rational surfaces. Before we start exploring the structure of the plane Cremona group, we use birational maps as a tool to study plane curve singularities in ChapterIII, Plane curves of fixed bidegree and their A_k-singularities. We are interested in the following question: For a given d, what is the maximalk such that there exists a curve of degree din the complex affine planeA²pCq that has anAk-singularity? We provide a tool how one can view a polynomial on the affine plane of bidegree pa, bq– by which we mean that its Newton polygon lies in the triangle spanned bypa,0q,p0, bqand the origin – as a curve in a Hirzebruch surface having nice geometric properties. As an application, we study maximal Ak-singularities of curves of bidegree p3, bqand find the answer for bď12. We embedd the curve in a well chosen Hirzebruch surface and resolve the singularities via elementary links. On the way, we discuss connections to knot theory. This chapter is published, see [Sch20b].

In ChapterIV,Relations in the Cremona group over perfect fields, we start to study thestructureof the Cremona group of rank 2, or more precisely, of the groupoid mentioned above. We leave the complex numbers and all algebraically closed fields behind and focus on perfect fields that are not algebraically closed.

The focus lies on birational maps whose base points form large Galois orbits, which is the analogue of the notion of covering gonality used in [BLZ19]. For this, we introduce the notion of Galois depth of a birational map. We provide relations of the groupoid, and with these we construct a group homomorphism from the plane Cremona group to a free product of direct sums of Z{2Z. For perfect fieldsk satisfyingrk:ks ą2, we prove that the group homomorphism is surjective and obtain new normal subgroups of the plane Cremona group. We follow the melody of the recent proof by Blanc, Lamy and Zimmermann that the Cremona group of rank n over (subfields of) the complex numbers is not simple forně3. In this way we get an elementary proof of non-simplicity of the plane Cremona group over many fields, without using any modern machinery.

This extends (a part of) the result of Lamy and Zimmermann to a larger class

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the ones obtained in [LZ19].

In Chapter V, Generators of the plane Cremona group over the field with two elements, we radicalise us even further: We work over the finite field with two elements,F², and describe generators of the plane Cremona group overF² explicitely. It is generated by three infinite families and finitely many birational maps with small base orbits: One family preserves the pencil of lines through a point, the other two preserve the pencil of conics through four points that form either one Galois orbit of size 4, or two Galois orbits of size 2. For each family, we give a generating set that is parametrized by the rational functions overF². Moreover, we describe the finitely many remaining maps and give an upper bound on the number needed to generate the Cremona group. Finally, we show that the plane Cremona group overF² is generated by involutions.

In ChapterII,Preliminaries, we give a selection of preliminary results that will be used in the coming chapters. Each of the ChaptersIII,IV, andVcontains an introduction and preliminary results specific to the topic. The numbering of the theorems, lemmas etc. is consistent with the published article [Sch20b], respectively the versions on arXiv [Sch19,Sch20a].

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Preliminaries

Our focus lies on smooth projective surfaces that are geometrically rational.

In fact, all surfaces in this thesis will be smooth, with one exception (see Lemma V.3.18).

Letk be a field and let sk be its (fixed) algebraic closure. A mapX Ñ Y between two varieties over k is always assumed to be defined over k if not mentioned otherwise. We denote rational maps X 99KY by dashed arrows in order to highlight that these are not maps in the usual sense as they are not defined everywhere. Recall that a variety X over a field k is rational if there exists a birational map from X to a projective spacePⁿ that is defined overk, and it isgeometrically rational if it is rational over the algebraic closure sk.

On a smooth projective varietyX, the notions of Weil and Cartier divisors coincide [Liu02, Proposition 7.2.16], so we will only speak of divisors. The set of divisors will be denoted by DivpXq. Two divisorsC andD onX arelinearly equivalent, denoted byC„D, if there exists a rational functionf PkpXqonX such thatD“C`divpfq, where divpfq “ pfq0´ pfq8withpfq0corresponding to the zeroes andpfq₈to the poles off. The Picard group Pic_kpXqis the group of divisors (defined overk) modulo linear equivalence.

On surfaces, where divisors are curves, we have the notion of theintersection number of two divisors:

Theorem 1 ( [Har77, Chapter V, Theorem 1.1]). Let S be a smooth projective surface over an algebraically closed field. There is a unique pairing DivpSq ˆ

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DivpSq ÑZ, denoted byC¨D for any two divisors C, D, such that (1) C ¨D “ ř

pPSIppC, Dq if C and D are distinct prime divisors, where I_ppC, Dqdenotes the local intersection ofC andD atp,

(2) C¨D“D¨C,

(3) pC₁`C₂q ¨D“C₁¨D`C₂¨D, and (4) if C1„C2, thenC1¨D“C2¨D.

OnP², the intersection number of two curves is given by the product of their degrees and the above statement is the classical Theorem of B´ezout.

1. Blowing up one point

As we concentrate on surfaces in this thesis, only blow-ups of points are con- sidered. For simplicity, assume here thatk is algebraically closed. Letp“ r1 : 0 : ¨ ¨ ¨ : 0s PPⁿ. We take coordinates rx₀ :. . . : x_ns in Pⁿ, and ry₁ :¨ ¨ ¨ : y_ns in P^n´1 and denote by BlpPⁿ the zero set of xiyj “xjyi fori, j “1, . . . , n in PⁿˆP^n´1. We say that the projectionπ: BlpPⁿ ÑPⁿ is theblow-up ofPⁿ at p. The strict transform Xr of a quasi-projective varietyX ĂPⁿ is the closure π^´1pXztpuq in Bl_pPⁿ, intersected with the preimage π^´1pXq. If p P X is a smooth point, we say that theblow-up of X atpis the restrictionπ|

XĂ:Xr ÑX (see also [Sha13, Theorem 2.15]). To blow up a different point q we make a linear change of coordinates ofPⁿ sendingqtop. One can show that blow-ups of points are unique up to isomorphism, so we can speak aboutthe blow-up of X atp.

Letπ: Y ÑX be the blow-up ofX at a smooth pointpPX. The preimage E“π^´1ppq ĂY is called theexceptional divisor. Note thatπ|_XzE: BlpXzEÑ Xztpu is an isomorphism. Hence, the blow-up of a point is a birational map.

Similar to above, the strict transform ĂW of a subvarietyW ĂX is the closure π^´1pWztpuq in Bl_pX, intersected with the preimage π^´1pWq. In particular, WĂ»W ifpRW.

Example 1.1. Taking the neighbourhood x₀ ‰0 in Pⁿ of p“ r1 : 0 :¨ ¨ ¨ : 0s, which is isomorphic toAⁿ, we see that locally the blow-up ofpis the projection π: Bl0AⁿÑAⁿ, where

Bl0Aⁿ“

!`

px1, . . . , xnq,ry1:¨ ¨ ¨:yns˘

|xiyj“xjyi @i, j )

ĂAⁿˆP^n´1. In the chart y₁ “1, we have x_i “x₁y_i and so locally, we can write the blow- up as a map Aⁿ Ñ Aⁿ given by px1, y2, . . . , ynq ÞÑ px1, x1y1, . . . , x1ynq. The exceptional divisor is thenx₁“0.

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Lemma 1.2( [Har77, Chapter V, Proposition 3.1]). Let S be a smooth projective surface and letpPS be a point. Letπ:SrÑS be the blow-up ofSatpwith exceptional divisorE. Then, Sris a smooth projective surface. The exceptional divisorE is isomorphic toP¹ and has self-intersection E²“ ´1.

Note that the following lemma holds for all smooth projective surfaces (see [Har77, Chapter V, Proposition 3.6]), however, since the points that we blow up always have a neighbourhood isomorphic to the affine plane, we only prove it in this case.

Lemma 1.3. Let S be a smooth projective surface and pPS a point. Assume that there exists a neighbourhood of pisomorphic toA². Letπ:S¹ ÑS be the blow-up at p with exceptional divisor E, and let C be a divisor on S. Then π^˚pCq “Cr`mppCqE, wheremppCqis the multiplicity ofC atp.

Proof. Letx, ybe local parameters inA²atp. So we can writeCas the zero set of the polynomialPmpx, yq ` ¨ ¨ ¨ `Pdpx, yq, wherePiPkrx, ysiare homogeneous polynomials of degree i and m “ m_ppCq. Locally, the blow-up is given by px, yq ÞÑ px, xyqand so the pre-image is given by the zero set of

Pmpx, xyq ` ¨ ¨ ¨ `Pdpx, xyq “x^mpPmp1, yq ` ¨ ¨ ¨ `x^d´mPdp1, yqq.

Sincex“0 is the exceptional divisor, the result follows.

Lemma 1.4. Let S be a smooth projective surface. Let π: S¹ Ñ S be the blow-up at pPS with exceptional divisorE. The following holds:

(1) PicpS¹q “π^˚pPicpSqq ‘ZE,

(2) π^˚pC1q ¨π^˚pC2q “C1¨C2, whereC1,C2 are two divisors on S, (3) E¨π^˚pCq “0 for divisorsC onS,

(4) Cr¨E“mppCq,

(5) Cr1¨Cr2“C1¨C2´mppC1qmppC2q.

Proof. Part (1) is [Har77, Chapter V, Proposition 3.2]. Part (2) and (3) are general properties of a birational regular map between smooth projective surfaces [Sha13, Theorem 4.7]. The other parts are [Sha13, Corollaries 4.2 and 4.3].

Lemma 1.5 ( [Har77, Proposition 3.3]). Let S be a smooth projective surface.

Let π:S¹ ÑS be the blow-up at p PS with exceptional divisor E. Then, the canonical divisor K_S1 is linearly equivalent to π^˚pK_Sq `E.

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The blow-up of a point is a particularly interesting example of a birational map. In fact, each birational map between (smooth projective) surfaces can be described in terms of blow-ups of points.

Theorem 2( [Sha13, Theorem 4.9] ). Letkbe an algebraically closed field. Let ϕ: S99KS¹ be a birational map between smooth projective surfaces. Then there exists a smooth projective surface T and two sequences of blow-ups of points π:T ÑS andπ¹:T ÑS¹ such thatϕ^˝π“π¹.

We say thatπ:T ÑSandπ¹:T ÑS¹are aresolutionof the birational map ϕ. We say that the resolution isminimal if for each other resolutionτ:T¹ÑS, τ¹: T¹ Ñ S¹ of ϕ there exists a surjective morphism ω: T¹ Ñ T such that τ “π^˝ω and τ¹ “π¹^˝ω. Taking a minimal resolutionπ, π¹ ofϕand writing π “ π1^˝¨ ¨ ¨^˝πn as a sequence of blow-ups πi: Si`1 Ñ Si of points pi P Si, we say that the pointspi PSi are thebase points of ϕ, and we denote the set of base points by Baspϕq. Note that two minimal resolutions only differ by an isomorphism ofT and so the set of base points is well defined.

Theorem 2is a corollary of the following two theorems:

Theorem 3(Elimination of points of indeterminacy [Sha13, Theorem 4.8]). Let S be a smooth projective surface and ϕ: S 99KPⁿ a rational map. Then there exists a sequence π:T ÑS of blow-ups of points and a morphism ψ: T ÑPⁿ such that the rational map ϕ^˝πis the morphism ψ.

Theorem 4(Factorization Theorem [Sha13, Theorem 4.10]). Letπ¹:T ÑS¹ be a birational morphism between smooth projective surfaces. Thenπ¹is a sequence of blow-ups of points.

Example 1.6. Letkbe any field. Thestandard quadratic transformation is the involutionσ:P²99KP²given by

rx:y:zs99Kryz:xz:xys.

This amounts to blowing up the three coordinate points r1 : 0 : 0s, r0 : 1 : 0s, r0 : 0 : 1s and then contracting the strict transforms of the three lines through each pair of points (that is,x“0,y“0, andz“0). A resolution ofσis given by σ“p₂^˝p^´1₁ , where p₁ and p₂ are the projections S Ñ P² of S ĂP²ˆP² given by

S“ tprx:y:zs,ru:v:wsq PP²ˆP²|xu“yv“zwu.

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2. Application of blow-ups: Ak-singularities

Before we continue, we show an application of blow-ups to plane curve singularities. In this section, we consider curves on smooth surfaces over the complex numbers. LetC be a curve on a smooth complex surfaceS. A point pPC is called asingularity of type A_k for some integerkě1 if there are local analytic coordinates in which C aroundp is given by the equationy²´x^k`1 “0. We have the following description of A_k-singularities using blow-ups that will be used extensively in ChapterIII(see Lemma III.2).

Lemma 2.1. Let π:S¹ Ñ S be the blow-up of a smooth complex surface S centered at the point pP S with exceptional divisor E ĂS¹. Let C Ă S be a curve reduced at p and let Cr Ă S¹ be its strict transform. The following are equivalent:

(1) mppCq “2 (2) Cr¨E“2

(3) C has anAk-singularity at pfor some kě1.

Moreover, if the above conditions hold, then also the following statements hold:

(a) CrXE contains two distinct points if and only ifk“1.

(b) CrXE“ tp¹uwherep¹ PCr is smooth if and only ifk“2.

(c) CrXE“ tp¹uwhere p¹PCr is a singular point of typeAk´2 if and only if kě3.

Furthermore, in case (b)the exceptional divisor E andCr are tangent at p¹. Proof. The statements(1)and(2)are equivalent because of(4)in Lemma1.4, and “(1) ùñ (3)” is [Wal04, Theorem 2.2.7]. As an Ak-singularity is locally given by y²´x^k`1 “0, which is of multiplicity 2 in the origin, statement(3) implies(1).

We assume now that conditions(1)to(3)hold. It is enough to considerCin a neighbourhood ofp0,0q PA², where it is given by the equationy²´x^k`1“0.

On one chart, the blow-up is given by η: A² Ñ A²,px, yq ÞÑ pxy, yq, so the exceptional divisor E is defined by y “ 0. The preimage η^´1pCq is given by y²p1´x^k`1y^k´1q “ 0, hence the strict transform Cr does not intersect the exceptional divisorE on this chart. On the other chart, the blow-up is given by π:A²ÑA²,px, yq ÞÑ px, xyq, so the exceptional divisorE is defined byx“0.

The preimageπ^´1pCqis given byx²py²´x^k´1q “0, hence the strict transform Cr is given byy²´x^k´1“0, which corresponds to anAk´2-singularity ifkě2.

Ifkě3, thenk´2ě1, so it is a singular point and we have(c). Ifk“2, then

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k´2“0, so it is a smooth point and we have(b). Ifk“1, we have thatCr is given byy²´1“0, so the exceptional divisor intersects the exceptional divisor E at two points, namely atp0,1qandp0,´1q, and we have(a).

Since the three cases (a),(b)and(c) cannot occur simultaneously, we have proved the “if and only if”-statements in(a),(b), and(c).

To conclude the proof, note that in case(b)there is only one point onCrXE but by(2) we haveCr¨E“2. Thus,Ip¹pC, Eq “r 2, so E andCr are tangent at p¹. This achieves the proof.

3. The absolute Galois group over a perfect field

A fieldkisperfect if every algebraic extension ofkis separable, or equivalently if every irreducible polynomial overkhas distinct roots, or equivalenty ifkhas either characteristic zero or positive characteristicpand every element ofkhas a p-th root. Many fields are perfect: All fields of characteristic 0, all algebraically closed fields and all finite fields are perfect, and algebraic extensions of a perfect field are again perfect. Not perfect is, however, the field of rational functions over a finite field. A field extension L{k is called Galois if it is normal and separable.

Assume from now on thatkis perfect. This assumption implies in particular thatsk{kis a Galois extension, whereskdenotes the algebraic closure ofk. Then the(absolute) Galois groupGof kis

Galpsk{kq “Autkpskq.

The absolute Galois group G acts on sk and satisfies sk^G “ k, so the field k is exactly the fixed field under the Galois action on sk. The Galois orbit of an elementαPskoverk is the set ofd(distinct) roots of the minimal polynomial minkpαqofαoverk, wheredis the degree of minkpαq.

The absolute Galois group can be very complicated, for example, the absolute Galois group over the rational numbers is an active area of research in number theory. Note that the action of the absolute Galois group on αis determined by the action of the Galois group GalpL{kq, where L is the splitting field of minkpαqover k. In particular, to determine the Galois orbit of αit is enough to consider a finite extension ofk.

Example 3.1 (Galois action on finite fields). Let q“pⁿ be some power of the prime pand consider the finite field F^q. Let αPFs^q with minimal polynomial f over F^q of degree d and denote by L the splitting field of f over F^q. The Galois group of a finite extension of F^q is the cyclic group generated by the

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Frobenius endomorphism x ÞÑ x^q [Cox12, Theorem 11.1.9]. Hence, the roots α“α1, . . . , αd off can be ordered asαi`1“α^q_i, hence

αi“α^q^i´1

fori“1, . . . , d. In particular, the splitting field isL“F^qpαq. SinceL{F^q is an extension of degree dand finite fields have a unique extension of degree d, we also have L“Fq^d. SoF^qpαq “Fq^d.

4. Geometry over perfect fields

Letkbe a perfect field. The action of the absolute Galois groupG“Galpsk{kq onskinduces an action on the projective spacePⁿ (via coordinates) and on the ring of polynomialskrx0, . . . , x_ns(via coefficients): ForσPGthe action on Pⁿ is given by

p“ rx0:¨ ¨ ¨:xns ÞÑσppq “ rσpx0q:¨ ¨ ¨:σpxnqs, and the action on krx0, . . . , xnsis

f “ÿ

a_Ix^I ÞÑf^σ“σ^˝f^˝σ^´1“ÿ

σpa_Iqx^I, where I“ pi₀, . . . , i_nqandx^I “xⁱ₀⁰¨ ¨ ¨xⁱ_nⁿ.

We say that a subsetX ofPⁿpskq(orS a set of polynomials with coefficients in sk) isdefined overkifσpXq “X (orf^σPS for allf PS) for allσPG.

Remark 4.1. Consider the projective plane with coordinatesrx:y:zs. Assume that the pointsp_i“ r1 :α_i:β_is PP²pkqs fori“1, . . . , dform a Galois orbit over a perfect field k. Hence, the αi respectively βi are the roots of an irreducible polynomial f_αrespectivelyg_β. We can writep_i as the zero setVpf_i, g_iq, where fi“αix´y,gi“βix´z. We can write the Galois orbit as the zero setVpHq, whereH consists of all polynomialsh₁¨ ¨ ¨h_d, whereh_i P tf_i, g_iufori“1, . . . , d.

Example 4.2. Letk“Rork“Q. The set of the two pointsp1“ r0 : 1 : is,p2“ r0 : 1 :ísis Galois invariant. We can writep₁“Vpx, zíyq,p₂“Vpx, zìyq, and so the union of the two points is given by the zero setVpx, y²`z²q.

Example 4.3. LetωPCbe a root of the irreducible polynomialx²`x`1PQrxs, hence it is a 3rd root of unity. The Galois group of Qpωq{Qis generated by σPAut_QpQpωqqthat mapsωontoω²“ ´ω´1. The two pointsp₁“ r1 :ω:ω²s andp2“ r1 :ω²:ωs form a Galois orbit. As in Remark4.1, set

h₁“x²`xy`y², h₂“x²`xz`z²,

h₃“ω²x²´ωxpy`zq `yz, h₄“ωx²´ω²xpy`zq `yz,

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and so the orbit can be written as the the zero setVph₁, h₂, h₃, h₄q. Note that h1, h2 are defined over Q, and thath^σ₃ “ h4, h^σ₄ “ h3. So the set of the four polynomials is defined overQ.

Lemma 4.4. The action of G “ Galpsk{kq on the zero set Vpfq Ă Pⁿpskq of a homogeneous polynomial f P skrx0, . . . , xns is given by σpVpfqq “ Vpf^σq, implying that Vpfq is defined over k exactly if there exists λ P ks such that λf Pkrx0, . . . , xns.

Proof. We writef “ř

aIx^I. LetxPVpfqand set y“σpxq. So f^σpyq “ÿ

σpaIqσpxq^I “σ´ÿ aIx^I

¯

“σp0q “0.

Hence, y P Vpf^σq. Conversely, lety P Vpf^σq and set x “σ^´1pyq. Note that σ^´1 exists becauseσPAut_kpskq. By assumption,f^σpyq “0. Hence

0“σ^´1p0q “σ^´1pf^σpyqq “ÿ

aIσ^´1pyq^I “fpxq, and soy“σpxq PσpVpfqq.

Ifλf Pkrx0, . . . , x_nsthen for allσPGwe have that

σpVpfqq “σpVpλfqq “Vppλfq^σq “Vpλfq “Vpfq,

henceVpfqis defined overk. Conversely, letf Pkrxs 0, . . . , xnsand assume that Vpfqis defined overk. By multiplication with an element inskwe can assume that one of the coefficients off equals 1. HavingVpfq “σpVpfqq “Vpf^σqfor allσPGimplies that there existsλPsksuch thatf “λf^σ. The coefficient off that is 1 stays 1 under the action ofσ, soλ“1 and hencef “f^σ for allσPG, and sof is defined overk.

Lemma 4.4gives us a corollary about base points of a rational map:

Lemma 4.5. Let ϕ:Pⁿ99KP^m be a rational map defined overk, that is, ϕis given by

rx₀:¨ ¨ ¨:x_ns Ñ rf₀px₀, . . . , x_nq:¨ ¨ ¨:f_mpx₀, . . . , x_nqs,

where f_i P krx0, . . . , x_ns are homogeneous polynomials of the same degree for i“0, . . . , m. Assume that pPPⁿpskq is a base point of ϕ. Then each point of the Galois orbit of pis a base point ofϕ.

Proof. LetσPGalpsk{kq. SincepPBaspϕq, we havepPVpfiqfori“0, . . . , m.

Hence,σppq PσpVpf_iqq “Vpf_i^σq, by Lemma 4.4. Since the polynomialsf_i are defined over k, this is the zero set offi. Hence, σppq PVpfiqfor i“0, . . . , m and so it is a base point ofϕ.

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Recall that to determine the Galois orbit it is enough to consider the action on a finite extension. For example, the smallest field over which a point p“ r1 :a₁:¨ ¨ ¨:a_ns PPⁿpskqis defined isL“kpa1, . . . , a_nq, and its Galois orbit is completely determined by GalpL{kq.

Example 4.6. (1) The Galois orbit ofr0 : 1 : is overQ(or R) istr0 : 1 :˘isu, and the Galois group acts on this orbit likeZ{2Z.

(2) OverQ, the Galois orbit ofr1 :?

2 : isconsists of the four pointsr1 :˘? 2 :

˘is, and the Galois group acts on pas Aut_QpQp?

2,iqq »Z{2ZˆZ{2Z. The Galois orbit can be written as the zero setVp2x²´y², x²`z²q.

(3) The minimal polynomial of ?³

2 over Q is X³´2, so its Galois orbit is t³

?2,?³ 2ω,?³

2ω²u, where ω is a primitive 3rd root of unity. The Galois orbit of the point r1 : ?³

2 : ?³

4s is of size 3 (it also contains r1 : ?³ 2ω :

?3

4ω²sandr1 :?³ 2ω²:?³

4ωs), and the action of the Galois group on this orbit is isomorphic to Sym₃.

Remark 4.7. Note that the last of the above examples gives a Galois orbit of size 3 such that the three points lie on the irreducible conic y²´xz “0, and so they are not collinear. More generally, the rootsα₁. . . , α_d of an irreducible polynomial in krxs of degree ddirectly give a Galois orbit of size d such that no three points are collinear, namely the points r1 : α_i : α²_is form an orbit.

Conversely, one can show that with a change of coordinates defined overk, any Galois orbit of size 4 with no three of its points collinear can be written as r1 :αi:α²_iswhereαi are the roots of an irreducible polynomial of degree 4 (see Lemma IV.6.15).

Example 4.8 (Geometry over finite fields, see also Example3.1). As the action of the Galois group over a finite fieldF^q is cyclic, the Galois orbit of the point p“ r1 :a:bsin P²pFs^qqis the (finite) settr1 :a^qⁱ :b^qⁱs |iPNu. Let d₁ be the size of the orbit of a in sk, and d2 the one of b. Then, the size of the orbit of p is the least common multipled ofd₁ andd₂. As we have F^qpaq “Fq^d1 and F^qpbq “Fq^d2, andF^qpa, bqis the smallest field that contains both fields, we find that F^qpa, bq “F^q^d. Therefore, the size of the orbit ofpis exactly the power d ofq such thatF^qpa, bq “Fq^d.

Observe that in Example 4.6,(2) we had two elementsa, bPQs with orbits of size 2 such that the orbitp“ r1 :a:bsis of size 4‰lcmp2,2q

5. Blowing up a Galois orbit of points

Over a perfect field, we can consider Galois orbits as closed points. Let X be a smooth projective surface defined over a perfect field k. So the blow-up

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π:Y Ñ X of a Galois orbit of size n in Xpskq is defined over k, which is a sequence of blow-ups of points defined over sk. The exceptional divisor ofπ is E“E₁` ¨ ¨ ¨ `E_n, whereE_iis the exceptional divisor of the respective blow-up of one point (defined oversk). Note that thep´1q-curvesE1, . . . , En onY form a Galois orbit. In particular,E is Galois invariant. The irreducible components Ei ofE are pairwise disjoint, soE²“ ´n.

In this setting, by replacing “blow-up of a point” by “blow-up of a Galois orbit” we can recover many statements from Section1. First, we show that the rank of the Picard group increases by one when blowing up one Galois orbit of points.

Lemma 5.1. Let S be a smooth projective surface over a perfect field k. Let π:S¹ Ñ S be the blow-up at a Galois orbit of size n with exceptional divisor E“E₁` ¨ ¨ ¨ `E_n. Then,Pic_kpS¹q “π^˚pPic_kpSqq ‘ZE.

Proof. Applying Lemma1.5ntimes, we have over the algebraic closure Pic_k_spS¹q “π^˚pPic_s_kpSqq ‘ZE1‘ ¨ ¨ ¨ ‘ZEn.

The absolute Galois group Gacts transitively on the Galois orbit E1, . . . , En. Hence there existsσPGsuch thatσpEiq “E1. Therefore, ifD¹“D`a1E1`

¨ ¨ ¨ `anEn is a divisor on S¹ defined overk(so it is Galois-invariant), we have a_i“a₁for alli“1, . . . , n.

Now, we recover the Factorization Theorem (Theorem4) and the elimination of indeterminacy points (Theorem3), and obtain an analogue of Theorem2.

Theorem 5(Elimination of points of indeterminacy). Letk be a perfect field.

Let S be a smooth projective surface and ϕ:S 99K Pⁿ a rational map over k.

Then there exists a sequence π: T ÑS of blow-ups of Galois orbits such that the rational mapϕ^˝π:T 99KPⁿ is a morphism.

Proof. From Theorem3 we obtain a sequenceπ: T ÑS of blow-upsπi:Si Ñ S_i`1of points defined oversksuch thatϕ^˝π:T ÑPⁿis a morphism defined over sk. The points blown up byπare the points in the base locus ofϕ. Hence, they form Galois orbits (Lemma4.5). So we can group the blow-upsπ_iinto blow-ups of Galois orbits, which are defined over k, and so we find thatπ:T Ñ S is a sequence of blow-ups of Galois orbits and hence the sequenceπof blow-ups and the surface T are defined over k. Therefore, ϕ^˝π is a morphism defined over k.

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Theorem 6(Factorization Theorem). Letπ¹:T ÑS¹be a birational morphism between smooth projective surfaces defined over a perfect field k. Then π¹ is a sequence of blow-ups of Galois orbits.

Proof. By the Factorization Theorem over algebraically closed fields (Theo- rem 4) we know that there exists a sequence of blow-ups πi: Si Ñ Si`1 of pointspi defined overk. The pointss pi are the base points of the inverse ofπ¹, which is defined overk. Therefore, the pointsp_i form Galois orbits. So we can group the blow-upsπi into sequences of blow-ups of Galois orbits.

These two theorems imply directly:

Theorem 7. Letϕ:X99KX¹ be a birational map between (smooth projective) surfaces over a perfect fieldk. Then there exists a surface Y overk such that

ϕ“π¹^˝π^´1,

where π:Y ÑX and π¹:Y ÑX¹ are sequences of blow-ups of Galois orbits.

A more general statement of these theorems can be found in [Liu02, Section 9.2.1].

Example 5.2. Consider the rational map

ϕ:rx:y:zs99Kry²`z²: 4xy:´4xzs

overQ(or overR). It is an involution that is not defined on r1 : 0 : 0sand the Galois orbit r0 : 1 :˘is. In fact, ϕ“α^˝σ^˝α^´1, whereαPAut_QpiqpP²qis given by the matrix

A“

´_{1 0 0}

0 1 1 0 i ´i

¯

PPGL3pQpiqq

andσis the standard quadratic transformation. The birational mapϕfactorizes as the blow-up at r1 : 0 : 0s and the Galois orbittr0 : 1 :˘isu, followed by the contraction of the three lines through two of the points (that is, x “ 0 and z“ ˘iy).

6. Mori fiber spaces

A curveE isomorphic toP¹that has self-intersection´1 is called ap´1q-curve.

In particular, the exceptional divisor arising from blowing up a point on a surface is ap´1q-curve. The following classical theorem states that anyp´1q-curve on a surface can be contracted (for a proof see [Har77, Chapter V, Theorem 5.7]):

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Theorem 8 (Castelnuovo’s contraction theorem). Let k be an algebraically closed field. Let X be a (smooth projective) surface and let E be a p´1q-curve on X. Then there exists a surface Y and a morphism π:X Ñ Y that is the blow-up ofY at a point with exceptional divisorE.

Assume that there are two disjointp´1q-curvesE1 andE2on a surface X.

ContractingE1gives a morphismπ:X ÑX1. SinceE1andE2are disjoint, the imageπpE2qis ap´1q-curve onX1. Hence, we can repeat the process and, since the Picard rank decreases in each step (Lemma 1.5), we stop when there are no morep´1q-curves on the obtained surface. In this way, any set of pairwise disjointp´1q-curves can be contracted. So it is natural to consider a “minimal model” of a surface with respect to blowing up. We say that a smooth projective surfaceSisminimal (overk) if each birational morphism defined overkfromS to a smooth projective surface is an isomorphism. Over perfect fields, a surface is minimal if and only if there is no Galois orbit of disjointp´1q-curves on it.

Theorem 9 ( [Isk79]). Let k be a perfect field and let S be a geometrically rational surface that is minimal overk. Then

(1) either S is a del Pezzo surface (that is,´KS is ample) of Picard rank 1, or

(2) there is a surjective morphismSÑC, whereC is a smooth curve of genus 0, such that a general fiber is isomorphic toP¹ overk.s

Over an algebraically closed field, the minimal rational surfaces areP²and theHirzebruch surfaces

Fⁿ“ tprx:y:zs,rs:tsq PP²ˆP¹|ytⁿ“zsⁿu

forn ‰1; F¹ is not minimal because there is a contraction to P². (We give a different description of Hirzebruch surfaces in ChapterIIIon page26.)

Here, we are interested in a relative version of minimality: We equip (geometrically) rational surfaces with a fibration (over a point or a curve), and ask for minimality of the (relative) Picard rank with respect to this fibration.

Definition 6.1. LetX be a surface over a perfect field k and let π:X ÑB be a surjective morphism with connected fibers to a smooth variety B. The relative Picard group is the quotient Pic_kpX{Bq:“Pic_kpXq{π^˚Pic_kpBq.

Note that we use the notation PickpXqinstead of PicpXqto emphasize that we work over fields that are not necessarily algebraically closed.

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Definition 6.2 (Mori fiber spaces in dimension 2). A surjective morphism π:X ÑBwith connected fibers, whereX is a smooth surface andBis smooth, is called a Mori fiber space, or Mori fibration, if the following conditions are satisfied:

(1) dimpBq ădimpXq,

(2) rk PickpX{Bq “1 (relative Picard rank),

(3) ´K_X¨Dą0 for all effective curvesD onX that are contracted byπ.

We say that two Mori fiber spacesX ÑB andX¹ ÑB¹ are isomorphic if there exists an isomorphism betweenX andX¹ that induces an isomorphism of B andB¹ such that the diagram

X X¹ B B¹

»

» commutes.

Over an algebraically closed field, the Mori fiber spaces are exactly

‚ P²Ñ t˚u, and

‚ FⁿÑP¹ for allně0.

Contrasting the notion of minimal surfaces, hereF¹ÑP¹is included. We give now some additional examples over fields that are not algebraically closed.

Example 6.3. Over a field admitting a Galois orbit α1, . . . , α4 of size 4, the blow-up X ÑP² centered at the four points r1 :α_i :α_i²s fori“1, . . . ,4 gives rise to a fibrationX ÑP¹, where the fibration is given by the pencil of conics through the four points. The relative Picard rank ofX ÑP¹is 1 because there is no Galois orbit of pairwisely disjointp´1q-curves contained in fibers. Hence, X ÑP¹ is a Mori fiber space.

To provide more examples of Mori fiber spaces of the form X Ñ t˚u, we make a short detour to del Pezzo surfaces: These are smooth surfaces S with

´K_Sample. The ampleness criterions of Kleiman respectively Nakai-Moishezon (see [Laz04, Theorem 1.4.23] and [Har77, Chapter V, Theorem 1.10]) state that a divisorD on a surfaceS is ample if and only if

(1) D¨Cą0 for allCPNEpSqzt0u(Kleiman), if and only if

(2) D¨Cą0 for all effective curvesC onS andD²ą0 (Nakai-Moishezon), where NEpSqis the cone of 1-cycles (here curves) up to numerical equivalence.

This leads to a well-known description of del Pezzo surfaces (see [Dem80, Theorem 1]).

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Lemma 6.4. Letkbe an algebraically closed field. Letπ:SÑP²be a birational morphism, whereS is a smooth projective surface. The following conditions are equivalent:

(1) ´KS is ample (that isS is a del Pezzo surface of degreeK_S²);

(2) the morphismπ is the blow-up at0ďrď8 proper points onP² such that no 3are collinear, no6 are on the same conic, no8 lie on a cubic having a double point at one of them;

(3) K_S² ą0and any irreducible curve of S has self-intersection ě ´1;

(4) K_S² ą0andC¨ p´KSq ą0for any effective divisor C.

Moreover, if Sis a del Pezzo surface as in Lemma6.4, then thep´1q-curves onS are exactly the strict transforms of

‚ the lines through two of the rpoints,

‚ the conics through five of the rpoints,

‚ the cubics through seven of therpoints that have a singularity at one of them,

and, ifr“8 there are also the strict transforms of

‚ the quartics through all eight points that have singularities at three of them,

‚ the quintics through all eight points that have singularities at six of them,

‚ the sextics that have a singularity at all eight points, one of which is of multiplicity 3 (see [Dem80, Table 3 on page 35]).

Lemma 6.5. Let X be a smooth projective surface over a perfect field k. Then X Ñ t˚uis a Mori fiber space if and only ifX is a del Pezzo surface with Picard rank rk Pic_kpXq “1.

Proof. LetX Ñ t˚ube a Mori fiber space, hence rk PickpXq “1. So PickpXqb_Z Q“Q¨K_X. For an ample divisor AonX (which exists sinceX is projective) there existsλPQsuch thatA„λKX. SinceAis ample, we have

0ăA²“λ²K_X²,

and henceK_X² ą0. By definition of Mori fiber space we also have that´KX¨ C ą0 for all effective curves C on X. So X is a del Pezzo surface with the Nakai-Moishezon criterion. The converse direction follows directly from the Nakai-Moishezon criterion.

This gives us examples for Mori fiber spacesX Ñ t˚u.

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Example 6.6 (Quadric inP³). OverQ(orR), consider the quadricQĂP³given by the equationxw“y²`z². There is an isomorphism that is defined over the algebraic closure between Qand P¹ˆP¹ but over Qthey are not isomorphic.

More precisely,Qis the image of the birational mapP²99KP³ rx:y:zs99Krx²:xy:xz :y²`z²s,

which is the blow-up of P²at the orbit r0 : 1 :˘is, followed by the contraction of the strict transform of the line x“0. The Picard rank ofQoverkis 1 and so QÑ t˚uis a Mori fiber space.

Example 6.7 (del Pezzo surface of degree 5). Over a perfect field admitting a Galois orbit α1, . . . , α5 of size 5, we consider the blow-up Y Ñ P² at the five points r1 : α_i : α²_is. So no three of the five points are collinear (see also Remark 4.7). Note that the strict transform of the conic y² “xz through the five points is a p´1q-curve. Contracting it gives a morphism Y Ñ D₅, where D5 is a del Pezzo surface of degree 5. The Picard rank ofD5 over kis 1, and so D₅Ñ t˚uis a Mori fiber space.

7. Sarkisov links

We are interested in birational maps between Mori fiber spaces such that its resolution of indeterminacy points into blow-ups is minimal with respect to the rank of the relative Picard group. To be more precise, we are interested in birational maps ϕ: X 99K X¹ between two Mori fiber spaces X Ñ B and X¹ Ñ B¹ (that are not isomorphisms of Mori fiber spaces) such that the two morphismsσ:Y ÑX andσ¹:Y ÑX¹ of the minimal resolution ofϕare both given as the blow-up ofat most one Galois orbit inX respectivelyX¹.

Definition 7.1. ASarkisov link (or simplylink) is a birational mapϕ:X₁99K X2 between two Mori fibrations πi: Xi ÑBi, i “1,2, that is part of a com- muting diagram of one of the following four types:

Type I X1 X2

t˚u “B1 B2

ϕ

whereϕ^´1:X₂ÑX₁ is the blow up of one orbit of Galpk{kq.

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Type II Z

X₁ X₂

B1 B2

σ1 ϕ σ2

»

where σi:Z ÑXi is a blow-up of one orbit fori“1,2.

Type III X₁ X₂

B1 t˚u “B2

ϕ where ϕ: X1 Ñ X2 is the blow-up of one orbit.

Type IV X1 X2

B₁ B₂ ϕ

»

where ϕ is an isomorphism that does not preserve the fibration and B1, B2

are curves.

Example 7.2. The following is an exhaustive list of all Sarkisov links over algebraically closed fields:

(III/I) The blow-up F¹ Ñ P² at one point is a link of type III, fitting into the diagram

F¹ P² P¹ t˚u

ϕ

, and the inverse ϕ^´1:P²99KF¹is a link of type I.

(II) Blowing up a point on a Hirzebruch surfaceFⁿ and contracting the strict transform of the fiber through the point gives a link of type II betweenFⁿ and F^n˘1, depending on whether the blown-up point lies on the minimal section (which is the curve r1 : 0 : 0s ˆP¹ in the coordinates of the Hirzebruch surface on page 16) or not. These are also calledelementary transformations, see also DefinitionIII.3.

(IV) The change of fibration ofP¹ˆP¹ is a link of type IV.

Over perfect fields that are not algebraically closed, there are more links. In fact, we have already seen some Sarkisov links: In Example 6.6we described a link P² 99K Q of type II, where Q ĂP³ is a quadric, and the birational map P²99KD5described in Example6.7is also a link of type II. In Example6.3we have seen the blow-up morphismX ÑP² of a Galois orbit of size 4, which is a link of type III.

Theorem 10 ( [Isk96]). Each birational map between two Mori fiber spaces in dimesion2 over a perfect field can be decomposed into Sarkisov links.

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Example 7.3. The birational mapϕ:P²99KP² given by ϕ:rx:y:zs ÞÑ ry²`z²: 4xy: 4xzs

can be written asϕ“π^˝ψ^˝π^´1, whereπ:F¹ÑP²is the blow-up atr1 : 0 : 0s given by the projection of F¹“ tprx: y: zs,rs:tsq |yt“zsu ĂP²ˆP¹ onto P², andψ:F¹99KF¹ is given by

ψ:prx:y:zs,rs:tsq Ñ prys`zt: 4xs: 4xts,rs:tsq.

Note that π^´1is a link of type I,ψis of type II andπis of type III. So we can decompose ϕinto three Sarkisov links:

P² F¹ F¹ P²

t˚u P¹ P¹ t˚u.

π^´1 ψ π

I „II III

For more properties of Sarkisov links, see also LemmaIV.2.11and LemmaIV.2.12.

An overview of all Sarkisov links between rational Mori fiber spaces can be found in ChapterIVon page110.

8. Why perfect fields?

We want to discuss shortly some things that are not true, or at least make some difficulties for imperfect fields k. The “problem” is that sk{k is not a Galois extension. Our prime example of non-perfect fields is k “ F²ptq. It is not perfect becausex²`tPkrxsis irreducible but not separable: Each element of a field in characteristic 2 has a unique square root, sox²`t“ px`?

tq². So the Galois orbit of r0 : 1 :?

ts is of size 1 but it is not defined over F²ptq. Hence, over imperfect fields k we can no longer identify Galois orbits of size 1 with points defined overk, we no longer have that a point (or a curve, or a map) is defined overkif and only if it is Galois-invariant.

For perfect fields, we identified Galois orbits with closed points (that is, zero sets of ideals). So blowing up a Galois orbit of the pointspα,0q PA², whereα are the roots of some irreducible polynomial f Pkrxs, is the same as blowing up the ideal pfpxq, yq Ăkrx, ys (see [Har92, Example 7.18] for blow-ups along subvarieties). In particular, the surface above is again smooth. The blow-up of the idealpx²`t, yq ĂF²ptqrx, ys, however, corresponds geometrically to the blow-up of a double point (see [EH00, Section IV.2.3]). The blow-up ofpx²`t, yq is given byZ “ t`

px, yq,ru:vs˘

PA²ˆP¹|uy“vpx²`tqu with the projection to A². Note thatZ is not smooth at the pointpp?

t,0q,r0 : 1sq PZ.

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There are more things that one has to be cautious of:

Lemma 8.1( [Liu02, Example 2.10]). Ifkis perfect, then any reduced algebraic variety is geometrically reduced.

This does not hold for the non-perfect fieldk“F²ptq: Letf “x²`tPkrxs.

Then the zero set Vpfq Ă A¹pkq is reduced and irreducible but over the field extensionF²p

?tq{F² we havef “ px`?

tq²so it is non-reduced (double point).

Recall that regularity (or non-singularity) of a variety is defined via its tangent space [Liu02, Definition 4.2.8], and can be detected with the Jacobian Criterion [Liu02, Theorem 4.2.19]. Smoothness, on the other hand, means regularity over the algebraic closure [Liu02, Definition 4.3.28].

Lemma 8.2. Let X be an algebraic variety over a perfect field k. ThenX is smooth over kif and only if it is regular.

Over k “F²ptq, the projective line P¹pF²p?

tqq is regular but not smooth [Liu02, Remark 4.3.34].

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Plane curves of fixed bidegree and their A _k -singularities

We provide a tool how one can view a polynomial on the affine plane of bidegree pa, bq – by which we mean that its Newton polygon lies in the triangle spanned by pa,0q,p0, bqand the origin – as a curve in a Hirzebruch surface having nice geometric properties. As an application, we study maximal A_k-singularities of curves of bidegreep3, bqand find the answer forbď12.

1. Introduction

We study algebraic curves on the affine planeA²pCq that have a singularity of type A_k, which means that there is an analytical local isomorphism such that the curve is given by y²´x^k`1 “0 in a neighbourhood of the singular point (c.f. Definition2). We ask:

Question 1. For dě1, what is the maximal k such that there exists a curve of degreedthat has an Ak-singularity?

We denote this byNpdqand can give answers for smalld:

d 1 2 3 4 5 6 7...

Npdq 0 1 3 7 12 19 ? ,

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where an explicit equation ford“5 can be found in [Wal96], and the example ford“6 was found by Yoshihara [Yos79]. Later, [Yan96] gave a classification of all simple singularities of sextic curves. (Note that the answers ofNp2q,Np3q and Np4q differ if we only consider irreducible curves.) The difficulty of the question increases rapidly for larger values ofd, so the asymptotic behaviour is studied and bounds for

α“lim sup2Npdq d²

are wanted, where we multiplied by 2 to obtain nicer numbers, as it is often done in the literature. Gusein-Zade and Nekhoroshev found that 1.5 ě αě ¹⁵₁₄ » 1.07142 [GZN00] and in the same year, Cassou-Nogu`es and Luengo refined the lower bound to 8´4?

3 » 1.07179 [CNL00]. A decade passed until Orevkov improved it even further to ⁷₆“1.16 [Ore12].

Question 1can also be approached through fixing a bidegree instead of the degree. We say that a polynomialF (or equivalently, the curve inA²pCqdefined by its zero set) has bidegree pa, bq if its Newton polygon lies in the triangle spanned bypa,0q,p0,0qandp0, bq. Note that this differs from the usual definition of bidegree. In particular, a polynomial is of bidegree pd, dqif and only if it is of degree at mostd. So we generalize Question1:

Question 2. Forpa, bq PN², what is the maximal ksuch that there is a curve inA²pCq of bidegreepa, bqwith anA_k-singularity?

Similar to above, we denote this byNpa, bq. For instance, one findsNp1, bq “ 0 for allb, and fixinga“2 yieldsNp2, bq “b´1 (c.f. Example2 respectively Lemma 11). We have studied the case where a “ 3 and found the following values ofNp3, bq:

Theorem 1. For smallb,Np3, bqis given by the following table:

b 3 4 5 6 7 8 9 10 11 12

Np3, bq 3 5 7 8 10 12 13 15 17 18 .

Moreover, for bě4 there are irreducible polynomials that achieve the maximal singularities.

Studying polynomials of bidegreepa, bqis interesting on its own, however it could also help to determine the asymptotical behaviour ofNpdq, thanks to the following result.

Proposition 1( [Ore12]). If Npa, bq `1ěk, thenαě ^2k_ab.