ontheweak lefschetzprinciple inbirationalgeometryontheweak lefschetzprinciple inbirationalgeometry...

On the WeakLefschetz Principle

in Birational Geometry

César Lozano Huerta and Alex MassarentiThe Lay of the LandAlgebraic geometers study the solution sets to finite col-lections of polynomial equations in ℂ𝑛 or in the complexprojective space ℙ𝑛. Often, the goal is to understand thebasic geometric properties of these sets such as connect-edness, smoothness, dimension, etc. The most importantand familiar examples are solution sets to finite collectionsof linear equations and, in this case, we notice that their

César Lozano Huerta is a CONACYT research fellow in mathematics at theNational University of Mexico. His email address is [email protected] Massarenti is a research fellow in mathematics at the Department of Math-ematics and Computer Science of the University of Ferrara, Italy. His emailaddress is [email protected].

For permission to reprint this article, please contact:[email protected].

DOI: https://doi.org/10.1090/noti2205

geometry exhibits many similarities to that of their ambi-ent spaces ℂ𝑛 or ℙ𝑛.

When one analyzes solution sets of higher degree equa-tions, the situation changes radically. These sets no longerhave an easily recognizable structure, such as that of a vec-tor space or a group. Unlike the linear case, they may havemany connected components and its dimension may bechallenging to compute. Despite all these difficulties, thisarticle will discuss scenarios in which the geometry of thesolution set is similar to that of the space in which it sits.Such situations are somewhat unexpected and have had aprofound impact in mathematics.

Our departing point is the influential work of SolomonLefschetz started in 1924 [11]. Consider a collectionof finitely many homogeneous polynomials whose solu-tion set in ℙ𝑛 has dimension bigger than one. Lefschetz

JANUARY 2021 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 47

analyzed the following: if one considers an additionalgeneric linear equation and looks at the resulting solutionset, the so-called hyperplane section, often one finds thatmany geometric properties of it are determined by the orig-inal solution set.

In what follows, we will look at the hyperplane theoremof Lefschetz in different contexts. Wewill revisit its originalformulation in algebraic topology, and build up to recentdevelopments of it in birational geometry. In doing so, wewill emphasize the main theme of this article: there aremany contexts in geometry in which analogous versionsof the Lefschetz hyperplane theorem hold. Even though itwas originally stated in the context of algebraic topology,there are versions of it in algebraic geometry, homotopytheory, complex geometry, and birational geometry. In-tuitively speaking, all these results focus on a solution set(or a submanifold, depending on the context) whose geo-metric properties are determined by the space in which itsits.

We have tried to present the development of the ideassomehow chronologically. We apologize for the omis-sions we unintentionallymade due to our lack of historicalunderstanding.

Notions from Algebraic TopologyStarted in the early 20th century and known as combina-torial topology until the early 1940s, the field of algebraictopology constructs algebraic invariants in order to distin-guish two topological spaces. Homology and cohomologygroups are two such invariants and central notions of thisarticle. These groups, whose definitions are subtle, turnout to be accessible to computations and are for this rea-son commonly used. Let us recall some of their properties.

The 𝑘th homology group of a topological space 𝑋 , withcoefficients in a field 𝔽, can be defined in terms of cyclesand boundaries when a triangulation of 𝑋 is available. In-deed, the 𝑘th homology group of 𝑋 is defined as the quo-tient

𝐻𝑘(𝑋, 𝔽) = {𝑘-cycles of 𝑋}/{𝑘-boundaries of 𝑋},

where the coefficients in question are in the field 𝔽. In thiscase, the group 𝐻𝑘(𝑋, 𝔽) has the structure of a vector spaceover 𝔽 and in the remainder of this article our homologygroups will have ℚ-coefficients.

A key feature of homology is the following: any contin-uous morphism between topological spaces 𝑓 ∶ 𝑋 → 𝑌induces a corresponding morphism

𝑓∗ ∶ 𝐻𝑘(𝑋,ℚ) → 𝐻𝑘(𝑌,ℚ)

which is a linear transformation between vector spacesover ℚ.

Alongside the 𝑘th homology group there is the 𝑘th co-homology group denoted by 𝐻𝑘(𝑋,ℚ). If 𝑋 is an oriented

manifold, the former is the dual vector space to the lat-ter, but the following property of cohomology is different:any continuous morphism between topological spaces 𝑓 ∶𝑋 → 𝑌 induces a corresponding morphism

𝑓∗ ∶ 𝐻𝑘(𝑌,ℚ) → 𝐻𝑘(𝑋,ℚ)

which is a linear transformation between vector spacesover ℚ.

Cohomology has a distinctive advantage over homol-ogy: not only can we add cohomology classes, we canmul-tiply them as well. When 𝑋 is an oriented manifold, thisproduct has a rich geometric meaning which is often givenby the intersection of two submanifolds. (We could do thisin homology, but the labels in cohomology work better.For one thing, they yield a graded ring: if 𝛼 ∈ 𝐻𝑘(𝑋,ℚ)and 𝛽 ∈ 𝐻𝑙(𝑋,ℚ), then 𝛼.𝛽 ∈ 𝐻𝑘+𝑙(𝑋,ℚ).) This is theso-called cup product and will appear a bit later.

Tracing Back the Lefschetz Hyperplane TheoremIn this section we formulate the theorem that sets thetheme of this article: the Lefschetz hyperplane theorem.We start by describing its context.

Let us consider a smooth complex algebraic surface 𝑆 ⊂ℙ𝑛; such an 𝑆 is a manifold of dimension 4 over ℝ. Weare interested in describing the homology group 𝐻1(𝑆, ℚ)of the surface 𝑆 with coefficients in the rationals.

Since the surface 𝑆 lies inside the projective space ℙ𝑛,one may take a smooth hyperplane section of it by throw-ing in an additional linear equation to the defining equa-tions of 𝑆 such that the result is a smooth algebraic curve𝐶 ⊂ 𝑆 of genus 𝑔. One can also consider a generic familyof hyperplane sections of 𝑆 containing 𝐶 as follows. Lin-ear polynomials in ℙ𝑛 form a family isomorphic to ℙ𝑛 fortheir 𝑛 + 1 coefficients can be thought of as coordinates.One point 𝑝 in this family gives rise to 𝐶. Pick anothergeneral point 𝑞 of this family and consider the unique line𝑝𝑞 = 𝐿 that contains them. We now may think of the hy-perplane sections of 𝑆 parametrized by 𝑡 ∈ 𝐿 ≅ ℙ1. Thatis, for each value 𝑡 ∈ 𝐿 there is a curve 𝐶𝑡 ⊂ 𝑆.1

The advantage of considering 𝐶 within this family isthat one may try to describe the homology of the surface𝐻1(𝑆, ℚ) as a linear combination of the 1-cycles of 𝐶𝑡. Thatis, we may consider a 1-cycle of 𝐶𝑡 and then think of it as a1-cycle of 𝑆. A result by Lefschetz says this is possible: any1-cycle of 𝑆 is homologous on 𝑆 to a 1-cycle lying on a generic 𝐶𝑡.Since the homology of the generic 𝐶𝑡 is 𝐻1(𝐶𝑡, ℚ) ≅ ℚ2𝑔,this yields 𝐻1(𝑆, ℚ) as a quotient of a well-known space.

1The family {𝐶𝑡} is called a Lefschetz pencil and its key property is that itsgeneric member is smooth and the singular members have all mild singularities.

48 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 68, NUMBER 1

The figure

𝜎1 𝜎2𝜎3

𝜎4𝜎5 𝜎6

depicts the generating 1-cycles of 𝐻1(𝐶,ℚ), where 𝐶 is asmooth curve of genus 3.

Let us state again the previous result. Let 𝑖 ∶ 𝐶 ↪ 𝑆 bethe inclusion of a smooth hyperplane section of 𝑆. Thenthe result above asserts that the induced map

𝑖∗ ∶ 𝐻1(𝐶,ℚ) → 𝐻1(𝑆, ℚ)is surjective. In this setting, this is the Lefschetz hyper-plane theorem. In order to fully describe the first homol-ogy group of 𝑆 we need to further investigate the kernel of𝑖∗, which consists of the so-called vanishing cycles. The nec-essary information about such cycles can now be extractedfrom thinking of 𝐶 in the family {𝐶𝑡}. From this analysis,one gets an important theorem: the dimension of𝐻1(𝑆, ℚ),called the first Betti number of 𝑆 and denoted by 𝑏1(𝑆), iseven.

In general it is quite challenging to deduce propertiesof the 1-cycles on 𝑆 from their behavior as 1-cycles on thecurves 𝐶𝑡. In fact, Zariski points out that such argumentsare the most difficult part of all the topological argumentsby Lefschetz [20, p. 136]. For example, Zariski commentsthat Lefschetz’s topological proof of the fact that the Bettinumber 𝑏1(𝑆) is even seems to be incomplete.2

This difficulty, however, can be overcome via Hodge the-ory in which case the key is the following claim: intersec-tion with the curve 𝐶𝑡 defines an isomorphism

̃𝜙 ∶ 𝐻3(𝑆, ℚ) → 𝐻1(𝑆, ℚ). (1)The proof that ̃𝜙 is an isomorphism was first completed

in [8]. We will come back to this map in the next section.Notice that ̃𝜙 factors through𝐻1(𝐶𝑡, ℚ). In fact, one can

deduce that ̃𝜙 is an isomorphism from the surjectivity of𝑖∗,

𝑖∗ ∶ 𝐻1(𝐶𝑡, ℚ) → 𝐻1(𝑆, ℚ), (2)by verifying that its kernel consists of vanishing cycles andis orthogonal to the cycles that generate 𝐻3(𝑆, ℚ).

The previous discussion is part of Lefschetz’s originalapproach to the study of the topology of an algebraic sur-face 𝑆. In what follows we will examine a generalization ofthis discussion which goes under the name of the weak Lef-schetz theorem. Before that, we decided to include a brief

2Zariski comments that Lefschetz’s proof is based on an unproven statement ofPicard.

description of the strong Lefschetz theorem so the readermay contrast. Strong Lefschetz will concern the map ̃𝜙while weak Lefschetz will concern the map 𝑖∗ above.

Strong Lefschetz TheoremThe previous isomorphism (1) admits a dual formulationinvolving cohomology which builds up to the strong Lef-schetz theorem. Indeed, the cup product with the coho-mology class𝜔 of the hyperplane section 𝐶𝑡 yields themap

𝜙 ∶ 𝐻1(𝑆, ℚ) → 𝐻3(𝑆, ℚ)

and 𝜙 is an isomorphism. A generalization now may beformulated as follows. Let 𝑋 be a smooth complex alge-braic variety of dimension 𝑛. Then the map defined by thecup product with the powers 𝜔𝑛−𝑘,

𝜙𝑛−𝑘 ∶ 𝐻𝑘(𝑋,ℚ) → 𝐻2𝑛−𝑘(𝑋,ℚ),

is an isomorphism [8]. This result by Hodge is carried outover the field of complex numbers. In characteristic 𝑝 > 0it was studied by Grothendieck, Artin, and Verdier in [2]and they called it the strong3 Lefschetz theorem.

Weak Lefschetz TheoremLet us keep using cohomology notation in this section sothe contrast between the weak and strong versions of theLefschetz theorem becomes clear. The weak Lefschetz the-orem, in our context, claims that if 𝑖 ∶ 𝐻 ↪ 𝑋 is the inclu-sion morphism of a smooth hyperplane section, then theinduced map on the 𝑘th cohomology groups

𝑖∗ ∶ 𝐻𝑘(𝑋,ℚ) → 𝐻𝑘(𝐻,ℚ) (3)

is an isomorphism for 𝑘 ≤ 𝑛 − 2 and an injection for 𝑘 =𝑛−1. For complex varieties, the theorem follows from theexact sequence

𝐻𝑘𝑐 (𝑈,ℚ) → 𝐻𝑘(𝑋,ℚ) → 𝐻𝑘(𝐻,ℚ) → 𝐻𝑘+1𝑐 (𝑈,ℚ),

where 𝑈 denotes the open set 𝑈 = 𝑋 ⧵ 𝐻 and 𝐻∗𝑐 (𝑈,ℚ)stands for cohomology with compact support. Since𝐻𝑘𝑐 (𝑈,ℚ) ≅ 𝐻2𝑛−𝑘(𝑈,ℚ), the weak Lefschetz theorem istherefore equivalent to the vanishing of the homologygroups: 𝐻𝑗(𝑈,ℚ) = 0 for 𝑗 ≥ 𝑛 + 1.

The analogous result in characteristic 𝑝 > 0 was provedby Grothendieck, Artin, and Verdier [2] in the context ofétale cohomology with ℚ𝑙-adic coefficients using exact se-quences and cohomology vanishing. They called it theweak Lefschetz theorem.

It is important to mention that the Lefschetz theorems,as formulated above, bear a close relation to the Weil con-jectures. We refer the reader to [6], in particular Kleiman’sexposé X, for an account about this fascinating part of al-gebraic geometry.

3This is also known as the “hard Lefschetz theorem.”


The isomorphism (3) is a crucial ingredient in what fol-lows. In fact, we will exhibit examples where this isomor-phism also preserves objects contained in the 2nd coho-mology group which encode birational information aboutthe varieties 𝑋 and 𝑌 . This is the “weak Lefschetz” of ourtitle. In the following section we will motivate the word“principle.”

Building up to a PrincipleA different approach to study the topology of a smoothprojective variety 𝑋 was developed by Bott [4] and by An-dreotti and Frankel [1], both in 1959. These two papersused Morse theory applied to a nondegenerate real 𝐶∞-function 𝑓 which satisfies 𝑓(𝑥) = 0 if 𝑥 ∈ 𝐻 and 𝑓(𝑥) > 0 if𝑥 ∉ 𝐻, where 𝐻 ⊂ 𝑋 is a smooth hyperplane section. Botteven describes a cellular structure on 𝑋 by attaching cellsto𝐻. This decomposition allows him to analyze the topol-ogy of 𝑋 in detail. In particular, Bott proves the statementdual to the weak Lefschetz theorem above: the inducedmorphism by the inclusion

𝑖∗ ∶ 𝐻𝑘(𝐻,ℚ) → 𝐻𝑘(𝑋,ℚ)

is bijective if 0 ≤ 𝑘 ≤ 𝑛 − 2, and

𝑖∗ ∶ 𝐻𝑛−1(𝐻,ℚ) → 𝐻𝑛−1(𝑋,ℚ)

is surjective.Bott actually proves more. He argues that one may fo-

cus on other topological invariants instead of homology orcohomology. For example, Bott applied this theory to ho-motopy groups and showed that the inclusion 𝑖 ∶ 𝐻 ↪ 𝑋yields isomorphisms for the homotopy groups

𝑖∗ ∶ 𝜋𝑘(𝐻) → 𝜋𝑘(𝑋).

This result, along with those for homology, starts lookingmore like a principle. Indeed, the relationship between thegeometry of a variety 𝑋 and that of a hyperplane section𝐻 seems to be robust as we are planning to exhibit in therest of this article.

The Weak Lefschetz PrincipleAfter the previous sections we have the following situation.Let 𝑋, 𝑌 be smooth complex projective varieties equippedwith embedding 𝑖 ∶ 𝑌 ↪ 𝑋 . This inclusion 𝑖 may inducean isomorphism on the cohomology, homology, or homo-topy groups. It turns out that it may also induce an isomor-phism on objects in some other contexts, for example thePicard groups, étale cohomology, or Hodge structures justto name a few. One thus may be led to think that the weakLefschetz theorem seems to be robust, to the extent of con-sidering it part of a principle. This principle says that oftenthe geometry of the ambient variety 𝑋 determines that ofthe subvariety 𝑌 . Finding the precise conditions for thisprinciple to hold is a subject of current research.

In the rest of the article, we aim to exhibit examples inthe context of birational geometry for which the weak Lef-schetz principle holds. The next section contains the pre-cise definitions to this end and is the technical part of thearticle. It may be skipped on a first reading.

Notions from Birational GeometryNow we change gears from topology to birational geome-try; that is, we return to studying zeros of polynomials. Theset of all algebraic varieties is vast, so one tries to organize itinto equivalence classes. Birational geometry studies alge-braic varieties up to birational equivalence and often seeksto find the “simplest” representative in each equivalenceclass.

From now on, we assume some familiarity with ele-mentary concepts from algebraic geometry such as Cartierdivisor, linear equivalence, blowup, and strict transform.Since the definitions introduced in this section might behard to grasp, we have included an explicit example thatcan be read independently in Boxes 1, 2, and 3 below.

Let us consider 𝑋 a smooth projective variety over ℂ. If𝐷 is a subvariety of codimension 1 in 𝑋 , then one can nat-urally think of the number 𝐷 ⋅ 𝐶, the result of intersect-ing 𝐷 with a curve 𝐶. Loosely speaking, we should countthe number of points in the intersection between 𝐷 and 𝐶.However, if 𝐶 is a subset of 𝐷 we need a different descrip-tion. Let us recall its formal definition so we are on firmground. Given a Cartier divisor 𝐷 on a normal variety 𝑋and a curve 𝐶 ⊂ 𝑋 , their intersection number is defined as

𝐷 ⋅ 𝐶 = deg𝒪𝐶(𝐷), (4)where 𝒪𝐶(𝐷) is the restriction to 𝐶 of the invertible sheaf𝒪𝑋(𝐷) associated to 𝐷. When 𝐷 does not contain 𝐶 thenumber 𝐷 ⋅ 𝐶 is the number of points of intersection be-tween 𝐷 and 𝐶 counted with multiplicity. Note that thisdefinition can be extended by linearity to ℚ-Cartier divi-sors, that is, Weil divisors having a multiple that is Cartier,and arbitrary linear combinations of irreducible curves,which are called 1-cycles of 𝑋 .

Now the correct notion of intersection reveals structurein the set of divisors. Indeed, we say that two Cartier di-visors 𝐷,𝐷′ ⊂ 𝑋 are numerically equivalent if 𝐷 ⋅ 𝐶 =𝐷′ ⋅ 𝐶 for any irreducible curve 𝐶 ⊂ 𝑋 . In this case, wewrite 𝐷 ≡ 𝐷′. The group of all Cartier divisors on 𝑋modulo numerical equivalence is a finitely generated freeabelian group, called the Neron-Severi group and denotedby 𝑁1(𝑋). Hence, the vector space

𝑁1(𝑋)ℝ = 𝑁1(𝑋) ⊗ ℝis finite dimensional [5, p. 8] and its dimension, denotedby 𝜌𝑋 , is called the Picard rank of 𝑋 .

The study of the Picard rank has deep roots in algebraicgeometry. When 𝑋 is a smooth surface over ℂ, the number𝜌𝑋 was carefully analyzed by Picard, Severi, and Lefschetz.


In fact, Lefschetz observed that curves in 𝑋 , up to numer-ical equivalence, can be thought of as elements in the co-homology group 𝐻2(𝑋,ℚ). From this he was able to showthat 𝜌𝑋 is less than or equal to the dimension of 𝐻2(𝑋,ℚ);hence finite. This enhances the analysis mentioned earlierin this article about the first Betti number 𝑏1(𝑆) of an alge-braic surface 𝑆.

So far we have mentioned three equivalence relationsthat can be imposed on Cartier divisors: linearly equiva-lent, numerically equivalent, and cohomologous. Thesethree relations may coincide, for instance if 𝑋 = ℙ𝑛, but ingeneral they are different; for example on a general quarticsurface in ℙ3. The first two relations give rise to the Picardgroup Pic(𝑋) and to 𝑁1(𝑋), respectively. Later in this sec-tion, we will restrict our study to spaces where linear andnumerical equivalence coincide.

Let us now recall the definition of some cones con-tained in the space 𝑁1(𝑋)ℝ which are important in bira-tional geometry and central for the rest of the article. SeeBoxes 1, 2, and 3 for a worked example.Definition 1. Let 𝐷 ⊂ 𝑋 be a ℚ-Cartier divisor, that is, adivisor such that 𝑚𝐷 is Cartier for some integer 𝑚 > 0.

- The divisor class is said to be effective if it representsan actual subvariety of codimension 1. The effectivecone of 𝑋 is the convex cone Eff(𝑋) ⊂ 𝑁1(𝑋)ℝ gener-ated by classes of effective divisors.- 𝐷 is very ample if it induces an embedding, and 𝐷 isample if 𝑚𝐷 is very ample for 𝑚 ≫ 0. The ample coneof 𝑋 is the convex cone Amp(𝑋) ⊂ 𝑁1(𝑋)ℝ generatedby classes of ample divisors.- 𝐷 is nef if 𝐷 ⋅ 𝐶 ≥ 0 for any irreducible curve 𝐶 ⊂𝑋 . Note that if 𝐷 is ample, then 𝐷 ⋅ 𝐶 > 0 for anycurve 𝐶 ⊂ 𝑋 . Hence nefness is a mild relaxation ofampleness. The nef cone of 𝑋 is the closed convex coneNef(𝑋) ⊂ 𝑁1(𝑋)ℝ generated by classes of nef divisors.- The stable base locus of 𝐷 is the set of points 𝑝 ∈ 𝑋such that for all 𝑚 > 0, if 𝑚𝐷 is integral, all the divi-sors in the linear system of 𝑚𝐷 pass through 𝑝. Themovable cone of𝑋 is the convex coneMov(𝑋) generatedby classes of movable divisors; these are divisors whosestable base locus has codimension at least two in 𝑋 .

We have inclusions among the previous cones:

Amp(𝑋) ⊂ Nef(𝑋) ⊂ Mov(𝑋) ⊂ Eff(𝑋) ⊂ 𝑁1(𝑋)ℝ.

Box 1: The effective cone and the Mori cone ofcurves

Let us work out explicitly the cone of effective di-visors and the Mori cone of curves of 𝑋 , the blow-up of ℙ𝑛 at two points 𝑝, 𝑞 ∈ ℙ𝑛, with 𝑛 > 1. Let𝐻,𝐻𝑝, 𝐻𝑞, 𝐻𝑝,𝑞 be the strict transforms, respectively,

of a hyperplane, a hyperplane passing through 𝑝,through 𝑞, and through both 𝑝 and 𝑞. Moreover,let 𝐸𝑝, 𝐸𝑞 be the exceptional divisors over 𝑝 and 𝑞,respectively. Note that 𝐻𝑝 ≡ 𝐻 − 𝐸𝑝, 𝐻𝑞 = 𝐻 − 𝐸𝑞,and𝐻𝑝,𝑞 ≡ 𝐻−𝐸𝑝−𝐸𝑞. Then 𝑁1(𝑋) ≅ ℤ[𝐻, 𝐸𝑝, 𝐸𝑞].We will denote by ℎ the strict transform of a gen-eral line in ℙ𝑛, and by 𝑒𝑝, 𝑒𝑞 classes of lines in 𝐸𝑝and 𝐸𝑞, respectively. The intersection pairing (5) isgiven by 𝐻 ⋅ ℎ = 1,𝐻 ⋅ 𝑒𝑝 = 𝐻 ⋅ 𝑒𝑞 = 0, 𝐸𝑝 ⋅ 𝑒𝑞 =𝐸𝑞 ⋅ 𝑒𝑝 = 0, 𝐸𝑝 ⋅ 𝑒𝑝 = 𝐸𝑞 ⋅ 𝑒𝑞 = −1. The last twointersection numbers might not be obvious from ageometrical point of view. To compute them onemay reason as follows: the divisor 𝐻 − 𝐸𝑝 repre-sents the strict transform of a general hyperplanethrough 𝑝, and ℎ − 𝑒𝑝 represents the strict trans-form of a general line through 𝑝. In the blow-up𝑋 these strict transforms do not intersect anymore,so 0 = (𝐻−𝐸𝑝)⋅(ℎ−𝑒𝑝) = 𝐻⋅ℎ−𝐻⋅𝑒𝑝−𝐸𝑝 ⋅ℎ+𝐸𝑝 ⋅𝑒𝑝and hence 𝐸𝑝 ⋅ 𝑒𝑝 = −𝐻 ⋅ ℎ = −1.Now, let 𝐶 ⊂ 𝑋 be an irreducible curve. Then either𝐶 is contained in an exceptional divisor and thenit is numerically equivalent to a positive multipleof 𝑒𝑝 or 𝑒𝑞, or it is mapped by the blow-down mapto an irreducible curve Γ ⊂ ℙ𝑛. Let 𝑑,𝑚𝑝, 𝑚𝑞 be,respectively, the degree and the multiplicities of Γat 𝑝 and 𝑞. Then 𝐶 ≡ 𝑑ℎ − 𝑚𝑝𝑒𝑝 − 𝑚𝑞𝑒𝑞. We maywrite 𝐶 ≡ 𝑑(ℎ − 𝑒𝑝 − 𝑒𝑞) + (𝑑 −𝑚𝑝)𝑒𝑝 + (𝑑 −𝑚𝑞)𝑒𝑞.Furthermore, 𝑑−𝑚𝑝 > 0; otherwise by Bézout’s the-orem Γ would contain a line through 𝑝 as a com-ponent, and similarly 𝑑 − 𝑚𝑞 > 0. Hence NE(𝑋)is closed and generated by the classes 𝑒𝑝, 𝑒𝑞, andℎ − 𝑒𝑝 − 𝑒𝑞. Note that the latter is the strict trans-form of the line in ℙ𝑛 through 𝑝, 𝑞. Similarly, it canbe shown that Eff(𝑋) is closed and generated by theclasses of 𝐸𝑝, 𝐸𝑞, and 𝐻𝑝,𝑞.

Similar to the case of divisors, two 1-cycles𝐶, 𝐶′ ⊂ 𝑋 arenumerically equivalent if 𝐷 ⋅𝐶 = 𝐷 ⋅ 𝐶′ for any irreducibleCartier divisor 𝐷 ⊂ 𝑋 . We will denote by 𝑁1(𝑋) the quo-tient of the group of 1-cycles by this equivalent relation,and consider the vector space 𝑁1(𝑋)ℝ = 𝑁1(𝑋) ⊗ ℝ.

Intersecting curves and divisors according to (4) inducea nondegenerate pairing

𝑁1(𝑋) × 𝑁1(𝑋) → ℤ (5)

which implies that 𝑁1(𝑋)ℝ is finite dimensional.The following cone is called theMori cone of curves and

was introduced by S. Mori. Let NE(𝑋) denote the closureof the convex cone in𝑁1(𝑋)ℝ of classes of effective 1-cycles,that is, classes in 𝑁1(𝑋)ℝ which represent actual curves in𝑋 .


Box 2: The Nef cone and movable cone (Box 1continued)

Let us work out the nef cone of 𝑋 when 𝑛 > 2. Thisis the cone of divisors intersecting nonnegativelyall the irreducible curves in 𝑋 . Since any curve in𝑋 can be written as a linear combination with non-negative coefficients of the generators of NE(𝑋), itis enough to check when a divisor intersects non-negatively these generators. Let us write 𝐷 ≡ 𝑎𝐻 +𝑏𝐸𝑝+𝑐𝐸𝑞. Then𝐷⋅(ℎ−𝑒𝑝−𝑒𝑞) = 𝑎+𝑏+𝑐,𝐷⋅𝑒𝑝 = −𝑏,and 𝐷 ⋅ 𝑒𝑝 = −𝑐, and Nef(𝑋) is defined in 𝑁1(𝑋)ℝ ≅ℝ3 by the inequalities 𝑎 + 𝑏 + 𝑐 ≥ 0, 𝑏 ≤ 0, 𝑐 ≤ 0.Hence Nef(𝑋) is generated by ⟨𝐻,𝐻𝑝, 𝐻𝑞⟩.Finally, we determine the movable cone of 𝑋 . Thedivisor 𝐻𝑝,𝑞 represents the hyperplanes of ℙ𝑛 pass-ing through 𝑝, 𝑞. Hence the stable base locus of𝐻𝑝,𝑞 consists of the strict transform of the linethrough 𝑝, 𝑞. The stable base locus of all divisorsin the cone generated by ⟨𝐻𝑝, 𝐻𝑞, 𝐻𝑝,𝑞⟩ is containedin such a strict transform as 𝐻𝑝, 𝐻𝑞 have no baseloci. Hence all the divisors in this cone are mov-able when 𝑛 > 2. On the other hand, all divisorsin the interior of the cone ⟨𝐻,𝐻𝑝, 𝐸𝑞⟩ contain 𝐸𝑞, alldivisors in the interior of the cone ⟨𝐻,𝐻𝑞, 𝐸𝑝⟩ con-tain 𝐸𝑝, and all divisors in the interior of the cone⟨𝐻, 𝐸𝑝, 𝐸𝑞⟩ contain 𝐸𝑝∪𝐸𝑞. Therefore,Mov(𝑋) is thecone generated by ⟨𝐻,𝐻𝑝, 𝐻𝑞, 𝐻𝑝,𝑞⟩.

We now introduce themain notion of this section: Moridream spaces in the context of the minimal model pro-gram. The goal of the minimal model program is to con-struct a birational model of any complex projective varietywhich is as simple as possible in a suitable sense. This sub-ject has its origins in the classical birational geometry ofsurfaces studied by the Italian school. In 1988 S. Mori ex-tended the concept of minimal model to 3-folds by allow-ing suitable singularities on them [16] and was awardedthe Fields Medal for his contributions on this topic. In2010 there was another breakthrough in minimal modeltheory when C. Birkar, P. Cascini, C. Hacon and J. Mc-Kernan proved the existence of minimal models for a suit-able class of higher dimensional varieties [3]. C. Birkar wasawarded the Fields Medal in 2018 for his contributions tobirational geometry.

Mori dream spaces, introduced by Y. Hu and S. Keel in2000 [9], form a class of algebraic varieties that behave verywell from the point of view of the minimal model pro-gram. In order to recall their definition, let us first defineℚ-factorial modifications.

We say that a birational map 𝑓 ∶ 𝑋 99K 𝑋 ′ to a nor-mal projective variety 𝑋 ′ is a birational contraction if its in-verse does not contract any divisor. We say that it is a small

ℚ-factorial modification if 𝑋 ′ is ℚ-factorial, that is, any Weildivisor on 𝑋 ′ has a multiple which is Cartier, and 𝑓 is anisomorphism in codimension one. If 𝑓 ∶ 𝑋 99K 𝑋 ′ isa small ℚ-factorial modification, then there is pull-backmap

𝑓∗ ∶ 𝑁1(𝑋 ′) → 𝑁1(𝑋)which sends Mov(𝑋 ′) and Eff(𝑋 ′) isomorphically ontoMov(𝑋) and Eff(𝑋), respectively. In particular, we have𝑓∗(Nef(𝑋 ′)) ⊂ Mov(𝑋).

The previous paragraph makes explicit the importancein birational geometry of the cones defined earlier in thissection. The next definition will restrict our study to spaceswhose birational geometry is encoded in finitely many ofsuch cones.

Definition 2 ([9, Definition 1.10]). A normal projectiveℚ-factorial variety 𝑋 is called a Mori dream space if the fol-lowing conditions hold:

- Pic(𝑋) ⊗ ℚ = 𝑁1(𝑋) ⊗ ℚ,- Nef (𝑋) is generated by the classes of finitely manysemiample divisors, that is, divisors having a multiplewith empty base locus,- there is a finite collection of smallℚ-factorialmodifi-cations 𝑓𝑖 ∶ 𝑋 99K 𝑋𝑖 such that each 𝑋𝑖 satisfies the sec-ond condition above andMov (𝑋) = ⋃𝑖 𝑓

∗𝑖 (Nef (𝑋𝑖)).

The collection of all faces of all cones 𝑓∗𝑖 (Nef (𝑋𝑖)) inDefinition 2 forms a wall-and-chamber decomposition ofMov(𝑋). If two maximal cones of this fan, say 𝑓∗𝑖 (Nef (𝑋𝑖))and 𝑓∗𝑗 (Nef (𝑋𝑗)), meet along a facet, then there exist anormal projective variety 𝑌 , small birational morphismsℎ𝑖 ∶ 𝑋𝑖 → 𝑌 and ℎ𝑗 ∶ 𝑋𝑗 → 𝑌 of relative Picard rankone, and a small modification 𝜑 ∶ 𝑋𝑖 99K 𝑋𝑗 such thatℎ𝑗 ∘ 𝜑 = ℎ𝑖. The fan structure on Mov(𝑋) can be extendedto a fan supported on Eff(𝑋) as follows. We refer the readerto [9, Proposition 1.11] and [17, Section 2.2] for detailsand to Box 3 for an example.

Definition 3. Let 𝑋 be a Mori dream space. A wall-and-chamber decomposition of the effective coneEff(𝑋), calledthe Mori chamber decomposition and denoted MCD(𝑋), isdescribed as follows. There are finitely many birationalcontractions from 𝑋 toMori dream spaces, denoted by 𝑔𝑖 ∶𝑋 99K 𝑌 𝑖. The maximal cones 𝒞 of theMCD(𝑋) are of theform: 𝒞𝑖 = 𝑔∗𝑖 (Nef(𝑌 𝑖))∗ℝ≥0 Exc(𝑔𝑖), where Exc(𝑔𝑖) is theexceptional locus of 𝑔𝑖. Here 𝐴 ∗ 𝐵 denotes the join of thecones 𝐴 and 𝐵. We call 𝒞𝑖 amaximal chamber of Eff(𝑋). Wethus have Eff(𝑋) = ⋃𝑗 𝒞𝑗.

The importance of the previous definition is that it tellsus precisely how the cones in 𝑁1(𝑋)ℝ defined earlier en-code birational information of a Mori dream space 𝑋 . Forone thing, it tells us the behavior of such cones when 𝑋undergoes a ℚ-factorial modification.


Box 3: The Mori chamber decompositionWeworked out explicitly the Mori chamber decom-position of the blow-up of ℙ𝑛 at two points 𝑋 =𝐵𝑙𝑝,𝑞ℙ𝑛 in Boxes 1 and 2. The following picture is atwo-dimensional cross-section of Eff(𝑋) displayingits Mori chamber decomposition:

𝐸𝑝 𝐻𝑝,𝑞

𝐸𝑞

𝐻

𝐻𝑝

𝐻𝑞

The divisors 𝐻,𝐻𝑝, 𝐻𝑞, 𝐻𝑝,𝑞 generate Mov(𝑋), and𝐻,𝐻𝑝, 𝐻𝑞 generate Nef(𝑋). The chamber delimitedby 𝐻,𝐻𝑞, 𝐸𝑝 corresponds to the contraction of 𝐸𝑝,similarly the chamber delimited by 𝐻,𝐻𝑝, 𝐸𝑞 corre-sponds to the contraction of 𝐸𝑞, and the chamberdelimited by 𝐻,𝐸𝑝, 𝐸𝑞 corresponds to the contrac-tion of both 𝐸𝑝 and 𝐸𝑞.In the case 𝑛 ≥ 3, then 𝑋 admits only one smallℚ-factorial modification 𝑋 ′ corresponding to thechamber delimited by𝐻𝑝, 𝐻𝑞, 𝐻𝑝,𝑞. In what follows(on this and the following page), we will investi-gate the geometry of 𝑋 ′.Consider a divisor lying on the wall delimited by𝐻𝑝, 𝐻𝑞, for instance 𝐷 ≡ 𝐻𝑝 + 𝐻𝑞 ≡ 2𝐻 − 𝐸𝑝 − 𝐸𝑞,and let 𝐿 be the strict transform of the line through𝑝 and 𝑞. Then 𝐷 ⋅ 𝐿 = 0 and the linear systemof quadrics in ℙ𝑛 through 𝑝 and 𝑞 induces a mor-phism ℎ𝐷 ∶ 𝑋 → 𝑌 contracting 𝐿 to a point.On the other hand, a divisor in the maximal cham-ber delimited by 𝐻𝑝, 𝐻𝑞, 𝐻𝑝,𝑞 must be ample on 𝑋 ′.We canwrite such a divisor as 𝑎𝐻𝑝+𝑏𝐻𝑞+𝑐𝐻𝑝,𝑞 with𝑎, 𝑏, 𝑐 > 0 and observe that (𝑎𝐻𝑝+𝑏𝐻𝑞+𝑐𝐻𝑝,𝑞)⋅𝐿 =−𝑐 < 0.Note that the curve 𝐿 prevents divisors in the cham-ber ⟨𝐻𝑝, 𝐻𝑞, 𝐻𝑝,𝑞⟩ from being ample.Let 𝑔 ∶ 𝑊 → 𝑋 be the blow-up of 𝑋 along 𝐿 withexceptional divisor 𝐸𝐿 ⊂ 𝑊 . Observe that 𝐸𝐿 is aℙ𝑛−2-bundle over 𝐿.There is a morphism 𝑔′ ∶ 𝑊 → 𝑋 ′ contracting 𝐸𝐿,in the direction of 𝐿, onto a subvariety 𝑍 ⊂ 𝑋 ′ suchthat 𝑍 ≅ ℙ𝑛−2. Consider the divisor 𝐷′ ≡ 𝐻𝑝 +𝐻𝑞 + 𝐻𝑝,𝑞 ≡ 3𝐻 − 2𝐸𝑝 − 2𝐸𝑞. The linear system of𝐷′ induces a rational map 𝜙𝐷′ ∶ 𝑋 99K 𝑋 ′, and wehave the commutative diagram

𝑊

𝑋 𝑋 ′

𝑌ℎ𝐷 ℎ

𝜑𝐷′

𝑔 𝑔′

where ℎ ∶ 𝑋 ′ → 𝑌 is a small modification contract-ing 𝑍 ⊂ 𝑋 ′ to ℎ𝐷(𝐿). The rational map 𝜙𝐷′ ∶ 𝑋 99K𝑋 ′ is an isomorphism between 𝑋 ⧵ 𝐿 and 𝑋 ′ ⧵ 𝑍and replaces 𝐿 with the variety 𝑍 which is coveredby curves having nonnegative intersection with alldivisors in the chamber ⟨𝐻𝑝, 𝐻𝑞, 𝐻𝑝,𝑞⟩. Concretely,in the case 𝑛 = 3 for instance, we can fix homoge-neous coordinates [𝑥 ∶ 𝑦 ∶ 𝑧 ∶ 𝑤] on ℙ3, assumethat 𝑝 = [1 ∶ 0 ∶ 0 ∶ 0], 𝑞 = [0 ∶ 0 ∶ 0 ∶ 1], andconsider the rational maps

𝛼 ∶ ℙ3 99K ℙ7

defined by 𝛼([𝑥 ∶ 𝑦 ∶ 𝑧 ∶ 𝑤]) = [𝑥𝑦 ∶ 𝑥𝑧 ∶ 𝑥𝑤 ∶𝑦2 ∶ 𝑦𝑧 ∶ 𝑦𝑤 ∶ 𝑧2 ∶ 𝑧𝑤], that is induced by thequadrics of ℙ3 passing through 𝑝 and 𝑞, and

𝛽 ∶ ℙ3 99K ℙ11

defined by 𝛽([𝑥 ∶ 𝑦 ∶ 𝑧 ∶ 𝑤]) = [𝑥𝑦2 ∶ 𝑥𝑧2 ∶𝑥𝑦𝑧 ∶ 𝑥𝑦𝑤 ∶ 𝑥𝑧𝑤 ∶ 𝑦3 ∶ 𝑦2𝑧 ∶ 𝑦2𝑤 ∶ 𝑦𝑧2 ∶ 𝑦𝑧𝑤 ∶𝑧3, 𝑧2𝑤], that is induced by the cubics of ℙ3 havingat least double points at 𝑝 and 𝑞. Then 𝑌 is theclosure of the image of 𝛼 and 𝑋 ′ is the closure ofthe image of 𝛽.Let us give a geometric description of 𝑋 ′. LetΠ ⊂ 𝑋be the strict transform of a 2-plane through the line𝑝𝑞. The planeΠ is contracted to a point by themap𝜋𝐻𝑝,𝑞 ∶ 𝑋 99K ℙ𝑛−2 induced by 𝐻𝑝,𝑞. Indeed, 𝜋𝐻𝑝,𝑞is induced by the linear projection ℙ𝑛 99K ℙ𝑛−2with center 𝑝𝑞. Observe that a divisor in the linearsystem of 𝐷′ has a base component when restrictedto Π; namely the curve 𝐿.Therefore, 𝜙𝐷′|Π is the rational map induced by thelinear system of conics through 𝑝 and 𝑞, hence itsimage is a smooth quadric surface 𝑄Π ≅ ℙ1 × ℙ1.The quadric 𝑄Π intersects 𝑍 at a point. The mor-phism 𝜋𝐻𝑝,𝑞 ∶ 𝑋 ′ 99K ℙ𝑛−2, induced by the stricttransform of 𝐻𝑝,𝑞 on 𝑋 ′, contracts 𝑄Π to the point𝜋𝐻𝑝,𝑞(Π) and maps 𝑍 isomorphically onto ℙ𝑛−2.We have the commutative diagram

𝑋 𝑋 ′

ℙ𝑛−2𝜋𝐻𝑝,𝑞 𝜋𝐻𝑝,𝑞

𝜑𝐷′


and 𝑋 ′ has a structure of a (ℙ1 × ℙ1)-bundle overℙ𝑛−2.Summing up, the birational model of 𝑋 corre-sponding to the chamber ⟨𝐻𝑝, 𝐻𝑞, 𝐻𝑝,𝑞⟩ is a quadricbundle over ℙ𝑛−2 and, as we already noticed,the other chambers ⟨𝐻,𝐻𝑝, 𝐸𝑞⟩, ⟨𝐻,𝐻𝑞, 𝐸𝑝⟩, and⟨𝐻, 𝐸𝑝, 𝐸𝑞⟩ correspond, respectively, to ℙ3 blown-up at 𝑞, ℙ3 blown-up at 𝑝, and ℙ3. The chamber⟨𝐻,𝐻𝑝, 𝐻𝑞⟩ corresponds to 𝑋 itself.

Birational Twin VarietiesOur aim in this section is to study the weak Lefschetz prin-ciple in the context of birational geometry. In other words,if we consider two varieties 𝑋, 𝑌 equipped with an embed-ding 𝑖 ∶ 𝑌 ↪ 𝑋 , then we ask about the objects encodingbirational information of 𝑌 which are fully determinedby 𝑋 . Such objects may include effective and nef cones,or finer information such as the Mori chamber decompo-sition. The following definition focuses on two possibleforms of the weak Lefschetz principle in this context. Theywere introduced and explored in [13], [10].

Definition 4. Let 𝑋, 𝑌 be ℚ-factorial projective varieties,and let 𝑖 ∶ 𝑌 ↪ 𝑋 be an embedding. These varieties aresaid to be Lefschetz divisorially equivalent if the pull-back 𝑖∗ ∶Pic(𝑋) → Pic(𝑌) induces an isomorphism such that

𝑖∗ Eff(𝑋) = Eff(𝑌), 𝑖∗Mov(𝑋) = Mov(𝑌),𝑖∗Nef(𝑋) = Nef(𝑌).

We say that 𝑋 and 𝑌 are birational twins if they are Lef-schetz divisorially equivalent Mori dream spaces and inaddition 𝑖∗MCD(𝑋) = MCD(𝑌).Example 1. Consider a linear subspace ℙ𝑘 ⊂ ℙ𝑛 passingthrough the points 𝑝, 𝑞. Let 𝑌 ⊂ 𝑋 = 𝐵𝑙𝑝,𝑞ℙ𝑛 be the stricttransform of such a ℙ𝑘 in 𝑋 . Then 𝑌 is isomorphic to theblow-up of ℙ𝑘 at two points, and via the embedding 𝑖 ∶𝑌 ↪ 𝑋 all the conditions of Definition 4 are satisfied when𝑛, 𝑘 > 1. In other words, 𝑋 and 𝑌 are birational twins.

The Grothendieck-Lefschetz theorem implies that asmooth variety 𝑋 of Picard rank one and dimension atleast four is Lefschetz divisorially equivalent to any of itseffective ample divisors. If the rank of the Picard group ishigher than 1, then B. Hassett, H-W. Lin, and C-L.Wang [7]exhibited an example of a variety 𝑋 with a divisor 𝐷 suchthat the inclusion 𝐷 ↪ 𝑋 induces an isomorphism of Pi-card groups but does not preserve the nef cone; hence, 𝐷and 𝑋 are not Lefschetz divisorially equivalent. In otherwords, the weak Lefschetz principle fails for the Nef conein this example.

In general, we do not know a classification of the vari-eties that admit a subvariety which is birational twin to it

nor do we know of natural conditions ensuring that vari-eties 𝑌 ⊆ 𝑋 are birational twins. There are examples wherethe natural choice fails. However, in the next section weexhibit a series of examples for which the ample cone, thenef cone, and even the Mori chamber decomposition sat-isfy the weak Lefschetz principle.

Examples: Complete Quadrics and CollineationsRecently in [12–15] the birational geometry of classicalspaces, called spaces of complete forms, have been stud-ied from the perspective of Definition 4. We finish up thearticle by mentioning some of the ingredients of such astudy.

Let 𝑉 be a 𝐾-vector space of dimension 𝑛 + 1 overan algebraically closed field 𝐾 of characteristic zero. Wewill denote by 𝒳(𝑛) and 𝒬(𝑛) the spaces of completecollineations and complete quadrics of 𝑉 , respectively.These spaces are very particular compactifications of thespaces of full rank linear endomorphisms and full-ranksymmetric linear endomorphisms of 𝑉 , respectively.

In [18], [19], I. Vainsencher showed that these spacescan be understood as sequences of blow-ups of the projec-tive spaces parametrizing (𝑛+1)×(𝑛+1)matrices moduloscalars along the subvariety parametrizing rank one ma-trices and the strict transforms of their secant varieties inorder of increasing dimension.

Recall that given an irreducible and reduced non-degenerate variety 𝑋 ⊂ ℙ𝑁 , and a positive integer ℎ ≤ 𝑁,the ℎ-secant variety secℎ(𝑋) of 𝑋 is the subvariety of ℙ𝑁obtained as the closure of the union of all (ℎ − 1)-planesspanned by ℎ general points of 𝑋 . Spaces of matrices andsymmetric matrices admit a natural stratification dictatedby the rank. Observe that a general point of the ℎ-secantvariety of a Segre, or a Veronese, corresponds to a matrixof rank ℎ. More precisely, let ℙ𝑁 be the projective spaceparametrizing (𝑛 + 1) × (𝑛 + 1) matrices modulo scalars,ℙ𝑁+ the subspace of symmetric matrices, 𝒮 ⊂ ℙ𝑁 theSegre variety, and 𝒱 ⊂ ℙ𝑁+ the Veronese variety. Sincesecℎ(𝒱) = secℎ(𝒮) ∩ ℙ𝑁+ , the natural inclusion ℙ𝑁+ ↪ ℙ𝑁lifts to an embedding 𝑖 ∶ 𝒬(𝑛) ↪ 𝒳(𝑛).

We finish up by citing our contribution to the vastrealm of Lefschetz-type theorems: the spaces of completequadrics and complete collineations are Lefschetz diviso-rially equivalent, for all 𝑛 > 1, via the embedding 𝑖 ∶𝒬(𝑛) ↪ 𝒳(𝑛) [13, Theorem A]. Furthermore, in the caseof complete quadric surfaces, we have that 𝒬(3) and 𝒳(3)are birational twins [13, Theorem B].

Example 2. The space 𝒳(3) is the blow-up of the projec-tive space ℙ15 along the Segre variety 𝒮 ≅ ℙ3 × ℙ3 ⊂ ℙ15,and also along the strict transform of sec2(𝒮). We willdenote by 𝐻 the strict transform of a general hyperplaneof ℙ15 and by 𝐸1, 𝐸2 the exceptional divisors over 𝒮 andsec2(𝒮), respectively. Similarly, 𝒬(3) is the blow-up of the


projective space ℙ9 along the Veronese variety𝒱 ⊂ ℙ9, andalso along the strict transform of sec2(𝒱). We will denoteby 𝐻+, 𝐸+1 , 𝐸+2 the divisors on 𝒬(3) corresponding to thestrict transform of a general hyperplane of ℙ9 and the ex-ceptional divisors over 𝒱 and sec2(𝒱), respectively. TheMori chamber decomposition MCD(𝒳(3)) is displayed inthe following two-dimensional cross-section of Eff(𝒳(3)):

𝐸1 𝐸3

𝐸2

𝐷2

𝐻 𝐷3

𝐷𝑀

where 𝐷𝑀 ≡ 6𝐻 − 3𝐸1 − 2𝐸2, 𝐷2 ≡ 2𝐻 − 𝐸1, 𝐷3 ≡ 3𝐻 −2𝐸1 − 𝐸2. Here, 𝐸3 ≡ 4𝐻 − 3𝐸1 − 2𝐸2 is the class of thestrict transform of sec3(𝒮). The movable cone Mov(𝒳(3))is generated by ⟨𝐻, 𝐷2, 𝐷3, 𝐷𝑀⟩.

The spaces 𝒳(3) and 𝒬(3) are birational twins. Hence,the Mori chamber decompositionMCD(𝒬(3)) is obtainedfromMCD(𝒳(3)) above by simply replacing 𝐻,𝐸1, 𝐸2 with𝐻+, 𝐸+1 , 𝐸+2 .

ACKNOWLEDGMENTS. We thank the referees fortheir careful reading of our manuscript. Their manyconstructive suggestions helped us shape this article.We also thank Omar Antolı́n Camarena for helpful dis-cussions. During the preparation of this article the firstauthor was partially supported by the CONACYT grantCB-2015/253061.

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[12] C. Lozano Huerta, Birational geometry of the space of com-plete quadrics, Int. Math. Res. Not. 2015 (2015), 12563–12589.

[13] C. Lozano Huerta and A. Massarenti, The Lefschetz prin-ciple in birational geometry: birational twin varieties, Bira-tional Geometry and Moduli Spaces, Springer INdAM Se-ries, vol. 39, Springer, Switzerland, 2020, pp. 109–131.

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[15] A. Massarenti, On the birational geometry of spaces of com-plete forms II: Skew-forms, J. Algebra 546 (2020), 178–200.MR4032731

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CésarLozano Huerta

Alex Massarenti

Credits

The opening image is courtesy of Alex Massarenti.Photo of César Lozano Huerta is courtesy of Adrián Lozano.Photo of Alex Massarenti is courtesy of Valentina Pastorello.

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