Topics in computation, numerical methods and algebraic  geometry

Topics in computation, numerical methods and algebraic geometry

DAVID EKLUND

Doctoral Thesis

Stockholm, Sweden 2010

TRITA-MAT-10-MA-13 ISSN 1401-2278

ISRN KTH/MAT/DA 10/06-SE ISBN 978-91-7415-770-3

KTH Matematik SE-100 44 Stockholm SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie doktorsexamen i matematik måndagen den 29 november 2010 klockan 13.00 i sal F3, Kungl Tekniska högskolan, Lindstedtsvägen 26, Stockholm.

© David Eklund, 2010

Tryck: Universitetsservice US AB

iii

Abstract

This thesis concerns computation and algebraic geometry. On the computational side we have focused on numerical homotopy methods. These procedures may be used to numerically solve systems of polynomial equations. The thesis contains four papers.

In Paper I and Paper II we apply continuation techniques, as well as symbolic al- gorithms, to formulate methods to compute Chern classes of smooth algebraic varieties.

More specifically, in Paper I we give an algorithm to compute the degrees of the Chern classes of smooth projective varieties and in Paper II we extend these ideas to cover also the degrees of intersections of Chern classes.

In Paper III we formulate a numerical homotopy to compute the intersection of two complementary dimensional subvarieties of a smooth quadric hypersurface in projective space. If the two subvarieties intersect transversely, then the number of homotopy paths is optimal. As an application we give a new solution to the inverse kinematics problem of a six-revolute serial-link mechanism.

Paper IV is a study of curves on certain special quartic surfaces in projective 3-space.

The surfaces are invariant under the action of a finite group called the level (2, 2) Heisen-

berg group. In the paper, we determine the Picard group of a very general member of this

family of quartics. We have found that the general Heisenberg invariant quartic contains

320 smooth conics and we prove that in the very general case, this collection of conics

generates the Picard group.

Contents

Contents iv

I Introduction and background 1

1 Introduction 3

2 Background 9

2.1 Numerical homotopy methods . . . . 9

2.2 Intersection theory . . . . 12

Bibliography 19 II Papers 21 3 Chern numbers of smooth varieties via homotopy continuation and intersection theory 23 3.1 Introduction . . . . 25

3.2 Intersection theory and homotopy continuation . . . . 26

3.3 Algorithms . . . . 29

3.4 Examples . . . . 34

3.5 Conclusion . . . . 37

4 Computing intersection numbers of Chern classes 43 4.1 Introduction . . . . 45

4.2 Definitions and background . . . . 46

4.3 Computing Chern numbers . . . . 49

4.4 Examples . . . . 51

4.5 The degrees of the Chern classes . . . . 54

4.6 Motivation . . . . 56

iv

v

5 Algebraic C ^∗ -actions and inverse kinematics 65

5.1 Introduction . . . . 67

5.2 Background . . . . 68

5.3 The method . . . . 76

5.4 An application to kinematics . . . . 80

6 Curves on Heisenberg invariant quartic surfaces in projective 3-space 93 6.1 Introduction . . . . 95

6.2 The family of invariant quartics . . . . 96

6.3 The Picard number . . . 102

6.4 Conics on the invariant surfaces . . . 108

6.5 Invariant surfaces containing lines . . . 109

6.6 The intersection matrix of the 320 conics . . . 113

6.7 The Picard group . . . 114

Part I

Introduction and background

1

1 Introduction

This thesis is a collection of four papers that center around the subject of alge- braic geometry. The papers are about computational algebraic geometry, numerical methods in algebraic geometry, kinematics and algebraic surfaces. In particular I have focused on numerical homotopy methods, also called numerical continuation methods. These procedures may be applied to numerically solve systems of poly- nomial equations. Polynomial systems arise naturally in science, engineering and applied mathematics as well as in pure mathematics and the applications in this thesis touch upon both the pure and the applied. For example, in Paper III we give a new solution to a classical problem coming from the kinematics of mecha- nisms and in Paper I and Paper II we apply continuation techniques, as well as symbolic algorithms, to formulate methods to compute Chern classes of smooth algebraic varieties. Intersection theory is another central theme, in part because it may be applied to count the number of solutions to systems of polynomials. In- tersection theory also connects to the theory of algebraic surfaces as well as Chern class computations.

Computing Chern classes

Chern classes are fundamental invariants of great importance, for example in the context of enumerative geometry. For some examples of how Chern classes relate to other interesting things, see the last section of Paper II. Enumerative problems in geometry ask for the number of geometric objects that satisfy certain conditions.

For example, the number conics in the plane tangent to five given conics in general position (3264) or the number of lines in 3-space that intersect four given lines in general position (2). Knowing the Chern classes of a smooth projective variety helps in solving such problems on the variety.

Paper I and Paper II deal with the computation of Chern classes. In Paper I we formulate a method to compute the degrees of the Chern classes of a smooth projective variety and in Paper II the approach is extended to include the degrees of intersections of Chern classes. Among the degrees of the Chern classes one finds for example the degree of the projective variety itself and the Euler characteristic of the underlying topological space.

3

4 1. INTRODUCTION

The approach in Paper I is based on an excess intersection formula, which could be seen a generalized Bézout’s theorem. By way of example, consider the twisted cubic C in P ³ . The curve C is cut out by three quadrics, that is it is the intersection of three surfaces of degree two. But if C is realized as a component of the intersection of two quadrics and cubic, then there will be an extra point outside of C included in the intersection of the three hypersurfaces. The Bézout number minus the number of extra points in such an intersection is called the equivalence of C in the intersection. The Bézout number is the product of the degrees of the three hypersurfaces, that is 12 in this case. If the three surfaces would have intersected each other in only a finite number of points, then we would get from Bézout’s theorem that the intersection consists of 12 points counted with multiplicity. Thus the equivalence is the contribution of the curve to the Bézout number, which enables us to formulate a generalized Bézout’s theorem. In the present context, the point is that the equivalence only depends on the degrees of the three hypersurfaces and certain invariants of C, namely its Chern classes. If we know the number of extra points in such an intersection, then we know a linear relation among the degrees of the Chern classes of C. Since C has two Chern classes, by computing the number of extra points in two realizations of C as a connected component of an intersection of three hypersurfaces, we may compute the degrees of the Chern classes of C.

The approach is valid for any smooth projective variety, of any dimension and codimension, as is shown in Paper I. Work in progress suggests that the method may be extended to compute Segre classes of singular projective varieties.

In Paper II we utilize polar classes and (in an indirect manner) Vogel-cycles to compute all possible intersection numbers of the Chern classes of a smooth projec- tive variety. The main point of the paper is that certain characteristic subvarieties, the equations of which can be written down explicitly, may be expressed in terms of the Chern classes. The characteristic subvarieties depend on parameters, and generically they intersect each other properly and therefore the intersection num- bers of the Chern classes may be computed by simply intersecting the characteristic subvarieties.

One feature that is emphasized in both papers is that the procedures are im-

plementable in a numeric setting as well as a symbolic setting. Numerical methods

benefit from parallelizability and relative insensitivity to sparseness but on the

other hand do not provide proof that the output is correct. The two papers are

in part built on the belief that mathematicians often use computers to get the an-

swer to their question only to pursue some other, more conceptual, way of proving

that the answer is correct. On the other hand, the procedures may very well be

implemented symbolically. Ongoing research suggests that the excess intersection

approach to Chern classes is effective in the symbolic setting. For example, among

the degrees of the Chern classes there is the topological Euler characteristic, which

is the degree of the top Chern class. There are implemented methods to calculate

the Euler characteristic, for example one can compute it as an alternating sum of

Hodge numbers using algorithms implemented in [11]. In this context our method

is seemingly an efficient way to compute this invariant. See also [1], [22] and Uli

5

Walther’s contribution to [12].

Torus actions and kinematics

Systems of polynomial equations are abundant in mathematics, science and engi- neering, but it is difficult to find approximations to their solutions. One approach that has proven to be effective for numerically solving systems of polynomials is us- ing so-called continuation methods or homotopy methods [14, 20]. These techniques provide a means of constructing a homotopy function and a finite set of start points such that paths emanating from the start points end in a finite set of endpoints that contain all isolated solutions of the given system of equations. The main idea is to deform the given system of equations to a simpler system, solve that system and then use a numerical step-method to track back to solutions of the original equations. For efficiency, it is desirable that the number of homotopy paths is as small as possible, preferably equal to the actual number of isolated solutions. This is particularly important for systems of equations that posses a rich structure. The main difficulty is to find a homotopy that deforms the system to something simpler, while retaining enough structure for the algorithm to be efficient. The systems of equations appearing in science and engineering often have a lot of structure and symmetry that can be exploited to increase efficiency.

A typical problem in the kinematics of mechanisms asks for a description of the possible motion of a given machine. Since the equations arising from such considerations are often polynomial, algebraic geometry provides tools to solve problems from kinematics. One example is a classical problem of robotics called the inverse kinematics problem of a 6R-robot. The 6R-robot has six revolute joints joining seven rigid links in a chain. One end link is attached to the ground and on the other end link there is a tool, or end-effector. The inverse kinematics problem is to find all sets of joint angles that place the end-effector of a robot in a desired location.

In Paper III a homotopy is introduced for numerically solving certain systems of polynomial equations. The main application of the method is a new solution to the inverse kinematics problem of a 6R-robot. More precisely, Paper III gives a method to compute numerical approximations to the intersection of two m dimensional al- gebraic subsets of a smooth 2m dimensional quadric hypersurface X in projective space. The homotopy is based on an action on X of the multiplicative group C ^∗ , which induces a cell decomposition of X known as the Bialynicki-Birula decompo- sition. The intersection theory on X is well understood and the intersection ring is generated by the closures of the Bialynicki-Birula cells. The C ^∗ -action is used to deform the subvarieties that we wish to intersect to copies of the cell closures.

This allows us to formulate a homotopy with the optimal number of solution paths

in the case where the subvarieties intersect transversely. Also, the paper includes

a sketch of how the procedure generalizes to the case where X is any algebraic ho-

mogeneous space G/P , where G is a connected semi-simple linear algebraic group

6 1. INTRODUCTION

and P a parabolic subgroup of G.

The inverse kinematics problem of a 6R-robot may be formulated exactly in these terms, where X is the so-called Study quadric hypersurface in P ⁷ . The con- nection between kinematics and projective geometry is the fact that the Euclidean group of rigid motions may be embedded as an open subset of the Study quadric.

The Study quadric is fundamental to robotics, and our geometric point of view may hopefully be applied to other problems in kinematics and elsewhere.

Quartic surfaces in 3-space

A quartic surface in P ³ is the zero locus of a homogeneous polynomial of degree four in four variables. There are many interesting unsolved problems about quartic surfaces. For example, how many irreducible conics can a smooth quartic in P ³ contain? (The corresponding problem for lines is solved and the answer is 64.) At the back of the 1990 edition of Hudson’s book on Kummer surfaces from 1905 [10]

it says that:

The theory of surfaces has reached a certain stage of completeness and major efforts concentrate on solving concrete questions rather than fur- ther developing the formal theory.

The text on the back seems to be a summary of the foreword written by W. Barth.

At any rate, I have worked with concrete questions regarding very special quartic surfaces in P ³ .

When studying curves on smooth surfaces one often considers formal integer combinations of curves as well as introduces some equivalence relation on the set of such combinations. It is desirable that the equivalence relation respects intersection numbers, which essentially means that if C, C ⁰ and D are curves with C and C ⁰ equivalent, then C and D intersect in the same number of points as C ⁰ and D. One such equivalence relation is linear equivalence. The free Abelian group on curves modulo linear equivalence is called the Picard group of the surface.

The subject of Paper IV is curves on certain quartic surfaces in P ³ . The surfaces are invariant under the action of the Heisenberg group of level (2, 2). This family of quartics appeared in the classical treatises [10, 13] and also in several later works [3, 15, 16, 21]. Abstractly, the Heisenberg group is isomorphic to (Z/2Z) ⁴ and the action on P ³ is given by coordinate permutations and sign changes. Among the invariant quartics are a threefold worth of (non-smooth) Kummer surfaces whose Abelian double covers are principally polarized. The action of the Heisenberg group arises naturally on a Kummer surface since it is the 2-torsion subgroup of an Abelian surface.

The general Heisenberg invariant quartic is however smooth. The main result

of the paper is the determination of the Picard group of a very general member of

this family of quartics. Here, very general means that the result holds outside a

countable union of proper Zariski-closed subsets of the parameter space. It turns

7

out that the general Heisenberg invariant quartic contains 320 smooth conics and that in the very general case, this collection of conics generates the Picard group.

Note: Paper I and Paper III are versions of the published papers [6] and [7], respectively.

Acknowledgements

I first want to thank my advisor Sandra Di Rocco. She has been very supportive and has spent a lot of time advising me. I’m grateful for her initiative to steer my research in an applied direction. She has also done an excellent job getting me involved with interesting work and connecting me to other people. Finally, she has been directly involved in the research as a coauthor [6, 7].

The next person on my list is Chris Peterson. To me he has functioned an extra advisor. His generosity and many ideas permeate this thesis. He is a coauthor on two of the papers that appear in it.

I also want to mention Andrew Sommese, who has likewise been very generous with his ideas as well as willing to share his vast knowledge. Paper III is based on ideas of his and he is a coauthor on Paper I.

Two more coauthors should be mentioned: Charles Wampler and Dan Bates.

Wampler’s ideas in kinematics in part gave rise to Paper III.

I want to take the opportunity to thank Kristian Ranestad for several inspiring discussions and in particular the semester I spent in Oslo in 2009. He suggested to look at the surfaces investigated in Paper IV and guided me in many ways through that project.

At KTH I have often gone to Wojciech Chacholski and Dan Laksov with math- ematical questions, and they were always willing and able to help. Carel Faber, Roy Skjelnes, Mats Boij and Stephanie Yang have also taken time to discuss my research problems.

I would like to mention my former and present officemates Oscar Andersson Forsman, Martin Blomgren, David Rydh and Mehdi Tavakol, all of whom have contributed to this thesis in one way or another. Also, I would like to thank Bengt Ek for the sessions of the set theory club.

Sist men inte minst: mamma, pappa, Jakob och Tobias.

2 Background

2.1 Numerical homotopy methods

Homotopy continuation is a numerical method to solve systems of equations. We will consider polynomial systems and isolated roots only. Below we give a short description of the ideas underlying this method, see [20] for further information.

Let F be a system of polynomial equations with the same number of equations as variables. To find the isolated roots of F we deform it to another system G, the isolated roots of which are known, and then use a step-method to track from the isolated roots of G towards roots of F . The system G is called the start system and its roots are called the starting points. Let f 1 , . . . , f n , g 1 , . . . , g n ∈ C[x 1 , . . . , x n ] and consider the maps

F : C ⁿ → C ⁿ : x 7→ (f 1 (x), . . . , f n (x)), G : C ⁿ → C ⁿ : x 7→ (g 1 (x), . . . , g n (x)).

We wish to approximate the isolated points of F ⁻¹ (0).

The simplest way to deform F to G is to define a linear homotopy H : [0, 1] × C ⁿ → C ⁿ H(t, x) = (1 − t)F (x) + tγG(x),

where γ is a random complex number on the unit circle. Then H(1, x) = γG(x) and H(0, x) = F (x). The constant γ is included to introduce sufficient genericity. For some γ the method may fail but there is only a finite number of such exceptional γ. If G is chosen appropriately, then, as t goes from 1 to 0, the starting points will follow continuous paths α : [0, 1] → C ⁿ in C ⁿ which are such that every isolated point of F ⁻¹ (0) is at the end of a path (see Figure 2.1). The paths are smooth on (0, 1), that is infinitely differentiable there. To track the paths from the starting points to roots of F we use the following numerical predictor/corrector heuristics.

Let H x denote the Jacobian of H with respect to (x 1 , . . . , x n ) and let H t denote the derivative with respect to t. Then H may locally be written

H(t + ∆t, x + ∆x) = H(t, x) + H x (t, x)∆x + H t (t, x)∆t + higher order terms.

For an appropriate choice of G we have that for generic t, H _x (t, x) is invertible for all x such that H(t, x) = 0. The prediction step is to solve for ∆x assuming that

9

10 2. BACKGROUND

Figure 2.1:

t=1 G = 0 t=0

C

[0, 1]

F = 0

H = 0 α

H(t + ∆t, x + ∆x) = H(t, x) = 0 and ignoring higher order terms:

∆x = −H x (t, x) ⁻¹ H t (t, x)∆t.

The correction step is to keep t constant, ∆t = 0, and solve for ∆x while assuming that H(t, x + ∆x) = 0 and ignoring higher order terms:

∆x = −H x (t, x) ⁻¹ H(t, x).

This may be viewed as a combination of Euler’s method for differential equations and Newton’s method to solve non-linear systems of equations. Let p be a point near the path α. As depicted in Figure 2.2, we make a prediction (Euler-step) by taking a step from p in the tangent direction of α, and then apply Newton’s method to increase the accuracy of the prediction (Newton-correction). This is done successively until a root of F is reached, or the path diverges, as determined by a preset tolerance.

We now turn to the choice of the start system G. The number of isolated roots

of G is also the number of paths of the homotopy. Thus, to get all the isolated

roots of F , G should have at least as many isolated roots as F . The simplest

way of achieving this is to set up a total degree homotopy. Let d _i = deg(f _i ) for

i = 1, . . . , n. We make use of the Bézout bound on the number of isolated roots of

2.1. NUMERICAL HOMOTOPY METHODS 11

Figure 2.2:

α

Euler-step

p Newton-correction

F , which is Π ⁿ _i=1 d _i . As G we choose

g 1 = x ^d ₁

− 1 g 2 = x ^d ₂

− 1

.. . g n = x ^d _n

− 1.

The roots of G are

x 1 = e ^2πk

√ −1/d

x 2 = e ^2πk

√ −1/d

.. .

x _n = e ^2πk

^{√ −1/d}

, where 1 ≤ k _i ≤ d i for i = 1, . . . , n.

Above, we try to illustrate the idea of numerical homotopy methods through the simple choice of a linear homotopy with the total degree number of paths and start system G as above. However, the method can be used with other homotopies as well. To reduce computing time it is desirable to follow as few paths as possible and thus we might want to model the start system G more closely on F than via the Bézout bound. Also, the corrector-predictor tracking method described above is a simplification. For one thing, one needs strategies for controlling the step length but more fundamentally the whole idea is in danger if the Jacobian matrix of the system drops rank (or the Jacobian matrix is nearly singular). One might encounter singularities at the end points since a root of the original system might be multiple.

There are several so called end-games for dealing with multiple end points [20].

We will state a few results that provide a theoretical basis for continuation

methods. Proposition 2.1.1 below follows directly from [20] Theorem 7.1.6. We use

the phrase for almost all t ∈ C to mean that there is a non-empty Zariski open

subset of C where the property in question holds, that is it holds for all but finitely

many t. That a path has multiplicity 1 means that the Jacobian has full rank all

along the path and that a solution path α is isolated means that for all t ∈ [0, 1],

α(t) is an isolated point of the fiber over t.

12 2. BACKGROUND

Proposition 2.1.1. Let U ⊆ C ⁿ be Zariski-open and let F (z; t) be a system of n polynomials in n + 1 variables. Assume that for generic t ∈ C, the Jacobian matrix of F with respect to z has rank n at all isolated points of {z ∈ U : F (z; t) = 0}. Let N (t, U ) denote the number of isolated solutions of F (z; t) = 0 in U as a function of t ∈ C.

1. N (t, U ) is finite and it is the same, say N (U ), for almost all t ∈ C. We denote the finite exceptional set of t ∈ C where N (t, U ) 6= N (U ) by E.

2. For all t ∈ C, N (t, U ) ≤ N (U ).

3. Let φ : [0, 1] → C be continuous such that φ(t) / ∈ E for t ∈ (0, 1]. The homotopy F (z; φ(t)) = 0 has N (U ) continuous isolated solution paths in U of multiplicity 1 defined on (0, 1].

4. Let φ : [0, 1] → C be continuous such that φ(t) / ∈ E for t ∈ (0, 1]. Every isolated root to F (z, φ(0)) = 0 in U is the limit of some isolated solution path of the homotopy F (z; φ(t)) = 0 in U as t → 0.

Regarding the third claim of Proposition 2.1.1, if φ is smooth on (0, 1) then the paths are smooth on (0, 1) by the complex analytic implicit function theorem.

Of course, the exceptional set E of Proposition 2.1.1 is not known in general.

Therefore one often chooses a path that contains a random constant. The following

“gamma trick” gives a recipe for this and is also a theoretical justification for such a move, see [20] Lemma 7.1.3.

Lemma 2.1.2. Fix a point q 0 ∈ C ^m , a proper algebraic set E ⊂ C ^m and a point q 1 ∈ C ^m , q 1 ∈ E. For θ ∈ R let γ = e / ^iθ denote the corresponding point on the unit circle. For all but finitely many θ ∈ [−π, π], the one-real-dimensional arc φ : (0, 1] → C ^m ,

φ(t) = s(t)q 1 + (1 − s(t))q 0 , s(t) = γt

1 + (γ − 1)t , t ∈ (0, 1]

is contained in C ^m \ E.

2.2 Intersection theory

In this section we give some background on intersection theory and discuss some particular results that are used in the thesis. A general reference is Fulton’s book [8].

2.2.1 The Chow ring

Let X be a complex projective variety of dimension n and fix an embedding X ⊆ P ^r .

The free Abelian group on k-dimensional irreducible subvarieties of X is denoted

2.2. INTERSECTION THEORY 13

Z k (X). The elements of Z k (X), which are called k-cycles, are thus finite formal linear combinations of k-dimensional subvarieties of X with integer coefficients. Let Z _∗ (X) = L n

k=0 Z k (X). For any subscheme Y ⊆ X we have an associated element [Y ] ∈ Z _∗ (X) taking the multiplicity of the components of Y into account. Consider now a (k + 1)-dimensional subvariety V ⊆ X × P ¹ and assume that the projection π : V → P ¹ is dominant. Then the fibers [π ⁻¹ (0)] and [π ⁻¹ (∞)] are elements of Z k (X) which are said to be rationally equivalent. The name rational equivalence derives from the fact that P ¹ is a rational curve. This induces an equivalence relation on Z k (X), called rational equivalence, by requiring that a cycle C ∈ Z k (X) is rationally equivalent to 0 exactly when there exist (k+1)-dimensional subvarieties V ₁ , . . . , V _t ⊆ X × P ¹ that dominate P ¹ and are such that

C =

t

X

i=1

([π _i ⁻¹ (0)] − [π _i ⁻¹ (∞)]),

where π i : V i → P ¹ is the projection. Two k-cycles are rationally equivalent if their difference is rationally equivalent to 0. The quotient group of Z k (X) by rational equivalence is denoted A k (X) and is called the k ^th Chow group of X. The Chow group of X is the direct sum

A _∗ (X) =

n

M

k=0

A k (X).

For any subscheme Y ⊆ X we have an associated element [Y ] ∈ A _∗ (X). Now assume in addition that X is smooth. In this case, the Chow group may be en- dowed with a product which has a geometric meaning in terms of intersections of subschemes of X. The resulting ring is called the Chow ring and the prod- uct on A ∗ (X) is often referred to as the intersection product. More precisely we require that if Y, Z ⊆ X are subvarieties that intersect properly, that is their intersection has the expected dimension, then Y · Z = [Y ∩ Z]. In this case [Y ∩ Z] = P m

j=1 i(Y, Z; W _j )W _j where W ₁ , . . . , W _m are the irreducible components of the intersection and i(Y, Z; W _j ) denotes the intersection multiplicity of Y and Z along W _j (see [9] Appendix A). Note that the Chow ring is graded by codimension, reflecting the circumstance that if Y and Z intersect properly and are of codimen- sion a and b, respectively, then Y ∩ Z is either empty or of pure codimension a + b.

The intersection product on A _∗ (X) can be defined using a Moving Lemma of the following type (see [17, 4, 5, 8] for proofs and further information).

Lemma 2.2.1. If C = P

i a _i V _i and D = P

i b _i W _i are k-cycles on X then there exists a k-cycle C ⁰ = P

i a ⁰ _i V _i ⁰ on X that is rationally equivalent to C and V _i ⁰ intersects W j properly for all i and j. Moreover, the rational equivalence class C · D = P

i,j a ⁰ _i b j [V _i ⁰ ∩ W j ] is independent of the choice of C ⁰ .

One can define the degree map A ₀ (X) → Z sending a 0-cycle class represented by P

i a _i p _i , where a _i ∈ Z and p i ∈ X, to its degree P

i a _i . Fixing an embedding

14 2. BACKGROUND

X ⊆ P ^r , we define a degree map on A k (X) by letting the degree of α ∈ A k (X) be equal to the degree of H ^k α ∈ A 0 (X) where H ∈ A n−1 (X) is the class of a hyperplane section of X.

Example 2.2.2. In the case of P ⁿ itself we have that A _∗ (P ⁿ ) ∼ = Z[X]/(X) ⁿ⁺¹ . A natural isomorphism is given by considering the map Z[X] → A ∗ (P ⁿ ) : X 7→ H, where H ∈ A n−1 (P ⁿ ) is the class of a hyperplane, and taking the quotient by the kernel.

Example 2.2.3. For a smooth surface X ⊆ P ^r , the intersection product together with the degree map induces a symmetric bilinear form A ₁ (X) × A ₁ (X) → Z, that is A ₁ (X) is not only a group but it has the structure of a lattice. For two curves that intersect properly, that is their intersection is finite, the intersection pairing records the number of points of intersection counted with multiplicity.

Example 2.2.4. Consider a smooth quadric surface Q ⊆ P ³ . In this case, A _∗ (Q) ∼ = Z[X ¹ , X 2 ]/(X ₁ ² , X ₂ ² ), where X 1 and X 2 correspond to classes L 1 and L 2 of lines from the two rulings of Q. Keeping in mind that Q ∼ = P ¹ × P ¹ , we let L 1 be the class of {p} × P ¹ and L 2 be the class of P ¹ × {q} where p, q ∈ P ¹ . The intersection form on Q is given by L ² ₁ = L ² ₂ = 0 and L 1 L 2 = 1.

Example 2.2.5. Consider the Fermat quartic surface X = {x ⁴ + y ⁴ + z ⁴ + w ⁴ = 0} ⊆ P ³ . The Picard group A 1 (X) is a free Abelian group of rank 20 which is generated by certain lines on X, see Paper IV of this thesis. The surface X is a so-called singular Kummer surface [18], where singular means that the rank of the Picard group is maximal among K3 surfaces.

2.2.2 Chern classes

We will give a geometric definition of Chern classes. Our approach is perhaps not the most elegant but it has the advantage of being fairly concrete.

Let Z ⊆ P ^r be a smooth irreducible subvariety of dimension n and let T _Z denote the tangent bundle of Z. The Chern classes of Z are the Chern classes associated to the vector bundle T Z . We will define the Chern classes associated to any vector bundle E of rank ρ on Z. These are cycle classes c 0 (E), . . . , c ρ (E), where c i (E) ∈ A n−i (Z). First suppose that E is generated by global sections and let s 1 , . . . , s ρ+1 be general global sections of E. Then, for every 0 ≤ i ≤ ρ, the locus

D ρ−i = {z ∈ Z : s 1 (z) ∧ · · · ∧ s i+1 (z) = 0}

is either empty or of pure codimension ρ − i, where s 1 (z) ∧ · · · ∧ s i+1 (z) = 0 means that the vectors s 1 (z), . . . , s i+1 (z) are linearly dependent. Moreover, for 0 ≤ p ≤ ρ, the class [D _p ] does not depend on the choice of global sections (see [8] Example 14.4.3). In the case where E is globally generated, we define the Chern classes by

c _p (E) = [D _p ].

2.2. INTERSECTION THEORY 15

If E is not globally generated we have to twist it by some line bundle to make it so.

Let L m be the line bundle corresponding to the sheaf O Z (m). If m is big enough E ⊗ L m is generated by global sections. To define the Chern classes of E we utilize a formula to express the Chern classes of E in terms of the Chern classes of E ⊗ L m

and the Chern class of L m (see [8] Example 14.4.3):

c p (E) =

p

X

i=0

(−1) ^p−i ρ − i p − i



c 1 (L m ) ^p−i c i (E ⊗ L m ).

It remains to define the first Chern class of a line bundle on Z, which is the corre- sponding divisor class. For the fact that the Chern classes defined this way do not depend on m we refer to [8]. The Chern classes of Z are denoted c 0 , . . . , c n , that is c p = c p (T Z ).

Remark 2.2.6. We list some examples of other incarnations of the Chern classes of Z in various dimensions.

1. For a line bundle on L on Z, c 1 (L) ∈ A n−1 (Z) is the corresponding divisor class.

2. We have that c 0 is the unit in the ring A _∗ (Z) and deg(c 0 ) = deg(Z).

3. The first Chern class is equal to the additive inverse of the canonical class, that is c 1 = −K Z .

4. The degree of the top Chern class is the topological Euler characteristic of Z, that is deg(c _n ) = χ _top .

5. For curves, deg(c 1 ) = 2 − 2g, where g is the genus.

6. For surfaces, we have for the sectional genus π that π = 1

2 (deg(c 0 ) − deg(c 1 ) + 2).

2.2.3 Equivalence

In this section we will define the equivalence of a connected component for an intersection of hypersurfaces and give an example treating the case of curves.

Let X 1 , . . . , X r be hypersurfaces in P ^r and let W = X 1 ∩ · · · ∩ X r . Let X = X 1 × · · · × X r and Y = P ^r × · · · × P ^r (r factors). There is a morphism g : W → X which sends a point p to (p, . . . , p). Let N X Y denote the normal bundle of X in Y , put N = g ^∗ (N X Y ) and let π : N → W be the projection map. Now consider the normal cone of W in P ^r , which may be constructed as follows. First cover P ^r by affine open subsets S

i U _i = P ^r , for example U _i = {x _i 6= 0} where (x 0 , . . . , x _r ) are

homogeneous coordinates on P ^r . Fix i, let U = U _i , let A be the coordinate ring of

U and let I ⊆ A be the ideal defining W ⁰ = W ∩ U . Then the normal cone to W ⁰

16 2. BACKGROUND

in U is defined by C W

U = Spec( L

k≥0 I ^k /I ^k+1 ). Since the coordinate ring A/I of W ⁰ is equal to I ⁰ /I we get a morphism C _W

U → W ⁰ . The pieces C _{W ∩U}

U _i are then glued together to form the normal cone C W P ^r of W in P ^r which comes with a morphism C W P ^r → W . Put C = C W P ^r and let C 1 , . . . , C t be the irreducible components of C. Note that C has pure dimension r ([8] Appendix B.6.6). Now, C embeds in N (see [8] Section 6.1) and therefore there is an associated cycle class [C] in A _∗ (N ). Let m 1 , . . . , m t ∈ Z denote the multiplicities of C 1 , . . . , C t , that is [C] = P t

i=1 m i [C i ]. For all i, put Z i = π(C i ), let N i denote the restriction of N to Z i and let s i : Z i → N i be the zero-section of of N i . The varieties Z 1 , . . . , Z t are called the distinguished varieties of the intersection. Finally, define

α i = s ^∗ _i [C i ] ∈ A 0 (Z i ).

If S ⊆ P ^r is a closed set and Z _i ⊆ S, then we may consider α _i to be an element of A ₀ (S). The part of (X ₁ · . . . · X _r ) supported on S is is a cycle class in A ₀ (S) defined to be

X

Z

⊆S

m i α i .

Now suppose that Z is a connected component of the intersection X 1 ∩ · · · ∩ X r . We define the equivalence of Z for the intersection (X 1 · . . . · X r ) to be the part of (X 1 · . . . · X r ) supported on Z and we denote it by (X 1 · . . . · X r ) ^Z . Note that the equivalence of Z is a cycle class in A 0 (Z). Sometimes the degree of (X 1 · . . . · X r ) ^Z is also called the equivalence of Z.

We will now state two theorems that explain how the equivalence of Z is the contribution of Z to the Bézout number of the intersection and how the equivalence of a smooth connected component may be expressed in terms of its Chern classes.

For proofs see [8] Proposition 9.1.1, [8] Proposition 9.1.2 and Paper I of this thesis.

Proposition 2.2.7. Suppose that X 1 ∩ . . . ∩ X r consists of a connected component Z and a finite set R. Put n i = deg(X i ) and for p ∈ R let m p = i(p, X 1 · . . . · X r ; P ^r ) denote the intersection multiplicity of p in X 1 · . . . · X r . Then

deg((X ₁ · . . . · X _r ) ^Z ) + X

p∈R

m _p =

r

Y

i=1

n _i .

Proposition 2.2.8. Let n _i = deg(X _i ) and let the k ^th elementary symmetric func- tion in n ₁ , . . . , n _r be denoted by σ _k and put

a _i =

n−i

X

j=0

(−1) ^j r + j j

 σ _n−i−j .

If Z is a smooth connected component of T r

i=1 X i then deg((X 1 · . . . · X r ) ^Z ) = P n

i=0 a i deg(c i ).

2.2. INTERSECTION THEORY 17

Example 2.2.9. Let 1 < r be an integer and suppose that C is a smooth con- nected curve in P ^r of genus g which is a connected component of an intersection of hypersurfaces X 1 , . . . , X r in P ^r . Assume further that T r

i=1 X i = C ∪ R where R is finite and R ∩ C = ∅. Let n i = deg(X i ). In this case, the preceding propositions state that

deg(C)(

r

X

i=1

n i − (r + 1)) + 2 − 2g + deg(R) =

r

Y

i=1

n i .

For illustrative purposes we will give an elementary argument for this (see the book by Semple and Roth, [19] Chapter IX). Forgetting about schemes for a while, note that X ₁ ∩ . . . ∩ X _r−1 = C ∪ C ⁰ , where C ⁰ is either empty or a curve. For the elementary argument that we want to apply to work we have to assume in addition that for p ∈ C \ C ⁰ , X ₁ , . . . , X _r−1 are non-singular at p and the embedded tangent spaces T p X 1 , . . . , T p X r−1 intersect properly, that is their intersection is the tangent line to C at p. Now,

deg(C) + deg(C ⁰ ) =

r−1

Y

i=1

n _i .

Note that deg(C ∩ C ⁰ ) + deg(R) = n r deg(C ⁰ ) and therefore

deg(C ∩ C ⁰ ) + deg(R) =

r

Y

i=1

n i − n r deg(C).

In follows that what we want to show is that

deg(C ∩ C ⁰ ) = deg(C)(n ₁ + · · · + n _r−1 − (r + 1)) + 2 − 2g.

Let µ 1 = µ 1 (C) denote the number of tangent lines to C that meet a general linear space of codimension 2 in P ^r . Given a general enough linear space L of codimension 2, the locus of points p ∈ C such that T _p C intersects L is called a first polar locus and the corresponding class in A _∗ (C) is called the first polar class, see [8] 14.4.15 (b). The number µ ₁ is the degree of the first polar class and it is called the first class of C. Let F ₁ , . . . , F _r be polynomials that define the hypersurfaces X ₁ , . . . , X _r and let a, b ∈ (P ^r ) ^∗ be two general hyperplanes defining a general linear space of codimension 2, a = (a 0 , . . . , a r ) and b = (b 0 , . . . , b r ). Let (x 0 , . . . , x r ) be coordinates on P ^r and for p ∈ C ^r+1 , put

J _p =









∂F

∂x

(p) ^∂F _∂x

0

(p) a 0 b 0

.. . · · · .. . .. . .. .

∂F

∂x

(p) ^∂F _∂x

r

(p) a r b r







 .

Define

J = {p ∈ P ^r : det J _p = 0},

18 2. BACKGROUND

and note that deg(J ) = P r−1

i=1 (n i − 1) and that J ∩ C is finite. Since, if p ∈ C \ C ⁰ then the Jacobian of {F 1 , . . . , F r−1 } has full rank, we get that

J ∩ C = {p ∈ C : T p C ∩ a ∩ b 6= ∅} ∪ (C ∩ C ⁰ ).

For general a and b, the corresponding first polar locus of C is disjoint from C ∩ C ⁰ . It follows that deg(J ∩ C) = deg(C)( P r−1

i=1 (n i − 1)) = deg(C ∩ C ⁰ ) + µ 1 . Hence deg(C ∩ C ⁰ ) = deg(C)(n 1 + · · · + n r−1 − r + 1) − µ 1 which reduces our problem to showing that

µ 1 = 2g − 2 + 2 deg(C),

a quite interesting observation in its own right. Now project C repeatedly from a general point to acquire a smooth curve D ⊆ P ³ with the same degree, genus and first class as C. Note that µ ₁ (D) is the number of tangent lines of D that meet a general line in P ³ . Now project D one more time from a general point to obtain a plane curve D ⁰ with only nodes as singularities and with the same degree as D.

The first class of D ⁰ can still be defined as the number of smooth points where the tangent line passes through a general point in the plane. Note that µ 1 (D ⁰ ) = µ 1 (D).

The genus of the normalization of D ⁰ is equal to the genus of D, which is also the genus g of the curve C that we started with. If m is the number of nodes of D ⁰ , then

2g = (deg(D ⁰ ) − 1)(deg(D ⁰ ) − 2) − 2m,

see [8] Example 9.3.3. Let (D ⁰ ) ^∗ denote the dual curve of D ⁰ , that is the closure in (P ² ) ^∗ of the set of tangent lines to D ⁰ at smooth points. Then,

deg((D ⁰ ) ^∗ ) = deg(D ⁰ )(deg(D ⁰ ) − 1) − 2m,

see [8] Example 4.4.4. Further, observe that µ ₁ (D ⁰ ) = deg((D ⁰ ) ^∗ ). Hence the above expressions for 2g and deg((D ⁰ ) ^∗ ) yield

µ ₁ (D ⁰ ) = 2g − 2 + 2 deg(D ⁰ ).

Since C and D ⁰ have the same degree, and the same first class, it follows that µ 1 = 2g − 2 + 2 deg(C), which is what we wanted to show.

Remark 2.2.10. Regarding Example 2.2.9, we call the class [C ∩ C ⁰ ] ∈ A _∗ (C) the

first Vogel class or the first Vogel cycle. In the curve case, the first Vogel class

and the first polar class are related by the fact that their sum is a multiple of the

hyperplane section corresponding to the intersection of C by the hypersurface J .

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Part II

Papers

21