Elementary Algebraic Geometry: Second Edition

CHAPTER I

Examples of curves

1Introduction

The principal objects of study in algebraic geometry are algebraic varieties. In this introductory chapter, which is more informal in nature than those that follow, we shall define algebraic varieties and give some examples; we then give the reader an intuitive look at a few properties of a special class of varieties, the complex algebraic curves. These curves are simpler to study than more general algebraic varieties, and many of their simply-stated properties suggest possible generalizations. Chapter II is essentially devoted to proving some of the properties of algebraic curves described in this chapter.

Definition 1.1. Let k be any field.

(1.1.1) The set {(x1, … , xn) | xi ∈ k} is called affine n-space over k; we denote it by kn, or by kX1, …, Xn. Each n-tuple of kn is called a point.

(1.1.2) Let k[X1, …, Xn] = k[X] be the ring of polynomials in n indeterminants X1, …, Xn, with coefficients in k. Let p(X) ∈ k[X]\k. The set

is called a hypersurface of kn, or an affine hypersurface.

(1.1.3) If {pα(X)} is any collection of polynomials in k[X] the set

is called an algebraic variety in kn, and affine algebraic variety, or, if the context is clear, just a variety. If we wish to make explicit reference to the field k, we say affine variety over k, k-variety, etc.; k is called the ground field. We also say ({pα}) is defined by {px}.

(1.1.4) k² is called the affine plane. If p ∈ k[X1, X2]\k, (p) is called a plane affine curve (or plane curve, affine curve, curve, etc., if the meaning is clear from context)

We will show later on, in Section III,3, that any variety can be defined by only finitely many polynomials px.

Here are some examples of varieties in ².

EXAMPLE 1.2

(1.2.1) Any variety (aX² + bXY + cY² + dX + eY + f) where a, …, f ∈ . Hence all circles, ellipses, parabolas, and hyperbolas are affine algebraic varieties; so also are all lines.

(1.2.2) The cusp curve (Y² − X³); see Figure 1.

(1.2.3) The alpha curve (Y² − X²(X + 1)); see Figure 2.

Figure 1

Figure 2

(1.2.4) The cubic (Y² − X(X² − 1)); see Figure 3. This example shows that algebraic curves in ² need not be connected.

(1.2.5) If (p1) and (p2) are varieties in ², then so is (p1) ∪ (p2); it is just (p1 · p2) as the reader can check directly from the definition. Hence one has a way of manufacturing all sorts of new varieties. For instance, (X² + Y² − 1)(X² + Y² − 4) = 0 defines the union of two concentric circles (Figure 4).

(1.2.6) The graph (Y − p(X)) in ² of any polynomial Y = p(X) ∈ [ [X] is also an algebraic variety.

(1.2.7) If p1, p2 ∈ [X, Y], then (p1, p2) represents the simultaneous solution set of two polynomial equations. For instance, (X, Y) = {(0, 0)} ⊆ ², while (X² + Y² − 1, X − Y) is the two-point set

in ².

Figure 3

Figure 4

(1.2.8) In ³, any conic is an algebraic variety, examples being the sphere (X² + Y² + Z² − 1), the cylinder (X² + Y² − 1), the hyperboloid (X² − Y² − Z² − 1), and so on. A circle in ³ is also a variety, being represented, for example, as (X² + Y² + Z² − 1, X) (geometrically the intersection of a sphere and the (Y, Z)-plane). Any point (a, b, c) in ³ is the variety (X − a, Y − b, Z − c) (geometrically, the intersection of the three planes X = a, Y = b, and Z = c).

Now suppose (still using k = ) that we have written down a large number of sets of polynomials, and that we have sketched their corresponding varieties in n. It is quite natural to look for some regularity. How do algebraic varieties behave? What are their basic properties?

First, perhaps a simple dimensionality property might suggest itself. For our immediate purposes, we may say that V ⊂ n has dimension d if V contains a homeomorph of d, and if V is the disjoint union of finitely many homeomorphs of i (i d). Then in all examples given so far, each equation introduces one restriction on the dimension, so that each variety defined by one equation has dimension one less than the surrounding space—i.e., the variety has codimension 1. (In kn, codimension means "n – dimension.) And each variety defined by two (essentially different) equations has dimension two less than the surrounding (or ambient") space (codimension 2), etc. Hence the sphere (X² + Y² + Z² − 1) in ³ has dimension 3 − 1 = 2, the circle (X² + Y² + Z² − 1, X) in ³ has dimension 3 − 2 = 1, and the point (X − a, Y − b, Z − c) in ³ has dimension 3 − 3 = 0. This same thing happens in n with homogeneous linear equations—each new linearly independent equation cuts down the dimension of the resulting subspace by one.

But if we look down our hypothetical list a bit further, we come to the polynomial X² + Y²; X² + Y² defines only the Z-axis in ³. This one equation cuts down the dimension of ³ by two—that is, the Z-axis has co-dimension two in ³. And further down the list we see X² + Y² + Z²; the associated variety is only the origin in ³. And if this is not bad enough, X² + Y² + Z² + 1 defines the empty set ∅ in ³! Clearly then, one equation does not always cut down the dimension by one.

We might try simply restricting our attention to the good sets of polynomials, where the hoped-for dimensional property holds. But one good polynomial together with another one may not yield a good set of polynomials. For instance, two spheres in ³ may not intersect in a circle (co-dimension 2), but rather in a point, or in the empty set.

Though things might not look very promising at this point, mathematicians have often found their way out of similar situations. For instance, mathematicians of antiquity thought that only certain nonconstant polynomials in [X] had zeros. But the exceptional status of polynomials having only real roots was removed once the field was extended to its algebraic completion, = field of complex numbers. One then had a most beautiful and central result, the fundamental theorem of algebra. (Every nonconstant polynomial p(X) ∈ [X] has a zero, and the number of these zeros, when counted with multiplicity, is the degree of p(X)) Similarly, geometers could remove the exceptional behavior of parallel lines in the Euclidean plane once they completed it in a geometric way by adding points at infinity, arriving at the projective completion of the plane. One could then say that any two different lines intersect in exactly one point, and there was born a beautiful and symmetric area of mathematics, namely projective geometry.

For us, we may find a way out of our difficulties by using both kinds of completions. We first complete algebraically, using instead of (each set of polynomials p1, …, pr with real or complex coefficients defines a variety (p1, …, pr) in n); and we also complete n projectively to complex projective n-space, denoted n( ). The variety (p1, …, pr) in n will be extended in n( ) by taking its topological closure. (We shall explain this further in a moment.) By extending our space and variety this way, we shall see that all exceptions to our dimensional relation will disappear, and algebraic varieties will behave just like subspaces of a vector space in this respect.

Hence, although in ², X² + Y² − 1 defines a circle but X² + Y² only a point and X² + Y² + 1 the empty set, in our new setting each of these polynomials turns out to define a variety of (complex) codimension one in ²( ), independent of what the radius of the circle might be. (The complex dimension of a variety V in n is just one-half the dimension of V considered as a real point set ; we shall see later that as a real point set, the dimension is always even. Also, even though the locus in ² of X² + Y² = 1 does not turn out to look like a circle, we shall continue to use this term since the ²-locus is defined by the same equation. Similarly, we shall use terms like curve or surface for complex varieties of complex dimension 1 and 2, respectively.)

In general, any nonconstant polynomial turns out to define a point set of complex codimension one in n( ), just as one (nontrivial) linear equation does in any vector space. A generalization of this vector space property is:

If L1 and L2 are subspaces of any n-dimensional vector space kn over k, then

For instance, any two 2-subspaces in ³ must intersect in at least a line. In n( ) this basic dimension relation holds even for arbitrary complex-algebraic varieties. Certainly nothing like this is true for varieties in ². One can talk about disjoint circles in ², or disjoint spheres in ³. These phrases make no sense in ²( ) and ³( ), respectively; the points missing in ² or ³ simply are not seen because they are either at infinity, or have complex coordinates. (This will be made more precise soon.) Hence it turns out that what we see in n is just the tip of an iceberg—a rather unrepresentative slice of the variety at that—whose true life, from the algebraic geometer’s viewpoint, is lived in n( ).

2The topology of a few specific plane curves

Suppose we have added the missing points at infinity to a complex algebraic variety in n, thus getting a variety in n( ). It is natural to wonder what the entire completed curve looks like. We consider here only curves in ² and in ²( ); complex varieties of higher dimension have real dimension 4 and our visual appreciation of them is necessarily limited. Even our complex curves live in real 4-space; our situation is somewhat analogous to an inhabitant of Flatland who lives in ², when he attempts to visualize an ordinary sphere in ³. He can, however, see 2-dimensional slices of the sphere. Now in X² + Y² + Z² = 1, substituting a specific value Z0 for Z yields the part of the sphere in the plane Z = Z0. Then if he lets Z = T = time, he can visualize the sphere by looking at a succession of parallel plane slices X² + Y² = 1 − T² as T varies. He sees a moving picture of the sphere; it is a point when T = − 1, growing to ever larger circles, reaching maximum diameter at T = 0, then diminishing to a point when T = 1.

Our situation is perhaps even more strictly analogous to his problem of visualizing something like a warped circle in 3-space (Figure 5). The Flatlander’s moving picture of the circle’s intersections with the planes Z = constant will trace out a topological circle for him. He may not appreciate all the twisting and warping that the circle has in ³, but he can see its topological structure.

Figure 5

To get a topological look at our complex curves, let us apply this same idea to a hypersurface in complex 2-space. In ², we will let the complex X-variable be X = X1 + iX2; similarly, Y = Y1 + iY2. We will let X2 vary with time, and our screen will be real (X1, Y1, Y2)-space. The intersection of the 3-dimensional hyperplane X2 = constant with the real 2-dimensional variety will in general be a real curve; we will then fit these curves together in our own 3-space to arrive at a 2-dimensional object we can visualize. As with the Flatlander, we will lose some of the warping and twisting in 4-space, but we will nonetheless get a faithful topological look, which we will be content with for now.

Since our complex curves will be taken in ²( ), we first describe intuitively the little we need here in the way of projective completions. Our treatment is only topological here, and will be made fuller and more precise in Chapter II. We begin with the real case.

¹( ): As a topological space, this is obtained by adjoining to the topological space (with its usual topology) an infinite point, say P, together with a neighborhood system about P. For basic open neighborhoods we take

We can visualize this more easily by shrinking ¹ down to an open line segment, say by x → x/(1 + |x|). We may add the point at infinity by adjoining the two end points to the line segment and identifying these two points. In this way ¹( ) becomes, topologically, an ordinary circle.

²( ): First note that, except for X, the 1-spaces Lα = (X + αY) of XY are parametrized by α; a different parametrization, Lα′ = (α′X + Y), includes X (but not Y). Then as a topological space, ²( ) is obtained from ² by adjoining to each 1-subspace of ², a point together with a neighborhood system about each such point.

If, for instance, a given line is Lα0, then for basic open neighborhoods about a given Pα0 we take

where |(x, y)| = |x| + |y|.

Similarly for lines parametrized by α′. (When α and α′ both represent the same line Lα0 = Lαó, the neighborhoods UN(Pα0) and UN(Pαó) generate the same set of open neighborhoods about Pα0 = Pαò.)

Again, we can see this more intuitively by topologically shrinking (R² down to something small. For instance,

Figure 6

maps ² onto the unit open disk. Figure 6 shows this condensed plane together with some mutually parallel lines. (Two lines parallel in ² will converge in the disk since distance becomes more concentrated as we approach its edge; the two points of convergence are opposite points. If, as in ¹( ), we identify these points, then any two parallel lines in the figure will intersect in that one point. Adding analogous points for every set of parallel lines in the plane means adding the whole boundary of the disk, with opposite (or antipodal) points identified. All these points at infinity form the line at infinity, itself topologically a circle, hence a projective line ¹( ). Since this line at infinity intersects every other line in just one point, it is clear that any two different projective lines of ²( ) meet in precisely one point.

¹( ): Topologically, the complex projective line is obtained by adjoining to an infinite point P; for basic open neighborhoods about P, take

Intuitively, shrink down so it is an open disk, which topologically is also a sphere with one point missing (just as is topologically a circle with one point missing). Adding this point yields a sphere.

²( ): As in the real case, except for the X-axis X, the complex 1-spaces of ² = XY are parametrized by α:

another parametrization, α′X + Y = 0, includes X but not Y. Then ²( ) as a topological space is obtained from ² by adjoining to each complex 1-subspace Lα = (X + αY) (or Lα′, = (α′X + Y)) a point Pα (or Pα′). A typical basic open neighborhood about a given Pα0 is

where |(z1, z2)| = |z1| + |z2| ; similarly for neighborhoods about points .

Intuitively, to each complex 1-subspace and all its parallel translates, we are adding a single point at infinity, so that all these parallel lines intersect in one point. Each complex line is thus extended to its projective completion, ¹( ); and all points at infinity form also a ²( ). As in ²( ), any two different projective lines of ²( ) meet in exactly one point.

The reader can easily verify from our definitions that each of , ², , ² is dense in its projective completion; hence the closure of ² in ²( ) is ²( ), and so on. We shall likewise take the projective extension of a complex algebraic curve in ² to be its topological closure in ²( ).

We next consider some examples of projective curves using the slicing method outlined above.

EXAMPLE 2.1. Consider the circle (X² + Y² − 1). Let X = X1 + iX2 and Y = Y1 + iY2. Then (X1 + iX2)² + (Y1 + iY2)² = 1. Expanding and equating real and imaginary parts gives

We let X2 play the role of time; we start with X2 = 0. The part of our complex circle in the 3-dimensional slice X2 = 0 is then given by

The first equation defines a hyperboloid of one sheet ; the second one, the union of the (X1, Y1)-plane and the (X1, Y2)-plane (since Y1 · Y2 = 0 implies Y1 = 0 or Y2 = 0). The locus of the equations in (2) appears in Figure 7. It is the union of the real circle (when Y2 = 0) and the hyperbola (when Y1 = 0). The circle is, of course, just the real part of the complex circle. The hyperbola has branches approaching two points at infinity, which we call P∞ and P′∞.

Figure 7

Figure 8

Now the completion in ²( ) of the hyperbola is topologically an ordinary circle. Hence the total curve in our slice X2 = 0 is topologically two circles touching at two points; this is drawn in Figure 8. The more lightly-drawn circle in Figure 8 corresponds to the (lightly-drawn) hyperbola in Figure 7.

Now let’s look at the situation when time X2 changes a little, say to X2 = ε > 0. This defines the corresponding curve

The first surface is still a hyperboloid of one sheet ; the second one, for ε small, in a sense looks like the original two planes. The intersection of these two surfaces is sketched in Figure 9. The circle and hyperbola have split into two disjoint curves. We may now sketch these disjoint curves in on Figure 8 ; they always stay close to the circle and hyperbola. If we fill in all such curves corresponding to X2 = constant, we will fill in the surface of a sphere. The curves for nonnegative X2 are indicated in Figure 10.

Figure 9

Figure 10

For X2 < 0, one gets curves lying on the other two quarters of the sphere. We thus see (and will rigorously prove in Section II,10) that all these curves fill out a sphere. We thus have the remarkable fact that the complex circle (X² + Y² − 1) in ²( ) is topologically a sphere.

From the complex viewpoint, the complex circle still has codimension 1 in its surrounding space.

EXAMPLE 2.2. Now let us look at a circle of radius 0, (X² + Y²). The equations corresponding to (1) are

The part of this variety lying in the 3-dimensional slice X2 = 0 is then given by

The first equation defines a cone ; the second one defines the union of two planes as before. The simultaneous solution is the intersection of the cone and planes. This consists of two lines (See Figure 11). The projective closure of each line is a topological circle, so the closure of the two lines in this figure consists of two circles touching at one point. This can be thought of as the limit figure of Figure 8 as the horizontal circle’s radius approaches zero.

When X2 = ε, the saddle-surface defined by εΧ1 + Y1Y2 = 0 intersects the one-sheeted hyperboloid given by . As before, their intersection consists of two disjoint real curves, which turn out to be lines (Figure 12); just as in the first example, as X2 varies, the curves fill out a 2-dimensional topological space which is like Figure 10, except that the radius of the horizontal circle is 0 (Figure 13). To keep the figure simple, only curves for X2 0 have been sketched ; they cover the top half of the upper sphere and the bottom half of the lower sphere, the other parts being covered when X2 < 0. Hence: The complex circle of "zero radius" (X² + Y²) in ²( ) is topologically two spheres touching at one point.

Figure 11

Figure 12

Figure 13

In the complex setting, we see that instead of the dimension changing as soon as the radius becomes zero, the complex circle remains of codimension 1, so that one equation X² + Y² = 0 still cuts down the (complex) dimension by one.

Incidentally, here is another fact that one might notice: In Example 2.1, (X² + Y² − 1), the sphere is in a certain intuitive sense indecomposable, while in Example 2.2, the figure is in a sense decomposable, consisting of two spheres which touch at only one point. But look at the polynomial X² + Y² − 1; it is indecomposable or irreducible in [X, Y].¹ And the polynomial X² + Y² is decomposable, or reducible— X² + Y² = (X + iY)(X − iY)! In fact, X² + Y² + γ is always irreducible in [X, Y] if γ ≠ 0. (A proof may be given similar in general spirit to that in Footnote 1.) Hence we should suspect that any complex circle with nonzero radius should be somehow irreducible. We shall see later that in an appropriate sense this is indeed true. By the way, X² + Y² = (X + iY)(X − iY) expresses that (X² + Y²) is just the union (X + iY) ∪ (X − iY). Each of these last varieties is a projective line, which is topologically a sphere; and any two projective lines touch in exactly one point in ²( ). This is a very different way of arriving at the topological structure of (X² + Y²).

EXAMPLE 2.3. Let us look next at a circle of pure imaginary radius, (X² + Y² + 1). Separating real and imaginary parts gives

At X2 = 0 this defines the part common to a hyperboloid of two sheets and the union of two planes. This is a hyperbola. Its two branches start approaching each other as X2 increases, finally meeting at X2 = 1 (the hyperboloid of two sheets has become the cone ). Then for X2 > 1, we are back to the same kind of behavior as for (X² + Y² − 1) when X2 > 0. Figure 14, analogous to Figures 10 and 13, shows how we end up with a sphere. Later we will supplement this result by proving:

Topologically, (X² + Y² + γ) in ²( ) is a sphere iff γ ≠ 0.

Figure 14

Do other familiar topological spaces arise from looking at curves in ²( )? For instance, is a torus (a sphere with one handle—that is, the surface of a doughnut) ever the underlying topological space of a complex curve? More generally, how about a sphere with g handles in it (topological manifold of genus g)? Let us consider the following example:

EXAMPLE 2.4. The real part of the curve (Y² − X(X² − 1)), frequently encountered in analytic geometry, appears in Figure 3. (The reader will learn, at long last, what happens in those mysterious excluded regions −∞ < X < − 1 and 0 < X < 1.)

Separating real and imaginary parts in Y² — X(X² − 1) = 0 gives

When X2 = 0, this becomes

Then either Y1 = 0 or Y2 = 0. When Y2 = 0, the other equation becomes . The sketch of this is of course again in Figure 3—that is, when X2 = Y2 = 0 we get the real part of our curve. When Y1 = 0, we get a mirror image of this in the (X1, Y2)-plane. The total curve in the slice X2 = 0 appears in Figure 15.

Note that in the right-hand branch, Y1 increases faster than X1 for X1 large, so the branch approaches the Y1-axis. Similarly, the left-hand branch approaches the Y2-axis. But in ²( ), exactly one infinite point is added to each complex 1-space, and the ( Y1, Y2)-plane is the 1-space Y = 0. Hence the two branches meet at a common point P∞. We may topologically redraw our curve in the 3-dimensional slice as in Figure 16.

Figure 15

By letting X2 = ε in (7) and using continuity arguments, one sees that the curves in the other 3-dimensional slices fill in a torus. In Figure 17, solid lines on top and dotted lines on bottom come from curves for X2 0. The rest of the torus is filled in when X2 < 0. The real part of the graph of Y² = X(X² − 1) is indeed a small part of the total picture!

We now generalize this example to show we can get as underlying topological space, a sphere with any finite number of handles; this is the most general example of a compact connected orientable 2-dimensional manifold. Such a manifold is completely determined by its genus g. (We take this up later on; Figure 19 shows such a manifold with g = 5.)

Figure 16

Figure 17

EXAMPLE 2.5. (Y² − X(X² − 1) · (X² − 4) · … · (X² − g²)). For purposes of illustration we use g = 5. The sketch of the corresponding real curve appears in the (X1, Y1)-plane of Figure 18. The whole of Figure 18 represents the curve in the slice X2 = 0.

Note the analogy with Figure 15. As before, the branches in Figure 18 meet at the same point at infinity. This may be topologically redrawn as in Figure 19, where also the curves for X2 0 have been sketched in.

We now see that looking at loci of polynomials from the complex viewpoint automatically leads us to topological manifolds! Incidentally, these last manifolds of arbitrary genus are intuitively indecomposable in a way that the sphere was earlier, so we have good reason to suspect that any polynomial Y² − X(X² − 1)· (X² − 4) … (X² − g²) is irreducible in [X, Y]. This is in fact so. Note, however, that a polynomial having as repeated factors an irreducible polynomial may still define an indecomposable object. (For example, (X − Y) = ((X − Y)³) is topologically a sphere in ²( ).) We also recall that if we take a finite number of irreducible polynomials and multiply them together, the irreducibles’ identities are not obliterated, for we can refactor the polynomial to recapture the original irreducibles (by uniqueness). The same behavior holds at the geometric level; each topological object in ²( ) coming from a (nonconstant) polynomial p ∈ (X, Y) is 2-dimensional, but it turns out that objects coming from different irreducible factors of p touch in only a finite number of points, and that removing these points leaves us with a finite number of connected, disjoint parts. These parts are in 1:1 onto correspondence with the distinct irreducible factors of p. For instance, (p), with

Figure 18

Figure 19

Figure 20

turns out to look topologically like Figure 20; it falls into three parts, the two spheres corresponding to the factors Y and (Y − 1), and the manifold of genus 2, corresponding to the 5th degree factor. The spheres touch each other in one point, and each sphere touches the third part in 5 points.

EXAMPLE 2.6. We cannot leave this section of examples without at least briefly mentioning curves with singularities; an example is given by the alpha curve (Y² − X²(X + 1)) (Figure 2). Separating real and imaginary parts of Y² − X²(X + 1) = 0 and setting X2 = 0 gives us a curve sketched in Figure 21. The two branches again meet at one point at infinity, Ρ∞, and the other curves X2 = constant fit together as in Figure 22. Topologically this is obtained by taking a sphere and identifying two points. Note that Y² − X²(X + 1) is just the limit of Y² − X(X − ε)(Χ + 1) as ε → 0. One can think of Figure 22 as being the result of taking the topological circle in Figure 17 between the roots 0 and 1 and squeezing this circle to a point. Also note that this squeezing process not only introduces a singularity, but has the effect of decreasing the genus by one ; the genus of (Y² − X(X² − 1)) is 1, while (Y² − X²(X + 1)) is a sphere (genus 0) with two points identified. One may instead choose to squeeze to a point, say, the circle in Figure 17 between roots —1 and 0; this corresponds to (Y² − X²(X − 1)). Its sketch in real (X1, Y1)-space is just the mirror image of Figure 2. Squeezing this middle circle to a point gives a sphere with the north and south poles identified to a point; the reader may wish to check that these two different ways of identifying two points on a sphere yield homeomorphic objects.

What if one brings together all three zeros of X(X + 1)(X − 1)? That is, what does (limε→0[Y² − X(X + ε)(X − ε)]) = (Y² − X³) look like? Of course its real part is just the cusp of Figure 1 ; the origin is again an example of a singular point. As it turns out, (Y² − X³) is topologically a sphere (Exercise 2.2).

Figure 21

Figure 22

After seeing all these examples, the reader may well wonder:

What is the most general topological object in ²( ) defined by a (nonconstant) polynomial p ∈ [X, Y]?

The answer is :

Theorem 2.7. If p ∈ [X, Y]\ is irreducible, then topologically (p) is obtained by taking a real 2-dimensional compact, connected, orientable manifold (this turns out to be a sphere with g < ∞ handles) and identifying finitely many points to finitely many points; for any p ∈ [X, Y]\ , (p) is a finite union of such objects, each one furthermore touching every other one in finitely many points.

We remark that a (real, topological) n-manifold is a Hausdorff space M such that each point of M has an open neighborhood homeomorphic to an open ball in n. For definitions of connectedness and orientability, see Definitions 8.1 and 9.3, and Remark 9.4 of Chapter II.

One of the main aims of Chapter II is to prove this theorem.

EXAMPLE 2.8. In Figure 23 a real 2-dimensional compact, connected, orientable manifold of genus 4 has had 7 points identified to 3 points (3 to 1,2 to 1, and 2 to 1).

Remark 2.9. We do not imply that every topological object described above actually is the underlying space of some algebraic curve in ²( ). However, one can, by identifying roots of Y² − X(X² − 1) … (X² − g²) manufacture spaces having any genus, with any number of distinct 2 to 1

Figure 23

identifications. But how about any number of 3 to 1, or 4 to 1 identifications, etc.? And in just how many points can we make one such indecomposable space touch another? Even partial answers to such questions involve a careful study of such things as Bézout’s theorem, Plücker’s formulas, and the like.

EXERCISES

2.1Show, using the slicing method of this section, that the completion in ²( ) of the complex parabola (Y − X²) and the complex hyperbola (X² − Y² − 1) are topologically both spheres.

2.2Draw figures corresponding to Figures 7–10 to show that the completion in ²( ) of (Y² − X³) is a topological sphere. Compare your figures with those for (Y² − X²(X + ε)) as ε > 0 approaches zero.

2.3Establish the topological nature of the completion in ²( ) of (X² − Y² + r), as r takes on real values in [−1, 1].

3Intersecting curves

The fact that any two indecomposable algebraic curves in ²( ) must intersect (as implied by the description in the last section), follows at once from the dimension relation

which means, in our case, cod( (pi) ∩ (pj) 2, or

EXAMPLE 3.1. For two parallel complex lines in ², the above amounts to a restatement that these lines must intersect in ²( ).

EXAMPLE 3.2. Any complex line and any complex circle in ²( ) must intersect. One can actually see, in Figure 7, how any parallel translate of the complex Y-plane along the X1-axis still intersects the circle (X² + Y² − 1), either in the (X1, Y1)-plane (as we usually see the intersection), or in the hyperbola in the (X1, Y2)-plane, for |X1| > 1.

EXAMPLE 3.3. Another case may be of some interest. Let us consider one curve which is a complex line, say (Y). Let another curve be (Y - q(X)| where q is a polynomial in X alone. Then the graph of Y = g(X) in XY is homeo-morphic to X; one can then easily check that (Y − q(X)) is a topological sphere in ²( ), as is (Y). By our dimension relation, these spheres must intersect, perhaps as in Figure 24.

Figure 24

Does this result sound familiar? It is very much like the fundamental theorem of algebra (every nonconstant q(X) ∈ [X] has a zero in ). This famous result can now be put into the ²( ) setting. If we do this, then in stating the fundamental theorem of algebra,

(a) There is no need to assume q(x) ∈ [X] is nonconstant ; if it is constant, we will have two lines which either coincide or which intersect at one point at infinity.

(b) There is no need to assume the given line is (Y)( =