Imagine (see Fig.6.1) that you are standing on a horizontal plane and (with just one eye) you observe the various points of this plane. When you look at a pointP of the plane, you can represent your line of sight by the line passing throughP and your F. Borceux, An Algebraic Approach to Geometry, DOI10.1007/978-3-319-01733-4_6,
© Springer International Publishing Switzerland 2014
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Fig. 6.1
eye. In doing so, you describe a bijection between all the points of the plane and all the non-horizontal lines passing through your eye. But what about the horizontal lines through your eye?
Imagine that a lined is drawn in the horizontal plane on which you stand. Fol- lowing that line, looking further and further away, as you focus on a very distant point on the lined, your line of sight becomes almost horizontal. In fact, your line of sight tends to the horizontal as you look further and further away alongd: your line of sight approaches the horizontal lined∞through your eye, which is parallel tod.
Now imagine that two parallel linesd anddare drawn in the horizontal plane, like the two rails of a train track. When you look very far along these lines, you have the impression that they meet somewhere “on the horizon”. In fact your line of sight, when following these two linesd andd, tends as we have seen to two horizontal linesd∞andd∞ passing through your eye and respectively parallel tod andd. But sinced is parallel tod, we must haved∞=d∞ . Is this not an elegant way to explain why you have the impression that parallel rails meet “on the horizon”?
In conclusion, we have described a bijection between:
• all the lines through your eye; and
• all the points of the horizontal plane, augmented by those “points at infinity”
where you have the impression that parallel lines meet.
Historically, the projective plane was defined as the ordinary plane to which one has added “points at infinity” where parallel lines meet (see Sect. 6.1 in [7], Tril- ogy I). An alternative approach is to present the projective plane as the set of all lines through a point (which in the above illustration corresponds to your eye) of
6.1 Projective Spaces over a Field 197 three dimensional space. Indeed, why not take your eye as origin of three dimen- sional space?
Definition 6.1.1 The projective spacePn(K)of dimensionnover a fieldKis the set of all vector lines of the spaceKn+1. Each such vector line is called a point of the projective space.
Of course “vector line” means sub-vector space of dimension 1. In general, to make the language less heavy, we shall simply refer toPn(K) without repeating each time “the projective spacePn(K)of dimensionnover a fieldK”.
Definition6.1.1is definitely the correct definition of a projective space. Perhaps, from the intuitive discussion above, you might be tempted to define the projec- tive plane instead as the set of all the half lines starting from your eye and passing through a point of the horizontal plane, plus all the horizontal half lines starting from your eye. Given two parallel linesd anddin the horizontal plane, they would then meet at two distinct points at infinity (two opposite horizontal half lines through your eye), depending on whether your line of sight followsdanddone way or the other. This is not the definition of the projective plane and moreover, such a defini- tion does not lead to a good geometric theory. Indeed, through two distinct points (your “opposite” points at infinity) you would be able to draw infinitely many dis- tinct lines, namely, all the “parallel lines” of the horizontal plane in this direction.
So you would lose at once the very basic geometrical axiom attesting that there is exactly one line passing through two distinct points.
On the other hand let us observe that:
Proposition 6.1.2 The projective spacePn(K)of dimensionnover a fieldK can equivalently be defined as the set of all vector lines of an arbitraryK-vector space V of dimensionn+1.
Proof Indeed, every choice of a basis inV provides an isomorphism betweenV and
Kn+1.
Having read this section carefully, the reader cannot be blamed for thinking of a projective point as being a vector line, but this is unlikely to develop into a useful intuition. After all, when one imagines a point of the real planeR2, one certainly doesn’t view it intuitively as a pair of equivalence classes of Cauchy sequences, or as a pair of Dedekind cuts. Instead, our intuitive view of such a point is precisely that of a point on a sheet of paper, exactly the intuitive idea which motivated the technical definitions in the first place. Of course when you have to prove the very first properties of the real line—such as its completeness, or the existence of an addition, and so on—you have to switch back to your formal definition of what a real number is. However one tends to forget such formal definitions, simply maintaining the intuitive picture of “actual points on a sheet of paper”, ensuring of course that your arguments only use those results that you have already formally proved. What you draw on the sheet of paper is not part of the proof: it is just a way to support the intuition.
The same is true with projective spaces. Your basic intuition of a projective plane should remain that of a sheet of paper, which extends infinitely, far away from the concrete edges of the sheet of paper, just as one thinks of the plane R2, but in the projective case, the sheet of paper should extend even further away in order to include the “points at infinity”, those points where the lines that you draw parallel on the sheet of paper eventually meet! Of course, although pictures on a sheet of paper remain the best way to support our intuition when working in a projective plane, one should not forget that the only acceptable arguments are those which rely formally on the definitions and on anterior results. One can rapidly learn to appreciate this intuition of the “projective sheet of paper”: its power and its limitations. Even when the fieldKis a finite field, or the field of complex numbers, or any fieldKsuch that P2(K)and evenK2do not resemble a sheet of paper, our intuition, as in the affine case, is still guided by the properties of this “projective sheet”.