MATHEMATICAL SCIENCES VIGRE SEMINAR
MATHEMATICAL SCIENCES VIGRE SEMINAR MATH 499
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Date Created: 10/19/15
Lecture Notes Math 499699 October 27 2004 No introductory course in Algebraic Geometry is complete without at least one days discussion on projective geometry So far we have been talking about af ne spaces usually ENC Today we will mix things up a bit Our rst topic is the real projective plane Pi Recall that in the plane R2 any two line intersect in a point unless they are parallel We can get rid or this 77unless77 if we say that parallel lines meet in some sort of point atoo Since we do not want lines which already intersect to intersect again at a point at in nity there should be different points of in nity for all possible slopes of lines Formally we introduce an equivalence relation N where L1 N L2 if L1 and L2 are parallel One can check that this is indeed an equivalence relation The equivalence class L consists of all lines parallel to the given line L From this discussion we can state our rst de nition for the projective plane De nition The projective plane over R denoted Pi is the set Pi R2 U one point at 00 for each equivalence class of parallel lines 7 If we let Moo denote the point at 00 of all lines parallel to L then the set L LiULLkO is the projective line corresponding to L Any two projective lines L1L2 meet in the projective plane as follows If m If a point in R2 if L1 and L2 are not parallel 1 2 L1 00 if L1 and L2 are parallel An easy way to imagine the points at in nity is to think of a straight road in the desert As you look down the road it appears to shrink into a point as it approaches the horizon In the theory of perspective the point at which that parallel sides of the road appear to be converging is called the vanishing point Furthermore if your road happens to have a median between the lanes traveling in different directions then the median will shrink as you look toward the horizon and the lanes will all converge to one point The same concept applies to any point on the horizon ie every point on the horizon is the in nity point of some parallel set of lines This horizon idea points out another interesting property of the real projec tive plane the points at in nity form a special projective line the line at 00 Thus the set of lines in Pi are those of the form I for lines L in the plane R2 and the line at in nity It is a fact that any two distinct lines in the projective plane determine a unique point and any two distinct points in Pi determine a unique projective line So far we have seen how to represent lines in the projective plane but how do you write points In R2 points are speci ed by their coordinates but in Pi points at 00 are speci ed by lines To avoid this asymmetry we introduce new homogeneous coordinates This requires a new de nition of Pi WE do this dy de ning a new equivalence on R3 by saying 11 yl 21 N 12 yg 22 if there is a nonzero real number A such that zly121 Azgy2 22 We denote zy2 the class of nonzero points 1 y 2 E R3 7 0 which are equivalent to z y With this relation we can give another de nition of PR De nition Pi is the set of equivalence classes zy2 We can write P2 R3 7 09 If a triple z y 2 E R3 7 0 corresponds to a point p 6 Pa we say that z y 2 are the homogeneous coordinates of p and 7r7r7r are all homogeneous coordinates of the same point in 1 e c n now de ne a projective line in terms of homogeneous coordinatesi Notice that homogeneous coordinates are not unique Foe example 111 2 2 2 a De nition Given real numbers AB C not all zero the set p 6 Pi l p has homogeneous coordinates zyz with Ar By C2 0 is called a projective line of Pi Notice that this is well de ned in Pi because if Ar By C2 0 then for all A E Rquot AAI BAy CA2 MAI By C2 0 We should now check to make sure our two de nitions of the projective plane are the same Using the second de nition of the projective plane consider the map go R2 A Pi de ned by zy gt gt 1y 1 It is easy to check that this map is injectivei Also Pi golR2 is the projective line H00 at in nity de ned by 2 0 Thus we have found that 19 R2 UH For this to be the same as the rst de nition we need to show that HDo contains all of the points at in nity Thus we need to investigate the relationship between lines in the real plane and in the real projective plane We have the following correspondence af ne line projective line point at 00 Lzymzb fzymzbz lm0 Lzzc Izzc2 010 Let7s try to understand this table For a point zy on the line L de ned by y mm b gozy 1y 1 which lies on the projective line I de ned by the equation y mm 122 Thus L is a subset of Z and the rest of the points in I come form when 2 0 This is just Z N Hoar At these points we nd that y mz so get the point zmz0 1721 0 E Pf In the case where our line is de ned by z c then just as above we nd that we get the projective line I czi Then at Z N H00 we nd that the ycoordinate must be 1 by the three coordinates cannot all be zero at the same time in projective spacer An important fact that comes out of the table is that two lines in R2 meet at the same point at 00 if and only if they are parallel One can check that this indeed is enough information to prove that HDo contains all of the points at in nity and the de nitions are equivalent There is even another way of thinking about the points in the projective planer Let zy2 be the homogeneous coordinates of a point p in Pf Then all of the other homogeneous coordinates of p are given by Azyz where A E R 7 Notice that if we think of these as points in R3 then zyz and Azyz lie on the same line through the origin in Rgi The fact that zy2 000 guarantees us that a point gives us a line through the origin in Rgi Conversely given a line L through the origin in real three space a point z yz E L 7 0 gives the homogeneous coordinates of a unique point in Pf us Pi 2 lines through the origin in R3 A way to visualize this is to think about drawing a three dimensional object on a piece of paper We can generalize the idea of the real projective plane to n dimensional projective space over any e i De nition ndimensional projective space over the eld k denoted Z is the set of equivalence classes of N on kn1 7 0 where we say that zoiuzn N zgiiizn if there is a nonzero element A E k such that r0iuzn riuz us wecanwrie A 6 Th 39t P W i 0 N Each nonzero n ltuple Ioiu1n E kn1 de nes a point p in P2 and we say that 10 i i In are homogeneous coordinates of p In projective space we use Zariski topology where closed sets are de ned by the zero sets of a nite collection of polynomials Thus some of the open sets are just where polynomials do not vanish we call these fundamental if they are de ned by the nonvanishing one a single polynomiali ln projective space we have special open setsi De nition The distinguished open sets denoted Ui fori 0i i i n in P2 are de ned Ui zouizn 6 P l I One can easily show that each of these distinguished opens is isomorphic to af ne nspace over 16 hni lso the complements Pg 7 U are just lP Zili Finally these sets actually form an open cover of P2 iiei Pg Ugo Uii Finally we want to extend the idea of varieties to projective spacer Recall that in af ne space a variety is de ned by the vanishing of a nite set of polyno mialsi Now if we simply extend this de nition to projective space we run into some problems For example consider the polynomial 11 7 1 6 H10 1112 The point p 1 4 2 satis es this polynomiali But p can also be written as 2 8 4 and 8 7 42 f 0 In order to avoid such situation we require that the polynomials de ning projective varieties be homogeneousi De nition Let k be a eld and let f1 i i fs 6 H10 i i i 71 be homogeneous polynomials We de ne Vf1uif5 uov wan 6 P fia0uian Ofor all 1 S i S s We call Vf1i i f5 the projective variety de ned by f1 i i f5 One can go further and de ne the ideal of a variety as before7 and extend the dictionary of af ne geometry to projective geometryi I Will not do this here
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