 Alright so you see so as you have just now seen we have these examples of quasi-affine varieties are not affine alright and then the what I am going to do is to tell you about more general varieties is called projective varieties and open subsets of such projective varieties which are called quasi-projective varieties okay then our definition of variety will include affine, quasi-affine, projective and quasi-projective okay and these projective varieties are a completely different class of specimens okay whose properties are very different from the properties of affine varieties okay but to tell you where they come from I just want to tell you that they come from process of gluing okay so let me give you some motivation so you see let us look at the usual topology okay let us look at the usual topology and look at well for example look at the complex plane or the real plane alright and so you know let us do the following thing we take the usual plane and then suppose I draw the sphere here okay so here is my sphere so this is the unit sphere in three space okay and assume that you are in R3 okay assume you are in R3 and what you do is well you have this is the origin this is the point 1, 1, 0 okay and or rather 1, 0, 0 and you know so what I do is if I take the if I call this point as a north pole and if I call this point as a south pole okay you would have heard of the so called stereographic projection so called stereographic projection in complex analysis which identifies so if I call this sphere as S2 so S2-the north pole can be identified homeomorphically with R2 okay by projecting from the north pole and S2-the south pole can be identified homeomorphically with R2 by projecting from the south pole okay so these are the so called Riemann stereographic projections from the Riemann sphere okay to the plane and therefore the sphere-the north pole is compactified the one point compactification of the sphere-the north pole is a sphere and that corresponds to that will correspond to the extended plane by adding a point at infinity under this homeomorphism so what it tells you is that the real plane has a one point compactification which is just the sphere and so there are these two there are these two stereographic projections. Now what you must understand is that if you consider each of these things they are you can you can call them as two open subsets of the sphere both are open subsets of the sphere okay and they cover the sphere and each open subset looks like R2 because looks like R2 means it is a homeomorphic R2 and the homeomorphism is via the stereographic projection. So what has happened is so what we say is this is a standard example of what is called as gluing so what you do is you take two copies of the plane okay and you glue them together okay so basically you take a copy of the plane and then you fold it to get the sphere-the north pole take the other copy of the plane fold it to get the sphere-the south pole and you know just glue them together and you get the sphere okay. Now this is a standard procedure you have some spaces you glue them together to produce new spaces but the point is when you do this the new space that you get will have completely new properties. So for example in this case we take the sphere the new topological property that you get is that it is compact whereas you know neither of the two copies that you originally started with to glue to get the sphere is compact of course you know both R2's are both copies of the plane are non-compact because you know in nuclear space something is compact a substance is compact definitely if it is both glues and bounded with respect to the usual topology. So the moral of the story is that you know actually you are able to by gluing spaces you are getting spaces with new properties okay and so this gluing process is the process that is used to produce new spaces from old spaces alright and basically another good example of gluing is well you know there are several examples for example you know I can take yeah so you know I can just take I can take the complex I can take the plane okay and thinking of this is a complex plane let me say something if you glue it correctly okay then you can make sense instead of thinking of S2 as just the real sphere you can think of S2 as a surface on which you can do complex analysis you can make it into a Riemann surface okay and you can make sense of holomorphic functions and then Luehl's theorem will tell you that there are no global holomorphic functions it will tell you that every global holomorphic function will be constructed. So the beautiful thing is that on the plane you will have so many holomorphic functions okay you have so many entire functions whereas this glued object there are no entire functions the only entire functions namely the functions the holomorphic everywhere are constant and you know so you are basically having two affine spaces you have glued them together to get this space and this space is compact and it has no global non-constant functions okay the same thing happens in algebraic function a projective space is gotten by gluing bunch of affine spaces okay and on the projective space you will see that there are no global regular functions the only global regular functions and the projective space will be constant okay and it is a complete analogy to what is happening here. So it is a gluing process so projective spaces are gotten by gluing affine spaces okay just like the sphere is gotten by gluing 2 copies of R2 alright of course some other examples of gluing are for example you know if you take a horizontal strip or a vertical strip for that matter and then you know if you are for that matter you know well if you glue the top edge and the bottom edge what will happen is that you will get a cylinder you will get a cylinder and that is by identifying the top edge and the bottom edge you cut off you cut the strip and then you identify the top edge of the strip with the bottom edge of the strip and fold it out you will get a cylinder you will get an infine cylinder. Now the original strip is topologically different from the cylinder because the original strip is simply connected. Any nice any simple closed curve in that original strip can be completely you know shrunk to a point where the cylinder is not simply connected because the any loop that goes around the cylinder that cannot be continuously shrunk to a point. So you see again you have produced by a gluing process you have produced a new topological space with topological properties are very different from the properties of original space alright. Another example is of course you know you could have taken you could have you could have done also something like this you could have taken just a parallelogram okay and then you could have glued the parallel opposing edges and the result is that you will get a torus because if you glue the upper edge of the lower edge you will get a cylinder with two circles on the two ends which need to be further identified if you identify them you will get a torus and the beautiful thing is that this is simply connected but this is not okay. So the gluing process is a very standard process it is a process that allows you to produce new spaces with new object with new properties okay and you must think of projective space also as coming out of gluing process okay. So I will explain how projective n dimensional complex space is gotten by gluing n plus 1 copies of n dimensional affine space okay and on what we are going to do is that we are going to define a Zariski topology on the projective space okay. So we are going to define algebraic sets we are going to define irreducible sets we are going to define close subsets of projective space call them projective varieties and then whatever we did for affine varieties lot of similar results like the nuisance etc will also work for the projective case okay. But of course certain things will go wrong alright and I will explain in the coming lectures in this and the coming lectures what is going to go wrong and what is not going to go wrong. So let me start with the definition of projective space so Tnc so this is complex projective n dimensional space here is complex projective n dimensional space and how is it defined it is the space of lines in affine space. So complex projective n dimensional space is a space of lines in An plus 1 okay and through the origin and that is true not only for complex numbers for any field and how do I get it as a space the reason I have to consider lines in n plus 1 space is because you see I want an n dimensional object okay and if I take lines in n space okay by taking lines I am cutting down by 1 dimension so the resulting space will be n minus 1 dimensional. So if I take space of lines in n space I will get an n minus 1 dimensional space okay because I am actually modding out by scalars. So if I want an n dimensional projective space I should take lines in n plus 1 space okay. So how does one get it one takes points in An plus 1 and then you go modulo and equivalence relation what is the equivalence relation the equivalence relation is very very simple so you know if you give me a point lambda 1 etc lambda n okay then the that point defines the same line as some other point if and only if the two points have coordinates which differ by a non-zero constant multiple okay. So you know if I take a point lambda 1 lambda n so I will call them as lambda not to lambda n right so I am in An plus 1 so the coordinates are n plus 1 coordinates which I am not labelling the coordinates 1 through n plus 1 I am labelling them from 0 through n which is the standard convention whenever you do studying projective space. So the line passing through this point so this is the line passing through the point lambda 1 lambda 0 lambda n this is the line passing through that and this is what I am going to do I am going to take this point lambda 0 lambda n I am simply going to map it to the line passing through lambda 0 lambda n and of course through the origin that is the line that joins the point this point to the origin I am simply mapping this point to that line okay and what I want to understand is that this is an equivalence relation in the sense that if you take this L lambda 0 lambda n is the same as L mu 0 etc mu n that means both these points lie on the same line and you know both these points lie on the same line if and only if this is non-zero multiple of that by a single scalar non-zero scalar okay so this is if and only if there exists p a non-zero element of the field such that mu i is equal to t lambda i for every i okay. So you know between n plus 1 tuples of points you put this equivalence relation this is an equivalence relation that 2 points 1 point is a multiple of the other point by a non-zero element of the field okay that is an equivalence relation and if you go model of that equivalence relation what you are going to get is precisely the projective space which is the space of lines okay 2 points here if they are they are equivalent namely they will differ by their coordinates differ by 1 in the same scalar non-zero scalar multiple if and only if the lines that they define through the origin are the same okay. So the space of so what has happened is that we have gotten the projective space as a quotient okay it is a set modulo and equivalence relation so it is a quotient and this is very very good because once you have this for example if I have a topology here I can transport the topology here by giving this the quotient topology alright. So it is always good that whenever you have a quotient kind of situation then you can transport from the source lot of things to the target right. So well you know I have this Zariski topology on this because this is after all this is just this is after all sitting inside a n plus 1 which has a Zariski topology this is a fine n plus 1 space and therefore this has a Zariski topology and I can put the quotient topology on this and that will give me a Zariski topology on p n alright and in fact it will happen that this map is this map will be an in fact even an open map okay and it will be of course continuous for the Zariski topology alright but then there is another way of defining the Zariski topology on this in a very in a slightly analogous way to the definition of Zariski topology on affine space. So you see so the point I want to make is that you know of course I can for example I could have put k equal to forget k being algebraically closed I could have taken k to be real numbers and then I will get real projective space okay and then I can study things on this I can simply take the usual topology on top and then you know give this the quotient topology alright I could have done that and similarly I could instead of k I can take complex numbers and then I will get complex projective space and on the complex projective space I could have again put the usual topology I have the usual topology here I could have given the quotient topology okay and because complex numbers is also an algebraically closed field I have another topology here which is the Zariski topology that also I can use to give a quotient topology here. So complex projective space has two topologies one is the topology which comes as a quotient topology for the usual topology and the other topology comes as a quotient topology for the Zariski topology on the on this on this punctured affine space above okay. So now let me go back let me go back to how we define the Zariski topology on A and A and A plus 1 see the Zariski topology on affine space was defined by giving closed sets and the closed sets were given as 0 sets of I mean common 0 loci of a bunch of polynomials okay. Now what we are going to do is we are going to imitate the same thing here what we are going to do is we are going to look at common 0 loci of a bunch of homogenous polynomials okay and then you see it will make sense you see if you take a polynomial function for a polynomial function of course if you take a polynomial in n plus 1 variables if I change if I evaluate that polynomial on a line alright of course the polynomial the values will change okay but if the polynomial is homogenous the property of it vanishing or not will not change okay. So this is the first observation if f of x0 xn is a homogenous polynomial okay say of say homogenous of degree d then either f restricted to line through lambda0, lambda n is identically 0 or otherwise okay u so what I am trying to say is because this is because you see f of lambda0, lambda n suppose I put t lambda0 not etc t lambda n is t power d f of lambda or not etc lambda n for t non-zero constant. So you know if the polynomial vanishes at one point of the line then it will vanish at every point of the line okay so in other words so I can make sense of whether a polynomial vanishes on a line or not but what is a line? A line is a point here so I can make sense of whether a polynomial vanishes at a point here or not okay and then I define the closed sets here to be common 0 low side of the bunch of points where the polynomials vanish where all those points where a bunch of homogenous polynomials vanish okay. So what you do in other words it makes sense to look at the 0 set of f in pn okay what is the 0 set of f in pn? Point in pn where f does not vanish is corresponds to a line on which f does not vanish point in pn where f vanishes corresponds to a line on which f vanishes okay so let me repeat I can make sense of the 0 set of a polynomial in projective space namely it is all those lines on which f vanishes all those lines on which f vanishes and I do not have to just do it for one polynomial I can do it for any collection of polynomials. So what you do is more generally we may define we may define a projective algebraic set in pn to be the common 0 locus of a subset of homogenous polynomials polynomials in k x0 etc xn okay and the fact is that we get topology on projective space that topology will be the so called Zariski topology okay you can check that this gives a check that we get a topology called Zariski topology on projective space by taking closed sets to be projective algebraic sets okay if you take projective algebraic sets to be closed sets you get a topology on projective space and that is called Zariski topology and the fact is that the fact is the fact is that topology is the same as the quotient topology that you get from the Zariski topology on top this topology is the same check also that this topology is same as the quotient topology for pi I will call this map as pi the projection using quotient topology for pi from the Zariski topology on an plus 1 okay. So the moral of the story is that you can get a Zariski topology you can get a topology on this Zariski topology on this that you learn two ways either you take the quotient topology I mean it is a topology that you put that makes this map continuous okay and that is one that is one definition okay. The other definition is that you define the topology directly on this to be given by closed sets which are given by common 0 loci of a bunch of homogenous polynomials the only difference with the projective case and affine case is that in affine case you consider all polynomials but in the projective space in the projective case you consider only homogenous polynomials and you know why you have to consider homogenous polynomials because only for a homogenous polynomial you can say for sure whether it will vanish uniformly on a line passing through the RJ okay. If it is a non homogenous polynomial it could vanish at some points on the line and it could be non vanishing at other points on the line okay. If you take a non homogenous polynomial and take it 0 set that 0 set will be a hyper surface okay which will be n dimensional in n person space and that hyper surface could hit the line at not at all points it may not contain the line so it could hit the line at some points and it could not hit the line at some points. So a non homogenous polynomial could vanish at some points on the line and not vanish at some other points of the line but if you have homogenous polynomial it either completely vanishes on the line or it vanishes at no point of the line okay. So if you take a homogenous polynomial it is very easy to define the 0 set of that in projective space and then if you take a bunch of homogenous polynomials then the common 0 locus of this bunch of homogenous polynomials is what is called an algebraic set and that is how a closed set is defined okay and this gives you the Zarisky topology on the projective space. And now you know lot of statements that we know for the usual affine space the same statements will carry over for projective space the only thing is for example in the affine case you deal with ideals general ideals and general polynomials in the projective space you deal only with homogenous polynomials and you will deal with ideals which are generated by homogenous polynomials and these are special they are called homogenous ideals okay. So just as in the affine case you have a bijection between radical ideals and algebraic subsets in the projective case also you will have a bijection between radical homogenous ideals of this ring polynomial ring in n plus 1 variables and algebraic projective algebraic subsets but you will have to throw out one ideal which is called the irrelevant maximal ideal and that is the ideal generated by all the variables that is the one that you have to throw out and you have to throw it out because on top you have thrown out the 0 set of that which is the origin okay you have to throw that out okay. So and just like in the affine case where you have the affine Nulstron and such which says that if you take an ideal which is you know proper ideal then the 0 set is non-empty similarly you will also see here that if you take a homogenous ideal which is not which is essentially not whose radical is not the irrelevant maximal ideal then it is 0 set will be non-empty so you get a projective version of the Nulstron and such. So lots of this correspondence between ideals and close subsets that you had for affine space will also hold for projective space and we will review that in the next lecture but there is one point of caution the point of caution is the following you can go and start defining regular functions on a projective variety okay on an open subset of a projective variety we will call an irreducible close subset of PN as a projective variety alright and it will turn out that it will be close subset will be irreducible if and only if its ideal is prime the ideal will be homogenous ideal but you will require that it has to be prime and what will happen is of course that you know in the projective case also if you look at so in the projective case also you can define regular functions the only thing is that in affine case your regular functions were quotients of polynomials okay now you have to define them as quotients of homogenous polynomials okay if you take quotients of two homogenous polynomials and assume that the both polynomials are homogenous of the same degree then that will define a proper function on affine on the projective space because you know if I divide two such polynomials then the t power d's and if they have the same degrees the t power d's will get cancelled and therefore a quotient of homogenous polynomials of the same degree will define a nice function on the projective space functions that look locally like this will be called regular functions on the projective space and then the beautiful thing is that if you try to look at any global regular function on projective space it will turn out to be constant just like if you try to look at a global polymorphic function on the Riemann sphere it has to be constant okay. So this projective and it is true for any projective variety if you look at any global regular function it will be a constant we will prove that okay and this is in sharp contrast with the case of affine variety when the global regular functions are given by all the polynomials restricted to that affine variety and there is so many of them whereas if you go to projective varieties there are no non-constant regular functions okay. So and of course I also forgot to tell you just like in this case S2 is a union of to R2 I will show in the next class that Pn is a union of n plus 1 copies of An. So the projective space locally it is covered by n plus 1 open sets each open set looks like An affine n space. So what you have done is you have taken n plus 1 copies of affine n space and glued them in a nice way to produce the projective space and the beautiful thing is that though each of the pieces that you glued with have lots of you know regular functions polynomials on this glued object there is no global regular function which is non-constant okay. So we will see all these aspects in the forthcoming lectures and let me also tell you one more point of difference that is that you know for an affine variety the coordinate ring of the affine variety is an invariant namely two affine varieties are isomorphic if and only if they are coordinate rings are isomorphic and now this is completely going to be false for projective varieties okay. So the same projective variety can be embedded into different projective spaces and if you try to define the ring of functions on that as this polynomial ring model of the ideal the homogenous ideal you will see that that ring is good is capable of changing. So the embedding of a projective variety in some projective space could be very different I mean alright. So you do not have the you do not have the beautiful analog of coordinate ring for affine varieties you do not have the correct analog in that sense for projective varieties okay and for that matter that is what leads one to study line bundles and sections of line bundles etc on projective varieties which are probably the content of a second course in algebraic geometry okay but I will stop here and we will continue in next lecture.