 You see a geometry on the projective space, so what this tells you is that philosophically the geometry of the projective space is controlled by the geometry of the affine space and that is going back to Euclid okay, so you know so let us recall that if you take a point in n dimensional projective space and you take a polynomial vanishing on the line above that point okay in the affine space n plus 1 dimensional affine space then we saw that every homogeneous component of that polynomial will also vanish on that line and in particular that polynomial will not have any constant term okay, so it will be a sum of homogeneous components and there won't be any homogeneous component of degree 0 alright. Now this leads into the study of what are called as homogeneous ideals and when you try to translate from the algebraic geometry of projective space to commutative algebra you end up studying properties of homogeneous ideals okay and the key to defining a homogeneous ideal is actually comes actually from this observation, so let me explain that in more detail, so the first thing is so the translation from projective geometry to commutative algebra is in the language of homogeneous ideals and graded rings okay, so this is a little bit of algebra that one needs to recall alright, so recall the following things the notion of a graded ring first a ring is called graded which is supposed to mean n graded or rather whole numbers graded if s is the direct sum of sd d greater than or equal to 0 okay with each s sub d an abelian group an abelian fact of course abelian a subgroup an additive subgroup subgroup of s such that sp into sq lands inside sp plus q for pq greater than or equal to 0 okay, so under multiplication. So you see what is a here the w corresponds to whole numbers which means that you include 0 and then along with the natural numbers which start from 1 okay and this indexing is on the whole numbers alright and the ring should break into a direct sum of pieces each piece is a additive subgroup of the additive group underlying the ring okay and there is a multiplication the multiplication should the multiplication of the pth piece the qth piece should land you inside the p plus q to p okay and of course you know well if my ring could my ring need not have one if you want okay when I want to make a general definition it need not even have one and it need not even be commutative and in that case I will also it also follows that sq into sp will also land into sp p plus q by this okay and therefore the pth piece and if you take an element in the pth piece and an element in the qth piece and you multiply them they land in the p plus q piece okay in we say that this is meant to this is also sometimes referred to as the multiplication preserves the gradation okay it respects the gradation it takes a so what you do is you think of elements of sd as elements as homogenous elements of degree d okay so the elements of each sd are called homogenous of degree d okay and what you are saying is that every element of the ring can be decomposed into homogenous elements of certain degrees and the decomposition is unique the uniqueness of the decomposition is because the direct sum okay direct sum tells you okay that every element of s can be broken down can be written as a sum of finitely many elements which have which are homogenous which belongs to certain homogenous pieces and for each homo for the each sum and in that sum is unique for that homogenous so the if you give me an element here the it is for any d it is dth homogenous piece is uniquely determined that is what the direct sum is supposed to mean alright and the multiplication preserves the it respects the homogeneity in the sense that a homogenous element of degree p multiplied by a homogenous element of degree q leads to a homogenous element of degree p plus q okay so this is the definition of what a graded ring is of course but so let me write this elements s d are called homogenous of degree d elements of sd are called homogenous of degree d alright and of course the particular case that we are interested in is polynomial rings and their quotients by prime ideals their quotients by ideals which are homogenous okay so what is a basic example the basic example is of course the polynomial ring in finitely many variables I will take the variables to be n plus 1 variables because I am always thinking of projective space so example take s equal to k x0 etc xn sd is equal to the subset of s consisting of homogenous polynomials of degree d okay and so you know that the whole polynomial ring is a direct sum of various degrees that is just a reflection of the fact that you take any polynomial you can break it down uniquely into homogenous components each component homogenous polynomial of certain fixed degree okay and so this is the example that we keep in mind okay and of course it is not just to work with it is not just enough to work with this but we need to also work with graded quotients of this okay so for that so you know the intuitive idea is very clear if you to get a quotient you have to go mod low and ideal okay but to get a graded quotient okay you have to go mod low what is called a homogenous ideal okay so what is a homogenous ideal an ideal ideal I in s which is graded where s is graded so I will draw a line here okay I will draw a line here and I am again going back to the old situation where I take a graded ring which is direct sum of homogenous pieces okay I should say it is a direct sum of pieces which correspond to homogenous elements of certain fixed homogenous degree okay so take an ideal s an ideal I in s it is called homogenous if I is I intersection sd direct sum d graded elements okay so look at this definition of what a homogenous ideal is so the definition is you take the ideal alright you intersect it with s sub d okay when you intersect the ideal with s sub d what you get is mind you are not an ideal because you are only intersecting with the additive subgroup and you know the ideal is also an additive subgroup therefore the intersection is again additive subgroup therefore each of these is an additive subgroup of the additive group underlying the ring s okay and now you take the direct sum okay of course it is a direct sum because all the sd's themselves are pieces of a direct sum you take the direct sum and that you should it is obvious that this will be contained inside I okay the right side every piece okay I intersection sd refers to all the elements of I which are homogenous of degree d what is an element of I intersection sd it is an element of I which is in sd but elements of sd are called homogenous elements of degree d so I intersection sd is those elements of I which are homogenous of degree d okay and of course if you take a direct sum of this mind you the direct sum and element in direct sum only consists of a finite sum even though the direct sum is over a collection of infinite infinitely many subscripts okay so an element here is certainly here by definition but the requirement is every element here comes from here that is the homogeneity definition okay so you know what it means it means that so so in other words in other words if f is in I okay if you take f is in f is an element of I then since s is a direct sum of all the sd's d greater than or equal to 0 what you will have is f will be f0 plus f1 plus etc up to fm okay you will get this alright where so you will get a finite expression like this okay you get a finite expression like this because it is a it is an expression a direct sum it will live only up to a finite index okay beyond this all the fj's will be 0 right so here f i are in si or fd is in sd you get a break up like this okay and so if you take an f and you break it into homogenous pieces then the condition is that this condition will tell you that each f i is also in f okay so you see the so implies that fd belongs to i for every d for every index d see why is that true that is because you see take an f here take an f here f is because f is in i and i is in this in this graded ring and this graded ring has this graded decomposition f has a decomposition where each of these pieces come or homogenous of the corresponding degrees alright on the other hand since you have written like this f also has a decomposition here because of the equality f belongs here so it corresponds to an element here so it also is it also has a decomposition in terms of homogenous elements okay but both decompositions are to be valid in s but in s there is only one decomposition the decomposition s is unique therefore what it forces is that each fd is already in i so the moral of the story is an ideal is homogenous saying that an ideal is homogenous is the same as saying that every element of that ideal you take any element in that ideal every homogenous piece of that element is also in that ideal one way of saying that ideal is homogenous is saying that it can you take any element in that ideal okay then every homogenous piece of that element is also back in that ideal okay and you see that is exactly a geometric that is exactly algebraic reflection of this geometric fact if your polynomial vanishes on a line then every homogenous component of that polynomial vanishes on that line so what you are saying is that if the line is in the zero set of an ideal suppose you take the line to be in the zero set of an ideal okay and you take an element of that ideal that means that is a polynomial which vanishes on that line then what you are saying is that every homogenous component of that polynomial is also vanishing on that line so if the zero set of an ideal contains a line what you are saying is that every element in that ideal okay every polynomial in that ideal is such that each of its homogeneous components is also again in the ideal of that line okay. So this is just a geometric reflection of this algebraic fact. So this is the key to defining what a homogeneous ideal is and the advantage of having a homogeneous ideal is that once you have a graded ring and you have a homogeneous ideal the quotient ring S mod i automatically gets a graded structure it becomes a graded ring okay. So the key to translating from projective geometry to commutative projective algebraic geometry to commutative algebra is that you have to change from ordinary ideals to projective ideals from ordinary ideals to homogeneous ideals and you have to change from ordinary rings to graded rings. So the language instead of just looking at rings and ideals commutative rings and ideals in them the language becomes a language of homogeneous ideals and graded rings okay that is the language that you should use that is the algebra that you should use for projective algebraic geometry okay and well now the fact is so let me tell you what happens there are a few nice facts so lemma that you can easily check the sum product intersection and radical so the sum product intersection of homogeneous ideals is homogeneous and the radical of a homogeneous ideal is also homogeneous. So this collection of homogeneous ideals in a graded ring is well behaved under the operation of taking some product intersection and radical okay this is a I mean this is a very straight forward verification algebraic verification which I leave you to do okay and therefore you know so you know we armed with this we can now translate from projective algebraic geometry to commutative algebra. So you know let me recall that as far as a final algebraic geometry was concerned what we did was if you recall we had affine space and we have this is the geometric picture the algebraic picture is the co-ordinate ring of affine space the algebraic picture is co-ordinate ring of affine space is polynomial ring in n variables okay and you know well you had you had a you had a map like this which is called as I and you had a map like this which is called as which we called as Z and what did these maps do well if you give me a subset T or if you give me a subset Y of affine space then I get I of Y the ideal of functions polynomials that vanish on Y and conversely if you give me an ideal I here I in the polynomial ring in n variables I get the closed subset Z of I and every closed subset is of this form okay and you know that so you get a correspondence between closed subsets here and on that side you have to take radical ideals okay and we had things like so on this side if you take sub varieties affine sub varieties which are reducible algebraic sets they corresponded on that side to prime ideals which were of course radical ideals and points here will correspond to maximal ideals there okay so we had this nice translation from algebraic geometry to commutative algebra alright this is for the affine space now we go to we can do that now for the projective space as well alright so in the same way what we do so I will have to make a statement here so here is one more lemma which I forgot to mention probably let me mention it here an ideal I in S is homogeneous if and only if it is generated by homogeneous elements so this is another definition of when an ideal is homogeneous this definition of an ideal being homogeneous requires its generators to be homogeneous okay so whereas the earlier definition of homogeneity is that you take the ideal is the sum of its components homogeneous components and which translates to saying that given any element in the ideal each of its homogeneous components is again back in that ideal okay so this is again a simple algebraic fact that you can verify as an exercise okay and the reason I need it is the following recall that we have defined in the projective space okay we have defined algebraic sets closed sets as 0 sets of a bunch of homogeneous polynomials okay and again instead of just taking the 0 set of a bunch of homogeneous polynomials you can take the 0 set of the ideal generated by these homogeneous polynomials and by this lemma that ideal will be homogeneous ideal okay so what you will do is so we will do the following thing we will use S we will use this notation S of P n this is the commutative algebraic picture this is k x0 to xn okay so this is thought of this is called the homogeneous co-ordinate ring of projective space okay projective space see in the affine situation okay A we use the word A to give you the affine co-ordinate okay which is the polynomial ring in as many variables as the dimension of the affine space now what you do is in the projective case the analog is the so called homogeneous co-ordinate okay and you know why they are called homogeneous co-ordinates because when you write a point in projective space your these co-ordinates are only when you put them together they are only common ratio okay I mean they are given by a set of ratios that is the reason we put a colon okay a point in projective space has co-ordinates x0 colon x1 colon etc xn and the colon means that there is a ratio involved okay and therefore it is a that is why it is called homogeneous and that is why it is called the homogeneous co-ordinate ring okay and for that matter each xi is a homogeneous polynomial of degree 1 right and what we do is well how did we how do we start we say closed set is of the form z of t where t is in s it is in this and I will put a I will put this h okay which means the union of the various degree d pieces namely all the possible homogeneous elements okay see this this homogeneous co-ordinate ring is a direct sum of its degree d pieces okay which is just trying to say that polynomial degree n is uniquely expressible as a sum of its homogeneous components but what you do is instead of taking if you take the direct sum you will get the homogeneous co-ordinate ring instead of taking the direct sum if you take union you will get all the homogeneous elements because by definition a homogeneous element is supposed to be an element in one of these pieces okay so where of course sd of pn is homogeneous polynomials of degree d in these variables that is what it means okay. So you take so what I am doing is why I am writing it like this I am taking I am taking a subset of homogeneous elements I am taking my t is a is not just any bunch of polynomials in this polynomial ring it is homogeneous elements that is the reason I have put the subscript the superscript h okay and that is just gotten by taking this union okay and what you do is that for this t you take the 0 set of t but now you see you are taking the 0 set in projective space okay mind you sometimes if you are working with both the affine space and the projective space at the same time you will have to worry about where you are taking the 0 sets you need better notation so sometimes it is if you do not want any confusion you put z sub pn of t which means you are looking at the 0 sets the 0 set of t and pn okay and this is how the close sets in projective space are defined this is how the Zariski topology is defined okay this was this was the second definition okay we had three definitions of Zariski topology the first one was a as quotient topology of the punctured n plus 1 dimensional affine space above the second one is this where the close sets are given by 0 sets of bunch of homogeneous polynomials and the third is of course the topology that is gotten by gluing the n plus 1 pieces though each of which is a which is isomorphic to an affine space okay of dimension n so well so this is how we have defined it okay and now what we can do is well so you have this so you have this just as in this case you have this you have this map z okay and there is also this map in this direction what is this map in this direction in the affine case you give me any set y then you look at all those polynomials which vanish on y okay and so you go like this and then this is automatically an ideal here okay so I land on the collection of ideals on this side okay so you know I also need to put an i here alright and you have to be careful that you should simply not say all the polynomials here which vanish on a given subset here mind you if a polynomial vanishes on a subset then it has to it has to be homogeneous I mean each homogeneous piece of that polynomial has to vanish on that subset you see what we just saw sometime ago was that you know if a polynomial vanishes on a line passing through the origin okay then each piece of that polynomial each homogeneous piece of that polynomial will also vanish on the line through the origin so it means that if you are so you must think of the polynomial vanishing on a line on the origin on a line through the origin as you must think of it like this take the point in projective space corresponding to that line and that point is a 0 of that polynomial in the projective space. So what you are saying is if your polynomial vanishes at the point in projective space then each of its homogeneous components will also vanish at that point in projective space and of course the constant term will not be there alright so if I want to make sense of a polynomial vanishing on a subset of projective space I need to make sure that every that first of all that it has no constant term and I also need to make sure that every homogeneous piece of that polynomial also vanishes on that subset of projective space alright. So what you do is you see finally everything reduces to vanishing of homogeneous polynomial so when you define this I you define it very carefully you keeping this in mind you define I of y to be the ideal in the homogeneous coordinate ring generated by all f in the all homogeneous f namely all homogeneous polynomials such that f of y is 0 for every y in y okay. So this is how you define when you define the ideal of y you define it as the ideal generated by all those homogeneous polynomials which vanish on y alright. So you see therefore this is an ideal which is generated by homogeneous elements therefore it is a homogeneous ideal because that is what the lemma above I have recalled says okay. So this is actually this is a homogeneous ideal this is a homogeneous ideal so what has happened is if you start with the set of homogeneous elements you get the 0 set of that which is a close subset of projective space and if you start with any subset of projective space you get the ideal of that subset and that will be a homogeneous ideal by definition and whatever happened here more or less will happen there except for 1 or 2 subtleties. So let me tell you what are the things that are going to happen you know you know a few things in the in the affine situation what do you know you know that you know if I take so the Luhl's sense says that I of z of I is rad I that is one fact okay then if I take z of script I of y I will get y bar the Zariski closure of y so z of script I of y is y bar okay and the fact is that the same thing will hold here except with one subtlety for the neutral set so let me state it here so what is going to happen here also I am going to get I of z of I is rad I okay and for of course I not x not etc xn so and well the other thing is Z of I of y will be y bar okay so both facts will be true here also except that here the ideal I start with should not be the maximal ideal corresponding to the 0 in the affine space above because you know I have thrown it out when I got the projective space below I have thrown out I have taken the punctured affine space and then I have gone modulo and equinox relation namely I have taken the lines in the punctured affine space passing through the origin okay I have thrown out the origin but the origin corresponds to this ideal in the affine space above the point 0 0 0 0 0 n plus on coordinates that corresponds to the maximal ideal generated by the variables and this is the ideal that you have to leave out it is a maximal ideal but you have to forget it and it will not so that maximal ideal will not it is also a homogeneous ideal because it is generated by the coordinates which are homogeneous functions they are all homogeneous of degree 1 okay so it is a certainly a homogeneous ideal but the point is that it is a maximal ideal it is a homogeneous ideal but it is not going to come into the picture okay so on this side you are only going to consider homogeneous ideals which are different from this particular maximal ideal that is ideal generated by all the variables and therefore this particular ideal generated by all these variables is given a very special name it is called the irrelevant maximal ideal okay so there is a name for this X0 through Xn is called the irrelevant maximal ideal it is the irrelevant maximal ideal okay and it is a homogeneous ideal but then the homogeneous ideals we are interested in are everything except that and that is why that is called irrelevant alright it is irrelevant with respect to the projective geometry right and so I what I want to tell you is that you can prove these statements from the corresponding statements for affine space if you just remember that the quotient on the projective space is the quotient topology given the Zariski topology on the punctured affine space above okay so all these statements can be proved by translating everything to the affine space above okay and by using the corresponding results in the affine case so what you must understand is you must understand the following I mean this is a picture that should help you to think of what is going on so you see you must always think of this is the affine space this is the punctured affine space and there is this projection onto the projective space okay and how you should think of it is that if you take if I draw a picture like this or the projective space and well I take the 0 set of a homogeneous ideal here okay and this is the 0 set in Tn of this ideal okay that is how close subsets in projective space look like then how you should think of it is if you take the affine space above so in the affine space above the diagram will be something like a cone so it will be so this is the diagram in the affine space above and what has happened is that for each of these lines they go down to a particular point so this line each of these generating lines L through a certain point lambda not through lambda n goes to the corresponding point in projective space lambda with homogeneous coordinates lambda not through lambda n and this is to be thought of as simply the line above this is just L of lambda not etc lambda n so you think of this point as a line above okay so what you will get is if you give me any projective any close subset of projective space take its inverse image here and then the only thing that will be missing is 0 which is what you will get when you take its closure you will get a close subset there okay 0 is the only thing that will be missed so if you add it you get this picture which is which you can easily think of as a cone over this close subset in projective space so you see this thing is the cone it is called the affine cone over this called the affine cone alright so and so the picture is something like this so if you give me any close subset of projective space and you take the inverse image in the affine space above and close it up so that you add the origin what you get is a cone and what is this cone what is it this is actually this is none other than this is just the z zeros of i in the affine space it is the same i take the same ideal the same ideal mind you the ideal is an ideal in the affine co-ordinate ring of An plus 1 which is thought of as projective co-ordinate ring homogenous co-ordinate ring of Pn note that An plus 1 is S of Pn and this is of course polynomial ring in these N plus 1 variables okay and i is sitting here okay so if you start with i homogenous homogenous here the 0 set is a close set in projective space if you take its inverse image and add the point 0 you will get the projective cone it is called the affine cone it is the cone in the affine space above and what is the affine cone it is just the 0 set of the same ideal considered as a 0 set in the affine space above okay. So any questions about z i in Pn can be translated to questions about z i in An plus 1 okay and then in the affine space of course I know I have a good dictionary I have the Null Schrodinger sets I have all that I need so I use that to prove things in get statements in the projective case okay so all so the point is somehow already the geometry that you know the affine geometry that you know that kind of helps you to get the projective geometry okay it controls the projective geometry. So now you see and well so what you will get so there are two facts that I want to say here you get a bijective correspondence between close subsets and radical ideals. So if you look at this situation the projective space and the homogeneous co-ordinatory you will get a bijective correspondence between close subsets and homogeneous radical ideals okay and in that collection you will have to get rid of this particular homogeneous radical ideal which is this maximal ideal corresponding to the origin above which you have thrown out okay so this is the irrelevant maximal ideal. So what you get in the projective case is a bijective correspondence between close subsets of projective space on one side on the other side you will have to take homogeneous ideals homogeneous radical ideals which are different from the irrelevant maximal ideal you take the collection of all homogeneous radical ideals which are different from the irrelevant maximal ideal that is in bijective correspondence with the close subsets of projective space okay. So let me write that check number one I and Z give inverse maps defining a bijective correspondence between close subsets of projective space and the set of and homogeneous maximal homogeneous radical ideals in S of P n except the irrelevant maximal ideal you will get this bijective correspondence okay. So the difference from the affine case is that there you simply consider radical ideals here you consider homogeneous radical ideals and there you consider of course all ideals but here you consider you leave out that particular irrelevant maximal ideal okay and that is one thing. Then the second thing is of course that in this case the correspondence in both directions is inclusion reversing it is an inclusion reversing correspondence because as the ideal grows bigger the 0 set becomes smaller okay and conversely so the same thing happens here as well okay. So this is an inclusion reversing correspondence the correspondence in one is inclusion reversing that is also true. Then of course whatever versions of the Null-Strahlen-Satz that you had for the affine case you also have corresponding version of Null-Strahlen-Satz for the projective case what is the Null-Strahlen-Satz for the affine case if a polynomial vanishes at every point of a variety then some power if a polynomial vanishes on the 0 set of an ideal then some power of the polynomial is in the rad some power of the polynomial is in the ideal that is the Null-Strahlen-Satz okay. The same statement will work for the projective case if you take a homogenous but the only thing is now you have to use homogenous polynomials and you have to use homogenous ideals. So if you have a homogenous polynomial which is positive degree and if it vanishes on the 0 set of a homogenous ideal then some power of that polynomial is certainly in that ideal that is the homogenous version projective version of the Null-Strahlen-Satz again the projective version of the Null-Strahlen-Satz can be you know derived from the affine version by going to the affine space above okay. So this is the whatever you want to do here you go above and do it okay because there you already have a clear picture you have affine geometry already there so you use that alright. So let me write that have a homogenous projective version of the Null-Strahlen-Satz and that is just if f is homogenous and f belongs to i of z of i where i is homogenous then f for m is in i for some m greater than or equal to 1. So this is the homogenous version of the Null-Strahlen-Satz and so the only thing that has not been said in all this is what happens to this irrelevant maximal ideal. So that is the only thing that I will have to tell you and that is pretty easy to state so here is the fact that it is a fact about the irrelevant maximal ideal but certainly it is not irrelevant to our discussion so you know. So here is a lemma which you can check i in S of P n a homogenous ideal the following are equivalent number 1 i is i contains S d for some d greater than or equal to for some d then 2 z of i in P n is empty okay number 3 rad i is irrelevant maximal ideal. So these are and this tells you why you throw out the irrelevant maximal ideal okay so these are all these 3 are these 3 are equivalent conditions and you know if i contains S d then it means that i will contain X i power d for every i therefore rad i will contain X i therefore rad i will contain the ideal generated by the X i and of course there is another there is one more possibility it is either this or it could be the whole ring so I should also write or S of P n itself so it can happen that see the ideal may contain S S not S not is homogenous polynomial of degree 0 they are the constants. So if the ideal constant it will contain non-zero elements of the field so it is the ideal will be a unit ideal and therefore the radical of the ideal will also be the whole ring will be the unit ideal so these 3 are equivalent conditions and this is the exact reason why you throw out the irrelevant maximal ideal to get a logic and okay so with that we have now we have now a nice dictionary between projective algebraic geometry and on the one side on the geometric side and on the algebraic side we have homogenous coordinate ring and homogenous ideal there. Now let me tell you a point of surprise you have seen for an affine variety that of course we so that reminds me we defined an affine variety to be an irreducible closed subset of affine space okay in the same way we define a projective variety to be an irreducible closed subset of projective space it will follow that by the same argument will follow that you know if you know a closed subset here is in affine space irreducible if and only if the corresponding its ideal is prime the same thing will hold also in projective space a closed subset of projective space is going to be irreducible if and only if the ideal its ideal is a homogenous prime ideal okay and the fact that you will have to remember when you go here is that under continuous map the image of an irreducible set is irreducible okay so that is a fact that is a topological fact that you have to remember and use okay so to get proof of the fact that a closed subset of projective space is irreducible if and only if ideal is a homogenous prime ideal okay and there are two big differences if you take an affine variety you know that the ring of regular functions is the same as its co-ordinate ring okay and the co-ordinate ring is just polynomials okay and there are a lot of polynomials okay there are a lot of polynomial functions at the worst if it is even a single point you have I mean you have constant functions but if it is not a point then you have many functions many non-trivial polynomial functions on your affine variety okay. However if you go to the projective space there also you can define regular functions and the amazing thing that will happen is on a projective variety namely an irreducible closed subset of projective space the only regular functions are constants okay so that is a major point of difference between affine geometry and projective geometry. The other major point of difference is the following we saw that two affine varieties are isomorphic if and only if they are affine co-ordinate rings are isomorphic as k-algebras okay but here the projective or homogenous co-ordinate ring is not such an invariant so what will happen is you can have two projective varieties which are isomorphic as projective varieties but their homogenous co-ordinate rings are not isomorphic which means that the way they are homogenous co-ordinate rings will depend on the way in which they are embedded in the ambient projective space okay. So of course here the definition of homogenous co-ordinate ring of projective variety is similar to the affine case namely in the affine case you take the all the polynomials on the ambient affine space and go modulo the ideal of the variety okay here also you do the same thing you take the homogenous co-ordinate ring of the ambient projective space and go modulo the ideal of the projective variety and you get what is called the homogenous co-ordinate ring or the projective variety but the fact is that this is not an invariant of the projective variety it will depend on which projective space into which you are putting the projective variety. So you see the geometry of projective varieties is far more complicated than the geometry of affine varieties this is what the complication is due to one is because there are no global regular functions which are different from constants there are no non-constant global regular functions that is one point of difficulty the second point of difficulty is that the homogenous co-ordinate ring of projective variety is not it is not an invariant okay. So this adds lot of richness to and variety to the geometry of projective varieties okay so and that is what more serious algebraic geometry is about studying projective varieties okay so I will stop here.