 Let us continue with our discussion of projective varieties alright so what we will do is recall from the previous lecture that we are taking K to be an algebraically closed field. So we always at least when I define the projective space I do not need this and in principle according to general the most general version of algebraic geometry which is scheme theory in fact for defining projective space K need not even be a field it can even be a ring but then let us not go we are not working in that generality but we assume that at least we are working with fields and since we are going to do geometry we are working with algebraically closed fields K. So at least for the initial part K just need be a field it need not even be algebraically closed so you know we have projective n space this is projective n space n space over K and what is this this is it is a quotient of it is a quotient of the affine space the affine n plus 1 space modulo and equivalence and what is the equivalence the equivalence is you identify points on a line of course I do not want to I do not want to I want really on 0 points so I throw out the origin and then I go modulo and equivalence relation and what is this equivalence relation this is the equivalence relation that identifies two points if they lie on the same line passing through the origin in other words this is just the space of lines in K n plus 1 and so you have a natural map K n plus 1 to P n K n plus 1 minus 0 this is the punctured affine space to projective space there is a natural map which is the quotient map okay you can think of so you can think of every point being sent to its equivalence class and equivalence class can be thought of as a line okay so you send a point which coordinates lambda not lambda n to the line passing through to its equivalence class which is the line passing through lambda not etc lambda n and the origin okay this is the same as the equivalence class of that point lambda not etc lambda n where the square bracket denotes equivalence class and this is also the same it is also written in this form it is written as lambda not semicolon dot dot dot lambda n but this colon is to be thought of as a ratio and we put square brackets to say that this is a set of n plus 1 scalars which have this given ratio okay so in other words what it means that is that you know if I multiply if I multiply this by any non-zero element of K okay which means I multiply each entry by the same non-zero element of K then the line will not change alright and the equivalence class will not change okay and this also will not change namely there will be the same scalar multiplied through each of these entries and the rule is that the idea is that whenever you have this whenever you have this colon you can cancel off the common factor so this is the same as t lambda 0 t lambda n for every t not equal to 0 in the field okay so that is because if you take t lambda 0 to 3 to through t lambda n that also lies on the same line joining lambda 0 lambda n through the origin it defines the same line so it is the same equivalence class and it is the same point but the idea here is that whenever you have a whenever you have ratio whenever you write a ratio as a is to b you also write this t a is to t b so you so the ratio a is to b is the same as the ratio t a is to t b okay and you can cancel the t off from the t from the ratio t a is to t b to think of it as a ratio a is to b that does for this is for the case of two ratio of two numbers but then this is a ratio of n plus 1 numbers okay and that is the meaning I mean that is the reason for this notation these are called homogeneous coordinates okay they are called homogeneous coordinates because you can cancel out a common you can cancel out a common divisor alright so well I have now the point is that we would like to do geometry on the space so first of all you know the starting point is always to first make it in a topological space and the two ways to do it okay so I will tell you one way the one one way is of course use the notion of quotient topology which comes from the from topology itself that is because the space above is actually the affine space affine n plus 1 space which is punctured at the origin okay and this is a Zerski topology okay so this is a topological space and this is a subjective map from a topological space on to another set okay and then you can make this into a topological space by giving this what is called the quotient topology for this map okay what is that quotient topology it is the open sets here are precisely those for which the inverse image under this map are open sets here okay and similarly for closed sets okay so what we do is that P n can be made into a topological space in two ways number one use the Zerski topology on a n plus 1 minus the origin okay a n plus 1 minus the origin which is the punctured affine n plus 1 space is an open subset of the affine n plus 1 space and any subset of a topological space automatically gets induced topology okay so this has an induced topology from the bigger space which is the whole affine space okay and so this has that topology and you can use the Zerski topology to give the projective space the quotient topology topology via the map you think of pi as a quotient map because it is quotient by an equivalence relation the after all what is projective space is the space of lines what are the lines they are the equivalence classes under this equivalence relation so this is a set of equivalence classes and you always call a map that goes from a set to a set of equivalence classes a quotient map and you call this set of equivalence classes as the set of you call this the you call the quotient of the you call this the quotient of the given set by the given equivalence relation. So this affine space here is being this projective space is here being thought of as the punctured affine space modulo going mod this equivalence okay and going modulo this equivalence is geometrically the same as looking at lines okay and every equivalence class here is a line passing through the origin okay so of course when I say space of lines in KN person I should say through the origin that is important if I simply say space of all lines that is not correct I meant space of lines in N person affine space through the origin alright and okay so this is a standard thing in topology whenever you have topological space and you have an equivalence relation on that then you can go to the set of equivalence classes that is a natural subjective map which associates to every point it is equivalence class and then you can always give the quotient the set of equivalence classes a quotient topology using that map okay and we can follow that and what is that that is a subset W in P N is open and respectively closed if and only if pi inverse of W is open respectively closed in the space above because in the space above which is a punctured affine space you know what closed or open means okay open means it is a complement of a closed set and a closed set means it is an algebraic set so a closed set here is an algebraic set in the affine N plus 1 space with the origin removed okay that is what closed sets here alright for the induced topology so this is one way you can give this quotient topology and you see in this quotient topology always in the definition of quotient topology automatically it is a it ensures that the quotient map becomes continuous because you are saying that a set here is open if and only if the inverse image is open so automatically a set is if a set is open then its inverse image becomes open that is the condition for verifying the continuity of a map okay so when you give the quotient topology automatically the map becomes continuous okay the what is the other way of doing things the other way of doing things is you try to indigenously define the Zariski topology on the projective space okay and how do you do that you imitate what you did for the affine space okay for the affine space how did we define the Zariski topology we define the Zariski topology by defining closed sets and whatever closed sets they were sets of common zeros of a bunch of polynomials in the right number of variables. Now what you do is you just adapt the same definition okay but now you say that you look at common zeros of a bunch of polynomials but not just any polynomials but polynomials are homogeneous okay because if a polynomial is not homogeneous then it is not it does not guarantee that if it vanishes at one point of a line it will vanish at other points of the line passing through the origin okay so you know what I am trying to say is that when we defined for example closed set in An or An plus 1 of course this sitting inside An okay I mean An plus 1 and how did we define a closed set here a closed set here was well we took a polynomial in N plus 1 variables alright and then not one of not one polynomial but several polynomials a collection of polynomials and look at the common zeros and define such sets to be closed sets here okay but the problem is that if you take a polynomial okay in N plus 1 variables trying to say that it vanishes at a point here is a little tricky which means effectively you are trying to say it is a you must think of it as a polynomial okay which vanishes on a point here corresponds to a line here a line in the affine N plus 1 space passing through the origin you want a polynomial to vanish on a line okay now that would not happen unless the polynomial is homogeneous okay so at least I should say that at least if the polynomial is homogeneous you have hope of the property that if it vanishes at a point on a line passing through the origin then it will vanish on the entire line passing through the origin okay so that is the reason that when you define the Zariski topology on projective N space you look at homogeneous polynomials and you imitate the definition of Zariski topology on affine space by in affine space a closed the closed sets are given by common 0 loci of a bunch of polynomials in projective space the closed sets or the algebraic sets are given by common 0 loci of bunch of homogeneous polynomials okay so here is the second definition define the Zariski topology on PN to be the one for which the closed sets are given as the common 0 common 0s of a collection subset of homogeneous in the all the you know functions on the affine space above which is just identified with k x0 through xn so you can define another topology on the projective space this is the Zariski topology on the projective space which imitates it is also given by common 0 loci of a bunch of polynomials but this k but in this case we consider only homogeneous polynomials okay and mind you homogeneous polynomial means that in the variables if you multiply each of the variables by t then that is if you substitute for each variable a constant multiple of that variable okay the same constant multiple then that constant comes out with the power it can be factored out okay and that power that comes out is called the degree of homogeneity of the polynomial okay. So the fact is that actually I think it is a good point to tell you one more thing there is in fact one more way of getting the topology on PN okay so I will tell you what that one more way there is yet another way so there is yet another way of getting the topology on projective space okay and of course you know what I am trying to tell you I am just going to tell you that you define there are three ways of defining the topology on projective space and they are all the same okay they are all going to give you they are all going to give you the same result alright. So what is this yet another way this is this is the standard way in which you think of a projective space as being gotten by gluing N plus 1 copies of affine space okay so you see I will tell you well let me put it as 3 in continuation with this via the gluing of N plus 1 copies of an now so I have to explain what this means so what we will do is let me do the following thing so you take projective space okay and you know the points in the projective space are given are written in terms of homogeneous coordinates like this okay and then what we will do is let us look at for each of the for each fixed position look at all those coordinates okay for which that particular coordinate does not vanish okay so what I mean is you define Ui to be the set of all points with homogeneous coordinates lambda 0 through lambda n such that lambda i is not same you define Ui like this okay this is for for i equal to 0, 1, etc up to n okay you define these sets alright and well you know you must guess that the sets are going to be open sets because you know they look like for example U0 is first coordinate first homogeneous coordinate not vanish okay U1 is second homogeneous coordinate not vanishing Ui is ith homogeneous coordinate not vanishing and whenever your coordinate does not vanish that should be an open set okay so you must you know you can see that these are going to be the open sets okay but then let us not worry about them as open sets you know well in fact I can say I can say that you know if you take the inverse image of this Ui okay under this under this map then I am going to get the complement of the 0 set of xi the ith coordinate the in fact I should call it i plus 1th coordinate because I am starting with 0 my numbering starts with 0 so you know if I take this Ui and take the inverse image above what I will get is I will get all the points with the i plus 1th coordinate not 0 okay and that is the I will get that set inside the punctured affine space and you know it is complement will be all those points with i plus 1th coordinate I mean with the with that coordinate 0 which is of course a closed set okay. So the inverse image of Ui under pi is certainly an open set and therefore you know this is certainly open according to the definition of the quotient topology okay I mean according to the definition of the quotient topology. It is also open according to the definition of the of this indigenous risk topology because it is actually the complement of the 0 locus of that coordinate where that coordinate is considered as a homogeneous polynomial of degree 1 okay see so note that note that pi inverse of Ui is actually z it is an plus 1 k minus z of xi intersection with an plus 1 k minus z in the origin okay this is what it is this is actually what this is just T xi it is this so the complement of the 0 set of xi is the basic open set defined by xi alright and that is an open set and if you intersect it with this subset it will give you an open subset of this okay so this is which is open which is open in the punctured affine space above of which the projective space is a quotient okay. So you know according to definition 1 this is an open subset of the projective space if you give it the if you look at the quotient topology then this suddenly an open set okay and what you must understand is that you see all these Ui's i equal to 0 to n there are n plus 1 of them and they cover the projective space because when you write a point with projective homogeneous coordinates all coordinates cannot be 0 because you thrown out this is the image of a point on the punctured affine space above okay so you are not so all the coordinates cannot be 0 alright so whenever you write homogeneous coordinates at least one coordinate is not 0 which tells you that therefore that point lies in one of the Ui's. So all the Ui's these n plus 1 Ui's they are a cover for the projective space okay and according to this definition if you give the map pi equation topology then these Ui's are open okay. According to this definition also it is open because you see note also that if you take away Ui from Pn what you will get is of course the 0 set in Pn of the homogeneous polynomial X i see you take the polynomial X i okay that is of course homogeneous polynomial it is homogeneous of degree 1 alright. So by definition it is 0 set is should define a close subset of projective space and therefore it is complement which is precisely Ui is open okay so this Ui is open according to this definition it is also open according to this definition alright. Now further all the Ui's union of all the Ui's i equal to 0 to n is actually Pn of course the projective space is covered by these sides okay but the third topology on Pn is something that I have not yet defined so but I need these Ui's to define it. So what I am going to do is now you take this Ui which is set of all homogeneous coordinates points in Pn such that the ith coordinate is not 0 which is subset of Pn and what you do is you define a map like this into An okay the point is that each Ui is automatically an affine N space each Ui is actually an affine N space okay and beautiful thing is it is complement which is Z X i is a projective space of dimension 1 less okay. So the beautiful thing is that each of these Ui's is actually an An okay and it is complement is a Pn minus 1 alright so let me explain that and that complement Pn minus 1 is called the hyperplanet infinity okay so let me explain that so you know this map is very easily defined so what you do is you take a point it coordinates lambda not etc so I will write this down there is a lambda i minus 1 there is a lambda i and then there is a well there is a lambda i plus 1 I will not write lambda i plus 1 because it might be the last then lambda n okay and you know what I am going to do see I know lambda i is not 0 alright and you know the homogeneous coordinate is not going to be affected if you know I multiply the whole set of homogeneous coordinates by non-zero number and what I am going to do is I am going to just multiply it by 1 by lambda i which makes sense because lambda is not 0 alright and then that will give me a 1 in the i th position okay in the place of lambda i will get a 1 so what you must understand is that this is the same as well let me write this lambda or not by lambda i lambda 1 by lambda i blah blah blah here I will get lambda i minus 1 by lambda i and I will get a 1 and so on I will get a lambda n by lambda i these two ratios are these two are one and the same point on projective space that is because this multiplied by this same lambda i which is a non-zero element of k okay and but then you see this point 1 is I mean this 1 is redundant so what you do is you get only the remaining n distinct scalars and these are the n scalars to which I mean these are the coordinates to these define the coordinates of the point which you are going to send it so what you are going to do is you are going to just send it to lambda not by lambda i comma lambda 1 by lambda i and lambda n by lambda i where you know the way I have written it is you omit lambda i by lambda i which is 1 okay so this is n plus 1 entries I omit that 1 and I get n entries okay now because of the way I have defined it you can check that this is a bijective map this is a bijective map that is the reason is because you know the moment one of the coordinates becomes 1 you see then the other coordinates are unique for that given line because if you try to change it by a scalar okay then this you cannot change it by a scalar without changing the 1 to something else okay so if you freeze the ith coordinate as 1 then the remaining then for each line for is ith coordinate is not 0 okay if you if you if you rationalize so that dieth coordinate becomes 1 for a representative point on the line then the remaining n coordinates are unique so the fact is that this is a bijective map so I know let me call this as phi i phi i is a bijection this phi i is a bijection and of course you know it is a bijection why is it surjective well phi i is injective it is injective and surjective for the reasons I just said but anyway let me if you want you can try you do not have to big deal you know if you have a phi i of one point given lambda 0 through lambda n is equal to phi i of another point mu 0 through mu n then so this will imply that lambda 0 by lambda i etc lambda n by lambda i is the same as mu 0 by mu i and so on mu n by mu i okay and what this will tell you is that it will just tell you that lambda not so it will tell you that lambda not is lambda i by mu i into mu 0 so each lambda j is lambda i by mu i into mu j and this lambda i by mu i is 1 the same it is fixed i is fixed okay lambda i is fixed mu i is fixed okay therefore this quotient is a fixed set scalar and what you are saying is that every lambda is a fixed set scalar times every mu the corresponding mu and that means that these two points define one and the same line that means they are one and the same point on projective space so this implies that the these two points are the same okay this is just injectivity I mean and of course surjectivity is very very simple again why is that so you give me a point in affine space a point etc xn in An is phi of well I can write this point you know what I am just going to put yeah I put x1 etc and then I put a 1 in the ith position and then I put of course I put same I have to put colons here because this is just homogenous coordinates so if I take this point this point will go to the point I started with so it is surjective so if you want a map from here to here you simply write these coordinates in the same order as n plus 1 entries homogenous coordinates but put do not fill the ith position the ith position you put 1 the remaining n positions you put these in the same order that is the that is the map in the reverse direction therefore you know these phi i's are all bijective maps the phi i's are all bijective maps and the fact is that now what I can do I can do the following thing. So you know in very much the same way in which you get a manifold by getting local charts into Euclidean space you think of trying to make projective space you know in a sense like a manifold by taking these as local charts which identify these essentially open sets with affine spaces so you have so you think of so in other words I am saying that you know I am saying use these phi i's which are bijective maps to transport the topology on the affine space back to ui see each affine each copy of affine space has a topology it is a Zariski topology so I can simply transfer the topology here namely what is it you give me a subset here subset here is open or respectively closed if and only if it is image here is open respectively closed for the Zariski topology on these affine space okay so with that definition also I can give you a topology on each ui okay and the ui's cover the ui's of course cover the affine space and the only thing that you will have to worry about is whenever there is an intersection the topologies agree okay. So that will happen okay that is analogous to the compatibility of charts which involves the transition functions being good okay and in this case the transition functions will be just xi by xj okay and therefore what will happen is that you will get a good all these topologies on each ui which are transported via the phi i so each of these topological spaces ui they will glue well on the intersections ui intersection uj to give a unique topology on pn okay and that is yet another topology on the projective space okay which comes from which comes via the gluing of n-toson copies of affine space is n-toson copies of affine space are given by the ui's which have been identified with the corresponding affine spaces via phi i's okay. So there are and the big deal is that the big deal here is that you know that all these topologies agree on the intersections okay that is a very important thing. So you know if you have a topological space okay and suppose you have two if you have a sub if you have just have a set okay and if you have two subsets and suppose on each subset I give you a topology alright then on the union of those two subsets this will give a topology provided on the intersection the topologies should agree okay the intersection a set in the intersection considered as a subset of one is open if and only if it is open with respect to when it is considered as subset of the other okay. So you know if you have two sets two subsets of a space of a set you give them topologies separately okay then when do you get a topology on the union you will get a topology on the union only when on the intersection the topologies agree that is a subset of the intersection is open respectively closed in one if and only if it is in the other okay this is the situation for two sets but if you have a collection of sets for example even a collection of subsets which is a cover for the given set on each subset if you give me a topology when will all these subsets glue together to give a global topology on the set that requires a compatibility okay it requires a compatibility that whenever you take a certain subcollection of subsets and you take their intersection a subset of that is open if and only or closed if and only if it is so in each topology coming from the corresponding members okay. So and the fact is that is a big fact the fact is that all these topologies on each UI okay which come from the topology on affine space they all glue together and give you a topology on the project space this is the third way of getting a topology on the project space and the fact is all the three are one and the same okay so there are three ways of giving a topology on project space and they are all the same okay. Now what I am going to do is so let me write that down so maybe I will write it here so I will write here Q each UI a topology by declaring PI to be a homeomorphism that is that is I subset W in UI is open respectively closed if and only if PIW in AM is open respectively closed. So if you give this topology to UI via PI then it becomes it is automatic that this PI becomes a homeomorphism of UI to affine space okay. So you know what you are doing is for every point on this UI you are giving me you are just taking n affine coordinates okay so each one is a coordinate map each one should be thought of a chart the analog of a chart for a manifold okay so each one is a coordinate map alright and what we are saying is that once you have a manifold structure each chart in the atlas if you take the corresponding coordinate map it is of course a good function it is of course an isomorphism okay. So in this case what is happening is we are gluing topological spaces all the UIs are glued together to give the topology on Pm so the topologies on each UI agree or compatible on the intersections and define a topology on Pn and for that topology on Pn when do you say a subset of projective space is open or closed it is open or closed if and only if its intersection with each UI is correspondingly open or closed in that UI and how do you check it you check it by taking its image under PI and see whether the set you get in the corresponding affine space is open or closed okay. So this has to be checked that you have to check that the topologies on each UI they are all compatible alright. So you know for example if I take two different UIs UI and UJ it should not happen that a subset of UI intersection UJ is an open subset of UI and is not an open subset of UJ such a thing should not happen it would not happen that is the compatibility you have to check okay. So in other words UI intersection UJ has a topological subspace structure from UI it also has a topological subspace structure from UJ and my claim is that these two are the same okay that is what has to be checked right it only if you check that then you are saying that these topologies glue together to give a global topology on the ambient space which is the full projective space okay and what is a big deal the big deal is whether you use one two or three the topology you get on projective space is one and the same and that is the that is the that is the topology on projective space okay. So let me write that down fact so here is a fact it needs a little bit of checking that it is pretty easy to do you can try it out fact or theorem if you want or each of these ways each of these three topologies on P n is the same as the others so the so what you must understand is that all these phi i's are actually homeomorphisms so if you think of projective space with the Zarisky topology if you think of this as an open subset and you take the induced topology then each phi i is automatically homeomorphism okay by this fact by this theorem alright. Now so this is something that I leave it to you to check but probably portions of it we will check as we go ahead now what I what I want to do is I want to you know the purpose of of course this course is to translate from geometry to algebra and back okay. So all this that we have been saying is more or less completely geometric and topological but then to go to the commutative algebra side we will have to worry about the polynomials that are involved okay and you know that is where the second definition comes in the second definition which says that the Zarisky topology on projective space is given by close by taking the close sets to be common 0 low psi of a bunch of homogenous polynomials in the right number of variables the number of variables will be one more than the than the subscript the superscript of appearing in the projective space okay. So we have to study these homogenous polynomials so for the first question is why homogenous polynomials okay and then that will lead to looking at the theory of homogenous ideals okay. So the fact is when we did just like when we did translation from the Zarisky topology on affine space to the polynomial ring we were worried about ideals okay we will when we do studies Zarisky topology on projective space you will also have to translate everything to ideals but the only thing is the ideals now will be so called homogenous ideals okay and one way of thinking of homogenous ideals is that these are ideals are generated by homogenous polynomials okay. So first of all I wanted to realize that the homogeneity property it can be defined in more than one way and to explain that let us look at an example okay so suppose you take a suppose we take your point lambda not lambda n in projective space and let f x not through x n vanish on the line through lambda not etc lambda n which is actually this point okay the point in projective space is the same as its inverse image which is the line okay. So you know well the way I have written it is that when I say that this point on projective space corresponds to this line this line can also be thought of as leaving it is actually being thought of as leaving above so actually you know this the inverse image of this point is this line thought of as a line above okay here you are thinking of the line as a point here okay but if you think of it as a subset of An plus 1 okay then the inverse image of this is this line itself okay. So you know pi inverse of this point is actually the line through lambda not lambda n considered as a subset of An plus 1 right that is what it is so you know you get another nice picture of this whole of this map you get the picture of what is called as a line bundle a line bundle is something for which you have a you have a map from one topological space to another topological space for every point below the inverse image is a line above okay. So you think of this as a bundle of lines above this and what is every line above goes to the corresponding point it represents in the projective space okay so you think of this vibration as it is called as a it is thought of as a it is sort of as a line bundle you think of given a point here a point here corresponds to a line above a line through the origin above and what is that line through the origin above in terms of the point here it is well if you take the inverse image of this point that is exactly that line above okay that is what I have written here alright. So you see if the polynomial vanishes on this line okay what it means is so this implies for every t which is non-zero scalar the field okay well f of t lambda not etc t lambda n should be 0 this should happen right because you see I have told you that the so you know if I draw a diagram here is my projective space okay and I have taken a point here with homogeneous coordinates lambda not through lambda n and under this map pi I go to the affine punctured affine space above what I get is well I get the line through lambda not etc lambda n minus of course the origin okay so if I take pi this is what so pi inverse gives me this inverse image do not confuse pi inverse for inverse map I get this line of course the origin is thrown out because I am considering it as a subset of the punctured plane then you see when I say the polynomial vanishes on this line it means that it should vanish at every point on that line and every point on that line looks like this okay but then you write this out you write this out what it will tell you is the following yeah so I should here also when I say pi inverse of the point is this line I should say this line minus the origin inside affine inside the punctured affine space because pi is not defined at 0 alright so this is the right way to write it but you do not worry about the point at 0 which has been intentionally removed okay so well so in any case now you see you write this polynomial f of x not xn you break it down into its various homogeneous components you see a polynomial in n variables can be broken up into pieces a sum of pieces each of which is homogeneous of a particular degree so there is a degree one term which will be linear expression in the x i's okay then there will be a degree two term which is a linear expression in product of two x i's namely it will involve x i squares and you will get x i x j okay and so on you will get so the polynomial breaks down into f not x not etc xn this is this degree 0 so this is actually a constant this is the 0th this is the constant term of the polynomial which is what you will get when you put everything 0 okay plus I will get f of f1 which is degree one term which is degree one this is the linear term then and I will get it and I will go on like this and I will get fd which is the degree d term this is the degree d term and of course when d is one it is linear d is two it is quadratic d is three it is q b k and so on and so forth and so on you go on up to f to the degree f this is the highest degree of the polynomial okay it is a degree of the highest weight monomial that monomials that occur okay and the point is that each of these guys is homogeneous of a corresponding degree so if you take fd this homogeneous of degree d f1 is linear it is homogeneous of degree one alright and now you know let us look at this condition f of t lambda not through t lambda n is 0 for every t non-zero so what it will tell you is 0 is equal to f of t lambda not t lambda n that will be that will tell you that you know when I plug this in I will get well this will just remain f not and t lambda not t lambda n plus when I plug t lambda not through t lambda n here then t will come out because it is degree one similarly when I plug it in fd t power d will come out because it is homogeneous of degree d so the expression will look like t times f of lambda not etc lambda n plus t squared f of lambda not etc lambda n and so on t to the d f of lambda not etc lambda n plus blah blah blah you will get t power degree f times f of lambda not etc lambda n this is what you get you get for every t for every t not equal to 0 in the field okay you get this expression sorry I should put sorry you are right I should put f1 should put f2 fd f sub degree fd yeah thanks these are not they should not be f they are the corresponding homogeneous complex thanks so this is t times f1 t squared times f2 t power d times fd and so on okay now you know I want to I claim that this implies that you know I claim that this implies that f each fd of lambda not etc lambda is 0 for every d greater than or equal to 0. I claim that each fd each fd vanishes and so fd of t lambda not etc t lambda n also vanishes for all t in a scalar which is non-zero this is my claim okay you know see therefore what I am trying to say is that if a polynomial vanishes on a line then each of its homogeneous components also vanishes on that line okay each of its homogeneous components also vanishes on that line okay that is what I am trying to say and this is the key to defining a homogeneous ideal alright and why is this true why this is true is because you know let me call this as let me call this scalar as a0 after all this is scalar of course here this is independent of t because this is the first term it is constant so let me call this so what I will get is I will get a0 plus ta1 plus t squared a2 plus t power d ad and so on up to plus t power degree f a sub degree f is 0. So I will get a polynomial in one variable okay with these coefficients k coefficients and what I am saying is that that polynomial identically vanishes on the affine line minus the origin okay but you know if a polynomial vanishes on an open set it vanishes everywhere okay if a polynomial is 0 on an open set it is 0 everywhere that is just because of continuity of the various interpolation therefore and of course mind you here is the first place where you are using k is algebraically closed because I want k to be an infinite field okay so probably I do not okay I need k an infinite field alright for that. So I am just saying the fact that you have a polynomial in one variable you know if it vanishes on an open set okay then it is identically the 0 polynomial so therefore this polynomial is identically the 0 polynomial therefore every coefficient is 0 that is exactly what I have written here okay. So the moral of the story is I want you to remember is that this is the key fact defining what a homogeneous ideal is. The fact is if a polynomial vanishes on a line then every each one of its homogeneous components that also vanishes on the line and in particular it is a non-constant power I mean the constant term is 0 in particular the constant term cannot be non-zero the constant term has to be 0 okay so this is the point I want you to remember when we go to the next lecture right so I will stop here.