ZQRCHPvpsetlengthheadheight0bpsetlengthheadsep0bpsetlengthtopmargin-36bpsetlengthtextheight704bpUniv at Albany: Math: W. F. Hammond: Courses: Math 825Spring Semester, 20061Note: If you found this document through a web search engine, you may not be aware of its href="http://math.albany.edu/math/pers/hammond/course/mat825s2006/"other presentation formats

1SU-1OutlineFri.,May.5:A 1949 paper by Andre Weil gave evidence for the existence of topological cohomology in algebraic geometry linked to the notion of zeta function for a nonsingular projective algebraic variety defined over a finite field Let X be a scheme of finite type over Z For each element xX the residue field at x is the fraction field of an algebra of finite type over Z Thus, the residue field at a closed element x is a field that is an algebra of finite type over Z, i.e., a finite field One defines the zeta function of X by Xsxclosed inX11Nxs where Nxx is the number of elements of the residue field of X at x (Ignore questions of convergence for now.) When XSpecZ, Xs is Riemanns zeta function When X is a scheme of finite type over Fq, each residue field at a closed element is a finite extension field of Fq, and, therefore, Nxqdx where dx is the extension degree With tqs one writes XsZXtxclosed11tdx With the condition ZX01 the Z form of the zeta function is determined by its logarithmic derivative ddtlogZXtxcloseddxtdx11tdx1tr1xcloseddxrrtr1tr1tr1rcrtr1tr1tr1rcrs1trs1t1rdividesrcrt1Nt1 where cr denotes the number of closed elements in X with dxr and N denotes the number of iref="fieldpoint"points of X with values in the degree extension of FqFor a beginning example, when XAn, one has Nqn, and, therefore, ZAnt11qnt Of course, An is not a projective variety for n1When F is a field, the set of Fvalued points of Pn is the disjoint union of A0F,A1F,,AnF Therefore, DlogZPnt (over Fq) is the sum of DlogZAjt for 0jn Hence, ZPnt11t1qt1qntFor P1P1, one has N1q2, and, therefore ZP1P1t11t1qt21q2tFor curves of genus 1 defined over finite fields, the shape of its Z function was established before the time of Weils conjectures For example, in the case of the curve E given by the Weierstrass equation y2x32x over the field F5, simply by counting points to see that EF510, it is a consequence of the theoretical framework that Zt is the rational function ZEt14t5t21t15tFor each of these last examples Pn, P1P1, and E one may observe that ZXt, relative to the field Fq is a rational function in one variable and that: the denominator is the product of polynomials whose degrees are the classical topological Betti numbers of the base extension XC of X for even dimensionsthe numerator is the product of polynomials whose degrees are the classical topological Betti numbers of the base extension XC of X for odd dimensionsthe polynomial factor corresponding to classical cohomology in dimension j has the form of the characteristic polynomial of a linear endomorphism of the form det1t with complex reciprocal roots all of absolute value qj2Wed.,May.3:Beyond the theory of curves of genus 1 a good bit of what is involved in the study of curves and of complete nonsingular varieties in general is studying the group DivXDivX For curves one has DivXDiv0XDivX where the quotient for the second step is the discrete group Z when Div0X is defined as the group of divisors of degree 0 It turns out that the quotient for the first step is a complete irreducible group variety of dimension g, and, thus, one cannot study curves in depth without studying varieties of higher dimension For varieties of dimension greater than 1, defining the degree of a divisor as the sum of its coefficients will not lead in the right direction One would like a definition of Div0X such that the first step is a complete irreducible variety and the second step a finitelygenerated abelian group, but there is no hope with these two conditions that the second step will always be cyclic since for the case XPk1Pk1 one will find that DivXDivXZZFor the purpose of gaining insight about DivXDivX in the theory of curves while at the same time beginning to understand what might be required for defining Div0X when dimX1, consider what is available with transcendental methods when kC Complex exponentiation provides the short exact sequence of abelian sheaves for the classical (locally Euclidean) topology on X: 0ZOholeOhol0 where efe2if In the long cohomology sequence the 0 stage splits off since H0X,OholC GAGA tells us that coherent module cohomology matches, and although O is certainly not an Omodule, its H1 in both algebraic and transcendental theories viewed through Czech theory classifies isomorphism classes of invertible coherent modules One has the exact sequence: 0H1X,ZH1X,OholH1X,OholH2X,Z If dimX1, then H2X,ZZ, and one finds that the last map in this sequence, a connecting homomorphism, sends the isomorphism class of an invertible Oholmodule to its degree Therefore, remembering that DivXDivXH1X,O, one has H1X,OholH1X,ZDiv0XDivX and, in fact, the left side is the quotient of a gdimensional vector space over C by a lattice Thus, Div0XDivX is a gdimensional complex torus; it is, moreover, a complete group variety over CFor dimX1 the kernel of the connecting homomorphism will provide a correct notion of degree 0For working over an arbitrary algebraically closed field, one sees that something is needed to replace classical cohomology Because constant sheaves are flasque in the Zariski topology, their Zariskibased cohomology cannot be usedMon.,May.1:Continuing with the discussion of the previous hour: If p,q,r are any three points of Xk, then the triple sum pqr, like any point of Xk is characterized by the linear equivalence class of the associated one point divisor One has the relation of linear equivalence pqrpqr2o Therefore, pqropqr3opqrdivh3ofor somehL3opqrdivsfor somesH0X,O3opqrdivaxu3byu3cu3,somea:b:cPk2pqrf1D,DdivaXbYcZDivPk2 where f:XPk2 is the projective embedding of X given by the invertible Omodule O3o In other words, taking multiplicities into consideration, three points sum to o in the group law on Xk if and only if the corresponding points of a Weierstrass model in Pk2, with o corresponding to the point on the line at infinity, are collinear From this description of the group law on Xk, in view of the fact that the third point of a cubic on the line through two given points (tangent if the two points coincide) depends rationally on the coordinates of the given points, it follows that Addition XXX and negation XX are morphisms of varieties over kIf F is the field generated over the prime field by the coefficients a0,,a6 of the Weierstrass equation, then The Weierstrass equation defines a scheme XF of finite type over F whose base extension to k is X For each extension E of F the set XFE is a group in a functorial way XFkXk Fri.,Apr.28:Continuing with curves of genus 1, we wish to change notation so that the projective embedding of the previous hour is given by the very ample invertible sheaf O3o,oXk This notational change notwithstanding, o is an arbitrary point Under the projective embedding given by O3o, one has fo0:0:1, the unique point of fX on the line at infinity We wish to show that there is a unique commutative group law on the set Xk for which the map :DivXXk DpXknppDpXknpp which is tautologically a group homomorphism, has the property that D1D2 whenever D1D2 (linear equivalence), and further the property that o is the zero element in Xk (This is not the strongest statement of this type that can be made.) Addition in Xk is defined by observing that since for given p,qXk the divisor pqo has degree 1, its complete linear system consists of a single nonnegative divisor of degree 1, i.e., r, and this unique rXk is defined to be pq Since pqor the properties specified for make this definition necessary if, indeed, it defines a group It is straightforward to verify that the addition is associative, that o is its identity, and that p is given by the unique member of the complete linear system 2op It is obvious that this group law on Xk is commutative and that is surjective If Div0X denotes the group of divisors of degree 0, then since DDdegDo, one sees that the restriction 0 of to Div0X is a surjective homomorphism Let DivX denote the group of divisors linearly equivalent to zero It is trivial that the map DDdegDo defines a homomorphism DivXDiv0X which, when followed with reduction provides a homomorphism DivXDiv0XDivX It is not difficult to verify that another homomorphism between this latter pair of groups is given by DDoDivX (That this is a homomorphism follows from reviewing the definition of D1D2.) Since these two homomorphisms agree on divisors of the form p which generate the free abelian group DivX , one has for all DDivX that DdegD0Do We know that degD depends only on the linear equivalence class of D as the first consequence of the iref="RRThm"RiemannRoch Theorem Since rXk is determined uniquely by the linear equivalence class of r, this formula tells us that D depends only on the linear equivalence class of D However, the formula also tells us that the linear equivalence class of D depends only on D and degD In particular, one has Div0XDivXXkWed.,Apr.26:Suppose that X is a complete nonsingular curve over an algebraically closed field k of genus 1 The range of degrees where a divisor D has H1OD0 is degD1, while we have dimH1O1 For each aXk the invertible module O2a has no base point, and, therefore, defines a morphism to Pk1 One has a two step filtration of the 3dimensional linear subspace L3a of kX: kL0LaL2aL3a Choosing xL2aL0 and yL3aL2a one obtains a filtrationcompatible basis 1,x,y of L3a, and if u is a rational section of Oa with divua, the morphism f:XPk2 given by fZ:X:Y,Zu3,Xxu3,Yyu3 provides a projective embedding of X by the theorem of the last hour Extending the filtration inside kX by the Lma, one sees that 1,x,y,x2,xy,x3 is a filtrationcompatible basis of L6a Since y2L6aL5a, one has a linear relation among monomials of degree 3 Y2Za1XYZa3YZ2a0X3a2X2Za4XZ2a6Z3 with a00 that characterizes fX as a nonsingular hypersurface in Pk2 One says that fX is in generalized Weierstrass form One regards Z0 as the line at infinity in Pk2, while one calls affine a point X,Y1:X:Y The intersection of fX with the line at infinity reduces to the equation a0X30 Therefore, the point 0:0:1 is the only point of fX on the line at infinity, and as the point of intersection of the line at infinity with fX it has multiplicity 3Mon.,Apr.24:Continuing with the case of a complete normal curve over an algebraically closed field k When D is a divisor with degD2g, then for each aXk one has degDa2g1, and, therefore, LDa is a hyperplane in LD Otherwise, said OD has no base point A coordinatefree interpretation of the morphism f:XPkN, where NdegDg, given by a basis of H0X,OD is that fa is the hyperplane H0X,ODa regarded as a point in the projective space of hyperplanes through the origin in H0X,OD If, moreover, degD2g1, then for ab in Xk it follows that H0X,ODab has codimension 2 in H0X,OD so that fa and fb must be different points, i.e., f is injective Since X is complete, fX must be a closed subvariety of dimension 1 in PkN The fact that H0X,OD2a also has codimension 2 in H0X,OD guarantees that daf:TaXTfaPkN has rank 1 for each a, and, therefore, that fX is itself a complete nonsingular curve Since morphisms of complete nonsingular curves are dual to the contravariant function field extensions, f must be an isomorphism, i.e., OD is very ample when degD2g1 As first example, when g0 and Da, the morphism f given by H0X,Oa is an isomorphism of X with Pk1Fri.,Apr.21:In the context of a complete normal variety X over an algebraically closed field k an invertible OXmodule L is called very ample if there is an integer N0 and a closed immersion f:XPkN such that LfOPkN1 (Recall the iref="funcptsprojsp"earlier description of the functor of points over k of PkN.) If L is very ample, then Lm is also very ample for each m1 One says that L is ample if there exists m1 such that Lm is very ample Finally, if there is an integer N0 and a morphism f:XPkN such that LfOPkN1, one says that L has no base point For a particular value of N if z0,zN are homogeneous coordinates in PkN, hence, a basis of H0PkN,OPkN1, then the sjfzj are elements of H0X,L that do not vanish simultaneously at any point of Xk It follows that the members of any basis of H0X,L also have no common zero, but it does not follow that the sj form a basis When dimX1, recall that for a divisor D of negative degree iref="nonnegdeg"one has dimkH0X,OD0 If K is a canonical divisor and D a divisor with degDdegK2g2 then KD is a divisor of negative degree, and, consequently, by iref="serredimone"Serre duality dimkH1X,OD0 for any divisor D with degD2g1 When the genus g1, this means that dimkH1X,OD0 for any divisor D of degree at least 1 The Riemann Roch formula then implies that dimH0X,ODdegD In particular if Da for aXk, one sees that LaL0k while both have dimension 1 Hence, there can be no fkX with only a single simple pole The same type of reasoning shows that kX contains an element whose only pole is a double pole at a given point aXkWed.,Apr.19:When A is a ring and B an Aalgebra, the module BA is the Bmodule receiving an Aderivation from B that is initially universal for derivations from B to Bmodules When f:XY is a morphism of schemes there is an OXmodule XY that globalizes the module of differentials from commutative algebra A morphism f:XY of irreducible varieties over an algebraically closed field k is called smooth if (i) f is dominant, i.e., fXY, and (ii) XY is a locallyfree OXmodule of rank dimXdimY A nonsingular variety over k is a variety X that is smooth over k (An irreducible variety of dimension 1 is nonsingular if and only if it is normal.) When X is a nonsingular variety, one defines Xp to be the pth exterior power pXk For ndimX the top exterior power XXn is a locallyfree OXmodule that is called the canonical OXmodule A form of Serre duality, which could be the subject of an entire course, is this: AssertLabel-1AssertLabel-1AssertDefaultTheoremIf X is a complete nonsingular variety of dimension n and F a coherent OXmodule, then HpX,F and ExtOnpF,X are dual vector spaces over k An important special case is that when F is a locallyfree Omodule Then ExtOnpF,XExtOnpO,XFHnpX,XF where F denotes the O dual of F serredimone In the case of a complete normal curve a canonical divisor is any divisor K for which OKX When FOD for an arbitary divisor D, the vector spaces HpX,OD and H1pX,OKD have the same dimension for p0,1 In particular one has gdimH1X,OXdimH0X,X, and application of the RiemannRoch formula to a canonical divisor leads to the conclusion that any canonical divisor must have degree 2g2Mon.,Apr.17:Continuing with the case of a complete normal curve X over an algebraically closed field, some observations: nonnegdeg If H0X,OD0, then degD0 since D is linearly equivalent to a nonnegative divisor divfD for some fLDThe set DEDivXE0,ED is called the complete linear system determined by D It may be bijectively identified with the projective space of lines through the origin in the vector space LDH0X,OD A linear system is a projective subspace of a complete linear system. One has DdivssH0X,ODLooking at the cohomology sequence associated with the short exact sequence 0ODaODiiOD0 one sees that in going from Da to D either the dimension of H0 goes up by 1 or the dimension of H1 goes down by 1 but not bothTo go further with complete normal curves we want to talk about Serre dualityFri.,Apr.7:When X is a complete normal curve over an algebraically closed field k, aX a closed point, a the corresponding divisor, and i:aX the corresponding closed immersion of a subvariety, one has the exact sequence of coherent Omodules 0IaOiOa0 and, remembering that IaOa, then tensoring this exact sequence with the invertible Omodule OD, D an arbitary divisor on X, one obtains 0ODaODiiOD0 The third term above is a skyscraper that is rank 1 on Oaak The relation among Euler characteristics given by the last short exact sequence reduces to X,DX,Da1 for every divisor D and every closed point aX, and, thus, the observation that X,DdegD is a constant depending only on X where degDznzwhenDznzz This provides a substantial portion of the RRThmRiemannRoch Theorem: X,DdegD1g where g, the genus of X, is defined as dimkH1X,O As a corollary of this, together with the observation that X,D depends only on OD, one sees that degD depends only on OD, and, therefore, degdivf0 for each fkX, a result that corresponds to the statement for compact Riemann surfaces that the number of zeroes of a meromorphic function equals the number of its poles For an initial understanding of the genus of a complete normal curve, consider the exact sequence of Omodules 0OkXkXO0 from which ensues the sequence of vector spaces over k 0kkXH0X,kXOH1X,O0 where the last 0 is H1 of the constant, hence flasque, sheaf kX and H0X,kXO is the vector space of principal part specifications Thus, g0 if and only if every principal part specification is realized by an element of kX Thereby it is clear that the genus of Pk1 is 0Wed.,Apr.5:For DDivX, X a normal variety, one defines LDfkXdivfD00 LD is an OXmodule that is isomorphic to the module of global sections of OD While a (regular) section of a locallyfree Omodule of rank 1 is not represented by a single element of kX, it does have local pieces that are unique up to multiplications from O and, consequently, has a globally welldefined divisor If sf0 is the section of OD corresponding biuniquely with fLD, one has divsfdivfD One sees that dimkH0X,OD0 if and only if D is linearly equivalent to some nonnegative divisor A nonnegative divisor D determines an Oideal ID that is locally the principal ideal generated by a local equation for D It follows that ID is a rank 1 locallyfree Omodule, and one sees easily that it is isomorphic to ODWhen X is a complete variety over a field k and M a coherent Omodule the kmodules HqX,M are finitedimensional over k for all q This is a consequence of the more general fact that direct images and higher direct images of a coherent module under a proper morphism are coherent (see the text) One defines the Euler characteristic of a coherent Omodule by X,Mq0dimX1qdimkHqX,M When 0MMM0 is an exact sequence of coherent Omodules on X, one has X,MX,MX,MMon.,Apr.3:When X is a normal variety, the affine coordinate ring OU of an open affine subvariety U is the intersection of its localizations at the prime ideals corresponding to the irreducible closed sets in U of codimension 1 Hence OX is the kernel of the homomorphism div Given a divisor DDivX and an open covering Ui of X that principalizes D, say, DUidivUifi, it follows from the computation of the kernel of div on the open subvariety UijUiUj that fiuijfj (all elements of kX) where uijOUij The Cech 1cocycle uij determines an element OD of the group HCech1X,O of locallyfree Omodules of rank 1, the map DOD is a group homomorphism, and the sequence 1OXkXDivXHCech1X,O1 is exact One says that two divisors D1 and D2 are linearly equivalent (and one may write D1D2) if D2D1divf for some fkX or, otherwise stated, if OD1OD2Fri.,Mar.31:For an irreducible variety X over an algebraically closed field k, a divisor is an element of the free abelian group DivX generated by the irreducible closed sets of codimension 1 When X is normal, the local ring at each irreducible closed set Z of codimension 1 is a principal valuation ring, and, therefore, each element f0 in the function field kX gives rise to a divisor divfZordZf which is called a principal divisor The map div:kXDivX is a homomorphism of abelian groups Since an open set U in X is also a variety, the functor UDivU defines an abelian sheaf Div on X that is easily seen to be flasque When X is normal and Z an irreducible closed set of codimension 1, the divisor in an open neighborhood of Z of the unique prime in OZ is the generating divisor corresponding to Z Thus one sees that each divisor on X is locally principalWed.,Mar.29:If f:XY is an affine morphism of algebraic varieties over an algebraically closed field k, then for each quasicoherent OXmodule F one has an isomorphism of HqX,F with HqY,fF Finite morphisms and closed immersions present important special cases To know the cohomology of every coherent OPmodule on each projective space PPkN is to know the cohomology of every coherent OXmodule on every projective variety XMon.,Mar.27:On a Noetherian space the cohomological functor Hq for abelian sheaves vanishes when qdimX The E2 spectral sequence for composite functors is operative when application of the first functor to an injective object in its domain yields an object that is acyclic for the second functor This applies to the direct image functor followed by the global sections functor on abelian sheaves since the direct image of an injective abelian sheaf is flasqueFri.,Mar.24:On a Noetherian space (descending chain condition for closed sets) each of the sheaf cohomology functors Hq on the category of abelian sheaves commutes with direct limitsWed.,Mar.22:More on cohomology: Every abelian sheaf on a topological space X may be regarded as a Zmodule (sheaf of modules over the constant sheaf Z) As base cohomology one uses the derived functors of the global sections functor in the category of Zmodules An abelian sheaf is flasque if its restrictions between open sets are all surjective Every flasque sheaf is acyclic for cohomology, and every injective Amodule, for any sheaf of rings A on X, is flasque Consequently, sheaf cohomology in the category of Amodules is consistent with that in the category of ZmodulesMon.,Mar.20:If f:X,AY,B is a morphism of ringed spaces, for every Bmodule G there is an Amodule pullback fG which at stalk level satisfies fGxGfxBfxAx For a morphism of affine schemes pullback of quasicoherent modules on the target is the same thing as base extension For PPkN, k an algebraically closed field, the exact sequence OPN1x0,,xNOP10 given by f0,,fNf0x0fNxN spawns, via pullback, the funcptsprojsp functor of points of PkN over k: a morphism :XPkN is the same thing as an invertible OXmodule L and an N1tuple of sections s0,sN of L that do not vanish simultaneously, i.e., that provide the exact sequence OXN1s0,,sNL0 which is the pullback of the referenced exact sequence on PkN For a kvalued point xXk one has xs0x:s1x::sNxFri.,Mar.17:The isomorphism classes of locallyfree Amodules of rank 1 form a group The notion of an exact sequence of Amodules Amodules form an abelian category in which every object admits an injective resolution The global sections functor MMX is left exact The qth cohomology functor XHqX,M is defined as the qth right derived functor of Sideline example: the short exact sequence 0ZOholeOhol0 of Zmodules in complex analytic geometry, where efe2if is the complex exponentialWed.,Mar.15:Homomorphisms of Amodules when A is a sheaf of rings on a topological space Locallyfree Amodules of rank r and transition matrices relative to a trivializing covering An invertible Amodule is a locallyfree Amodule of rank 1Mon.,Mar.13:Class cancelledFri.,Mar.10:Properties and significance of the OP modules OPd on PPkn for dZ where k is an algebraically closed fieldWed.,Mar.8:The concept of sheaf of modules on a ringed space Quasicoherent and coherent modules on a scheme ExamplesMon.,Mar.6:If f:XY is a morphism of schemes with Y separated, then f is universally closed if every split base extension of f is closed Proper morphisms Valuative criteria for separated morphisms and proper morphismsFri.,Mar.3:Separated morphisms If f:XY is an Smorphism and Y is separated over S, then the graph of f is closed in XY and f is separated if and only if X is separated over S Henceforth, an algebraic variety will be assumed to be separated over its base field; consequently, all morphisms of varieties will be separated In a scheme that is separated over an affine base, the intersection of any two open affines is affineWed.,Mar.1:If x is an element of X, the scheme underlying an irreducible algebraic variety, the Krull dimension of the local ring Ox is the codimension of x in X When X is normal, the local ring at an irreducible subvariety of codimension 1 in X is a discrete valuation ring The set of closed points of a complete and normal irreducible algebraic curve correspond biuniquely with the nontrivial discrete valuation rings in its function field that contain the ground field, and the entire structure of such a curve as a scheme may be recovered from its function fieldMon.,Feb.27:Finite morphisms yet another class closed under composition and base extension The normalization of an irreducible variety Universally closed morphisms Finite morphisms are universally closedFri.,Feb.17:Any base extension of a morphism of finite type is also a morphism of finite type Case in point: the fibre of a morphism f:XY of finite type over an element yY is a scheme of finite type over the residue field y Over its image a morphism may be viewed as providing a family of varieties, though not a wellbehaved one without assumptions on the morphism The notion of affine morphism: another class of morphisms that is closed under compostion and base extensionWed.,Feb.15:The join of two Cartesian squares is another Cartesian squares provide shelter for both the geometric notion of product and the algebraic notion of base extension The notion of base extension of a morphism Example: The action of Galkk on Xk when X is a kscheme (and k is the algebraic closure of the field k)Mon.,Feb.13:Detailed examination of the functor of points for ESpecZx,yFx,y where Fx,y is the polynomial Fx,yy2xaxbxc, particularly in relation to base extensions of the coordinate ring Existence and uniqueness of products in the category of schemes over a given schemeFri.,Feb.10:The notion of morphism of a scheme over a base scheme globalizes the notion of homomorphism for algebras over a base ring If S is a scheme, the functor SchemesSopSets given by THomST,XXT is called the functor of points of X over S X is determined as an Sscheme by its functor of points If X is the scheme associated with a variety X0 over an algebraically closed field k, then XkXSpeck is the set underlying X0 If K is an extension field of k, a fieldpoint point XK determines an element xX (no longer called a point) that is called its center and a kalgebra homomorphism from the residue field at x to K In the affine case XK is precisely the set of naive points of X in KWed.,Feb.8:A morphism from a scheme to the affine scheme SpecA is dual to a ring homomorphism from A to the ring of global sections of the schemes structure sheaf The scheme associated with an affine variety over an algebraically closed field is characterized as a reduced scheme of finite type over (the spectrum of) the fieldMon.,Feb.6:The category of schemes Locally closed subschemes Morphisms; schemes over a base schemeFri.,Feb.3:The category of affine schemes as (1) a fully faithful subcategory of the category of localringed spaces and (2) as the opposite category of the category of commutative ringsWed.,Feb.1:The notion of an affine scheme as a topological space equipped with a sheaf of rings; morphisms between affine schemesMon.,Jan.30:The sheaf of rings associated with the spectrum of a commutative ring; the initial ring is the ring of global sectionsFri.,Jan.27:The spectrum of a commutative ring and its Zariski topologyWed.,Jan.25:Presheaves and sheaves; examplesMon.,Jan.23:Overview

2SU-2CommentsSU-2.1Things Spotted on the WebWikipedia There are a number of ways to enter href="http://en.wikipedia.org/wiki/AlgebraicGeometry"Algebraic Geometryhref="http://en.wikipedia.org/wiki/Scheme28mathematics29"Schemeshref="http://en.wikipedia.org/wiki/Special:Search?search=22algebraic+geometry22fulltext=fulltext"Search Wikipedia for algebraic geometry Notes on Lectures by Hartshorne These are notes by William Stein of 1996 lectures given by Robin Hartshorne at UC Berkeley: http://modular.ucsd.edu/AG.html Href="../"UP Href="/math/pers/hammond/"TOP Href="http://math.albany.edu/"Department