CLASSICAL ALGEBRAIC GEOMETRY

Daniel Plaumann Universität Konstanz Summer ￿￿￿￿

A brief inaccurate history of algebraic geometry

￿￿￿￿ - ￿￿￿￿ Projective geometry. Emergence of ’analytic’geometry with cartesian coordinates, as opposed to ’synthetic’(axiomatic) geometry in the style of Euclid. (Celebrities: Plücker, Hesse, Cayley) ￿￿￿￿ - ￿￿￿￿ Complex analytic geometry. Powerful new tools for the study of geo- metric problems over C.(Celebrities: Abel, Jacobi, Riemann) ￿￿￿￿ - ￿￿￿￿ Classical school. Perfected the use of existing tools without any ’dog- matic’approach. (Celebrities: Castelnuovo, Segre, Severi, M. Noether) ￿￿￿￿ - ￿￿￿￿ Algebraization. Development of modern algebraic foundations (’com- mutative ring theory’) for algebraic geometry. (Celebrities: Hilbert, E. Noether, Zariski) from ￿￿￿￿ Modern algebraic geometry. All-encompassing abstract frameworks (schemes, stacks), greatly widening the scope of algebraic geometry. (Celebrities: Weil, Serre, Grothendieck, Deligne, Mumford) from ￿￿￿￿ Computational algebraic geometry Symbolic computation and dis- crete methods, many new applications. (Celebrities: Buchberger) Literature

Primary source [Ha] J. Harris, Algebraic Geometry: A first course. Springer GTM ￿￿￿ (￿￿￿￿)

Classical algebraic geometry [BCGB] M. C. Beltrametti, E. Carletti, D. Gallarati, G. Monti Bragadin. Lectures on Curves, Sur- faces and Projective Varieties. A classical view of algebraic geometry. EMS Textbooks (translated from Italian) (￿￿￿￿) [Do] I. Dolgachev. Classical Algebraic Geometry. A modern view. Cambridge UP (￿￿￿￿)

Algorithmic algebraic geometry [CLO] D. Cox, J. Little, D. O’Shea. Ideals, Varieties, and Algorithms. Springer UTM (￿￿￿￿) [EGSS] D. Eisenbud, D. R. Grayson, M. Stillman, B. Sturmfels. Computations in Algebraic Geometry with Macaulay ￿. Springer (￿￿￿￿).

’Big books’ [GH] P. Griffiths, J. Harris. Principles of Algebraic Geometry. John Wiley & Sons (￿￿￿￿) [Hs] R. Hartshorne. Algebraic Geometry. Springer GTM ￿￿ (￿￿￿￿) [Sh] I. Shafarevich. Basic Algebraic Geometry I: Varieties in projective space. Springer (translated from Russian) (￿￿￿￿)

§￿ Projective Varieties Affine varieties Affine varieties Affine varieties algebraically closed field K algebraically closed field n Kn affine space AKn Kn aalgebraicallyffine space closed field n n A Kn affine space V =An is an affine variety if there is a set of polynomials M K x￿,...,xn such that V =A is an affine variety if there is a set of polynomials M K x￿,...,xn such that n V =AV is anMaffinep varietyn iff therep is￿ for a set all off polynomialsM . M K x￿,...,xn such that ⊂ V M p An f p ￿ for all f M . ⊂ [ ] ⊂ n ⊂ [ ] There⊂ V is a finiteM subsetp MA fMpsuch￿ thatfor all fM M . M (Hilbert⊂ [ Basis Theorem] ). If I is the= idealV( )generated= { ∈ by∶ M( ,) then= M ∈ }I . By the Hilbert Basis Theorem, there is a finiteIf I is thesubset=idealV( M)generated= {M ∈that′ by also∶ M( generates,) then= MI, so∈ in particularI} . Furthermore,′ M I is finitelyM . generated by the ⊂ V( V() =)V=( V)( ) Hilbert Basis Theorem′ . In particular, there is a finite subset M ′ M such that M M . ProV( jec) =tiveV( space) If I and J are two⊂ ideals in K x￿,...,,...,xn ,, then then V( ′ ) = V( ) ′ ⊂ V( ) = V( ) Let be a -vector space. VV I KV J V IJ [ I J ] V I V one-dimensionalJ V I J subspaces of , the projective space of P (V) ∪ ( ) = ( ) = V( ∩ ) V V where IJ isn the idealn ￿ generatedn ￿ by by all all products products fg,, f I,, g J.. ((P))∩= {P(K) = ( K+ ) ￿ Affine va}rieties Affine varieties where+ v w + λ K v λw. ∈ ∈ K algebraically= = ￿ closed￿ field{ }￿￿∼ n Pro×jective space PointsAKn K ofnPaalgebraicallyffiarene denotedspace∼ closed in⇐⇒homogeneous field∃ ∈Pro∶jec coordinates=tive spaceZ￿,...,Zn where LetAnV beKn aaKffi-vectorne space space. Zn￿,...,Zn λZ￿,...,λZn for λ K . [ ] VLet V=Abeis a anK-vectoraffine variety space. if there is a set of polynomials M K x￿,...,xn such that n × V =AP Vis an affione-dimensionalne variety if there subspaces is a set of of polynomialsV , the projectiveM K spacex￿,..., ofxnV such that [VP V M one-dimensional] =p[ An f p subspaces] ￿ for all∈ off VM, the. projective space of V ⊂ n n ￿ n ￿ ⊂ [ ] V(P) =M{PK p AKn f p￿ ￿ for all f M} . There⊂ isP anfinitePKsubsetn ￿ MKn ￿M such￿ that M M (Hilbert⊂ [ Basis Theorem] ). (= V) (= where{) = +{ v∈ w ∶+ ( ) =λ K v ∈λw}. } There is= aVfinite(= where)subset= +{ v=∈￿Mw′ ∶+ M￿( {such) }=λ￿￿ that∼K v M∈λw.} M′ (Hilbert Basis Theorem). = = ￿ ⊂ ￿ { }￿￿∼ × V( ) = V( ) ∼ ′ ⇐⇒ ∃ ∈ ×∶ = ′ Projective space ∼ ⊂⇐⇒ ∃ ∈Pro∶jecV(=tive) = spaceV( )

Let V be a K-vector space. P V one-dimensional subspaces of V , the projective space of V n n ￿ n ￿ (P) = {PK K ￿ } where+ v w + λ K v λw. = = ￿ ￿ { }￿￿∼ Points of n are denoted in homogeneous× coordinates Z ,...,Z where Points of P are denoted∼ in⇐⇒homogeneous∃ ∈ ∶ coordinates= Z￿,...,Zn where Z ,...,Z λZ ,...,λZ for λ K . Z￿,...,Zn λZ￿,...,λZn for λ K . [ ] × [ ] = [ ] ∈ Projective varieties

n A polynomial F K Z￿,...,Zn is not a function on P , since in general

F Z￿,...,Z∈n [ F λZ￿,...,] λZn . If F is homogeneous( ) ≠of degree( d, then) d F λZ￿,...,λZn λ F Z￿,...,Zn .

So given( a set M of homogeneous) = ( polynomials) in K Z￿,...,Zn , it makes sense to define n M p P f p ￿ for all f M , a projective[ variety] . V( ) = ￿ ∈ ∶ ( ) = ∈ ￿ The projective (resp. affine) varieties in Pn (resp. An) form the closed sets of a topology, the Zariski topology. Projective space is covered by the open subsets n n Ui Z￿,...,Zn P Zi ￿ Y￿,...,Yi ￿, ￿, Yi ￿,...,Yn P .

The map= ￿[ ] ∈ ∶ ≠ ￿ = ￿[ − + ] ∈ ￿ n Ui A , Z￿,...,Zn Z￿ Zi,...,Zi ￿ Zi,...,Zi ￿ Zi,...,Zn is a homeomorphism.→ [ The] inverse￿ ( map￿ is − ￿ + ￿ ) n A Ui, z￿,...,zi ￿, zi ￿, zn z￿,...,zi ￿, ￿, zi ￿,...,zn . n n Thus P is→ covered( by n −￿ copies+ of) A￿ [. − + ] + The Zariski topology

The projective (resp. affine) varieties in Pn (resp. An) form the closed sets of a topology, the Zariski topology. Projective space is covered by the open subsets n n Ui Z￿,...,Zn P Zi ￿ Y￿,...,Yi ￿, ￿, Yi ￿,...,Yn P .

The map= ￿[ ] ∈ ∶ ≠ ￿ = ￿[ − + ] ∈ ￿ n Ui A , Z￿,...,Zn Z￿ Zi,...,Zi ￿ Zi, Zi ￿ Zi,...,Zn is a homeomorphism.→ [ The] inverse￿ ( map￿ is − ￿ + ￿ ) n A Ui, z￿,...,zi ￿, zi ￿, zn z￿,...,zi ￿, ￿, zi ￿,...,zn . n n Thus P is→ covered( by n −￿ copies+ of) A￿ [. − + ] +

How to think about projective space

Projective line

Real topological picture (O. Alexandrov - Wikimedia Commons) Exact picture [Ha, p.4]

Intuitive picture How to think about projective space How to think about projective space

ProjectiveProjective plane plane

Real topological picture - Boy's surface (virtualmathmuseum.org)

Exact picture [Univ. of Toronto Math Network]

We think of projective space over Wean think algebraically of projective closed space field over just anas algebraically real affine space, closed together field just with asthe realidea affinethat space, taking together intersections with Intuitive picture thealwaysidea that works taking perfectly. intersections [freehomeworkmathhelp.com] always works perfectly.

Linear Spaces Linear Spaces

If W V is a linear subspace, then PW PV is a projective subspace, a linear space of dimen- sion dim PW dim W ￿ in PV. ⊂ ⊂ dim PW ￿ = −point dim PW ￿ line dim PW = ￿ plane dim PW = dim PV ￿ hyperplane = If L PW=, L PW −, write ′ ′ =LL P =W W . We have′ ′ We have = ( + ) dim LL dim L dim L dim L L . ′ ′ ′ = + − ∩ Dimension Dimension Dimension Let X be a variety. X is reducible if it is the union of two proper, non-empty closed subvarieties; otherwiseLet X be a variety.it is calledX isirreducible.reducible if it is the union of two proper, non-empty closed subvarieties; otherwiseLet X be a variety.it is calledX isirreducible.reducible if it is the union of two proper, non-empty closed subvarieties; otherwiseThe Dimension it is called of Xirreducible.is the largest integer k such that there exists a chain The Dimension of X is the largest integer k such that there exists a chain The Dimension of is the largest integer such that there exists a chain X￿ X￿ X Xk ￿ X k X￿ X￿ X X X X Xk ￿ X of irreducible￿ ￿ X￿ ￿ closedX￿ ￿ ￿ subvarieties.￿ Xk−￿ ￿ X of irreducible closed subvarieties.− of irreducible￿ ￿ ￿ closed￿ ￿ subvarieties.￿ ￿ In particular,￿ ￿ ￿dim P￿n￿ ￿dim−An￿ n. In particular, dim Pn dim An n. In particular, dim Pn dim An n. If X Pn or X An is= irreducible:= If X Pn or X An is= irreducible:= IfdimX X n ￿or X pointn is= irreducible:= X ⊂ P X ⊂ A dim X ￿ curvepoint dim⊂X ￿ ⊂point dim⊂X = ￿ ⊂surfacecurve dim X = ￿ curve dim X = ￿ threefoldsurface dim X = ￿ surface dim X = ￿ hypersurfacethreefold dim X = ￿n ￿ threefold dim X = ￿n ￿ hypersurface = hypersurface dim X = n − ￿ Theorem.= −The hypersurfaces in Pn are exactly the varieties defined by a single equation. Theorem. The hypersurfaces in n are exactly the varieties defined by a single equation. = − Pn TheTheorem. hypersurfacesThe hypersurfaces in P￿ are the in Pplaneare exactlyprojective the varieties curves. defined by a single equation. The hypersurfaces in P￿ are the plane projective curves. The hypersurfaces in P￿ are the plane projective curves.

Points Points Points Proposition ￿.￿. Any finite set of points in n is described by polynomials of degree at most . d Pn d Proposition ￿.￿. Any finite set of d points in Pn is described by polynomials of degree at most d. Proof. Let Γ p ,...,p . For q Γ, let L be a linear form with L p ￿ and L q ￿. Proof. Let ￿ d . For , let q,i be a linear form with q,i i and q,i . PutProof. Let Γ p￿,...,pd . For q Γ, let Lq,i be a linear form with Lq,i pi ￿ and Lq,i q ￿. Put = { } ∉ ( ) = ( ) ≠ Fq Lq=,￿{ Lq,d. } ∉ ( ) = ( ) ≠ Fq Lq,￿ Lq,d. Then Γ F q Γ . Then = q￿ . Then Γ = Fq￿ q Γ . = V( ∶ ∉ ) ￿ = V( ∶ ∉ ) n ￿ Definition. Let p￿,...,p P .Ifd n ￿, the points pi are independent if Definition. Let d n.If , the points are independent if Definition. Let p￿,...,pd P .Ifd n ￿, the points pi are independent if dim p￿ pd d ￿, ∈ ￿ + dim p￿ pd d ￿, ∈ ￿ + otherwise dependent. ∈ ￿ + otherwise( dependent￿ ) = .− otherwise( dependent￿ ) = .− If d n ￿, the p are in (linearly) general position if no n ￿ of them are dependent (i.e. lie in If , the i are in (linearly) general position if no of them are dependent (i.e. lie in aIf hyperplane).d n ￿, the pi are in (linearly) general position if no n ￿ of them are dependent (i.e. lie in a hyperplane).> + + > + + Theorem ￿.￿. Any collection of at most points in general position in n can be described by Theorem ￿.￿. Any collection of at most ￿n points in general position in Pn can be described by Theorem ￿.￿. Any collection of at most ￿n points in general position in Pn can be described by quadraticTheorem forms.￿.￿. Any collection of at most ￿n points in general position in Pn can be described by quadraticTheorem forms.￿.￿. Any collection of at most ￿n points in general position in P can be described by quadratic forms. Proof. Let n be such a collection. May assume , . Proof. Let Γ Pn be such a collection. WeMay may assume assumeΓ that￿n,ΓΓcontainsp￿,..., exactlyp￿n . ￿n points. Proof. MayΓ assumeP Γ ￿n, Γ p￿,...,p￿n . Γ ￿n Proof. LetMay assumen beΓ such￿ an, collection.Γ p￿,..., Wep may￿n . assume that contains exactly points. nΓ P Γ ￿n Let q Pn be⊂ such that ￿ ￿ = = { } Let q Pn be⊂ such that￿ ￿ = = { } Let q Pn be such that￿ ￿ = = { } Let qF P be￿⊂ such thatF q ￿ F∈ Γ ￿ F q ￿ F∈ Γ ￿ F q ￿ F∈ Γ ￿ F q ￿ holds∈ forΓ all quadratic forms F. We must show q Γ. holds for￿ = all quadratic￿⇒ forms( ) =F. We must show q Γ. holds for￿ = all quadratic￿⇒ forms( ) =F. We must show q Γ. (￿)If ￿ = with￿⇒ ( ) = , then spans a hyperplane∈ and . (￿)IfΓΓ ΓΓ￿￿ Γ￿Γwith￿ withΓ￿Γ￿ Γ￿Γ￿ n,n then, thenΓi spansΓi spans a hyperplane∈ a hyperplaneHi Hi Li Landi ,H defined￿ H￿ by aL linear￿L￿ . (￿)IfΓΓ ΓΓ￿￿ Γ￿Γwith￿ Γ￿Γ￿ Γ￿Γ￿ n,n then Γi spansΓi a hyperplane∈ Hi Hi Li Landi H￿ H￿ L￿L￿ . (Soform￿)IfLΓ￿L￿,Γ andq￿ Γ￿￿withbywith hypothesis.Γ￿ Γ￿ n Hence,.So then, thenqΓi spansHspans￿ H a hyperplane￿∈by. a hyperplane hypothesis.Hi HenceLi and,H defined￿ H￿. by aL linear￿L￿ . SoformL￿ΓL￿i, andqΓ￿ HΓ￿￿￿byH hypothesis.￿ Γ￿ LΓ￿￿L￿ Hence.Son Lq￿L￿ΓHiq￿ H￿￿∈by. hypothesis.H Hencei qLi H￿ H￿. So L￿L=￿= q∪∪ ￿ by hypothesis.￿ ￿￿ =￿ ￿= ￿￿ =￿ Hence= q H￿ H￿. = V=( V)( ) ∪ = V( ) form￿L=￿=i, and∪∪H￿ H￿ ￿￿￿ =￿ ￿= L￿￿￿=L￿￿=.SoL￿L￿ q￿ ￿￿by hypothesis.= HenceV=( V)(q )H￿ ∪H￿. = V( ) (￿) Let=(p￿,...,)∪= p ￿ Γ￿ =be￿ a￿ minimal= subset∈ of∪ Γ with the property= Vq( )p￿ p . ∪ = V( ) (￿) Let =(p￿,...,)∪= p∪k ￿Γ=be￿V=( minimala￿ minimal￿ =) with subset∈ the( ) ofproperty∪=Γ with the property. q= Vp(￿ ∈)pk.∪ By(￿) Let(￿), we(p￿ can,...,) = findpk suchΓ be a subset minimal with with∈ the. property∪ q p￿ pk. k By(￿) Let(￿), wep￿ can,..., findp∪k suchΓ=be aV subset( a minimal) with subsetk (n.) of=Γ with the property￿ k q p￿ ∈ p .∪ Claim: k ￿ ( k q p￿) k n k ByClaim: (￿), we{k can￿ ( find} suchq∈ ap￿ subset) with . ∈ ￿ TakeClaim:{k ￿ ( } q∈ p￿,) k . Byn hypothesis,∈ ￿ span a hyperplane that Take Σ{ Γ p￿,...,} ∈ pk ￿, Σ n k ￿￿. By hypothesis,∈p￿,...,￿ pk, Σ span a hyperplane H that Claim:Take Σ{k =Γ ￿ ( ⇐⇒p￿,...,} q∈ =pkp ,) Σ n k ￿￿. By hypothesis, p￿,...,pk, Σ∈span￿ a hyperplane H that TakedoesΣ notΓ= contain⇐⇒pp￿￿,...,,...,p￿.So=ppkkq￿,withΣH, sincenΣ knp￿ ￿.k Byqp￿ hypothesis,. Bypk hypothesis,. p￿,..., thepnk,pointsΣ spanp a￿ hyperplane,...,pk ΣHspanthat does not= contain⇐⇒￿ p￿.So=kq H, since p￿ ￿qp￿ pk. ￿ k aByTakedoes hyperplane (￿),Σ notq⊂liesΓ= contain￿ in{H⇐⇒p thethat,...,p hyperplane￿.So does=p q}with notH=, contain since spannedΣ− np+￿ pk by.qp Since the￿. Byp remainingpk hypothesis,. qp pn points. the, it followsn points It follows thatpq,..., thatH.pq liesΣ onspan the By (￿), q⊂lies￿ in{H the￿ hyperplane￿ k } = spanned− +￿ p￿ by the￿ remainingpk￿ qp￿ pnkpoints. It follows￿q thatH qklies on the hyperplaneaBy hyperplane (￿), q⊂lies=￿ spanned in{{⇐⇒ thethat hyperplane does by= }}∉ notand= contain￿ spanned any−￿ = +−∈ byof.+ Since the the￿ points remaining n points., it follows.It The follows that intersection{ that. q lies} of∪ all on such the hyperplaneBy (￿), q lies spanned inH the hyperplane by p￿∉and spanned any n p∈￿￿ byof the the￿ points remainingp￿ qpp￿k ￿p,...,nkpoints.p￿n.It The follows intersectionq thatH q lies of all on such the hyperplane⊂ ￿ spanned{ by} ￿∉and￿ any￿ = −∈ of+ the￿ points∈ k￿￿ ￿n. The intersection{ ∉ } of∪ all such hyperplaneByhyperplanes (￿), q lies spanned in is the just hyperplanep￿ by, hencep￿ andq spanned anyp￿.n ￿ byof the the points remaining∈ pk￿￿,...,n points.p￿n.It The follows intersection∉ that q lies of all on such the hyperplanes is just p￿, hencep￿ q p￿.n ￿ pk+￿,...,p￿n hyperplanehyperplanes spanned is just p￿ by, henceandq anyp￿. − of the points∈ ￿+ . The intersection∉ of all such hyperplanes is just p￿, hencep￿ q p￿.n ￿ pk+￿,...,p￿n = − + ￿ hyperplanes is just p￿, hence q = p￿. − ￿ = − + ￿ = ￿

Projective equivalence Projective equivalence Projective equivalence

n n The group PGLn ￿K GLn ￿K K I acts on Pn. Two varieties X, Y Pn are projectively The group PGLn ￿K GLn ￿K K I acts on P . Two varieties X, Y P are projectively equivalent if there exists A PGLn ￿×K such that A X Y. equivalent if there+ exists A +PGLn ￿×K such that A X Y. Any two ordered+ sets= of(n ￿+points)￿ in general position in Pn are projectively⊂ equivalent. Any two ordered sets= of(n ￿ points)￿+ in general position in Pn are projectively⊂ equivalent. ∈ + ⋅ = ￿∈ ￿ ⋅ = The group PGL￿K acts on+P￿ A￿ through Möbius transformations: The group PGL￿K acts on+P A through Möbius transformations: ab = az∪ {∞b} ab PGL￿K induces=z az∪ {∞b}. cd PGL￿K induces z cz d . cd cz +d ￿ ￿ ∈ ￿ + ￿ ￿ ∈ ￿ + Two sets of four points in ￿ are projectively+ equivalent if and only if they have the same cross- Two sets of four points in P￿ are projectively equivalent if and only if they have the same cross- ratio, defined by P ratio, defined by z￿ z￿ z￿ z￿ λ z￿, z￿, z￿, z￿ z￿ z￿ z￿ z￿ . λ z￿, z￿, z￿, z￿ z￿ z￿ z￿ z￿ . (z￿ − z￿)(z￿ − z￿) ( ) = ( − )( − ) ( ) = ( − )( − ) ( − )( − )

Cross-ratio [Krishnavedala - Wikimedia Commons] The twisted cubic

￿ ￿ ￿ ￿ ￿ ￿ Let v P P , X￿, X￿ X￿, X￿X￿, X￿X￿ , X￿ . The image C v P￿ is the twisted cubic in P￿. It is defined by ∶ → [ ] ￿ [ ] C F=￿, F(￿, F)￿ where = V( ) It is not defined by any two of these. ￿ F￿ Z￿Z￿ Z￿ For example, F￿ and F￿ define the union of C F￿ Z￿Z￿ Z￿Z￿ and the line Z￿ Z￿ ￿ . = − F Z Z Z￿. ￿ = ￿ ￿ − ￿ { = = } It is not defined= − by any two of these. Exercise (1.11. in [Ha]) For example, F￿ and F￿ define the union of C and the line Z￿ Z￿ ￿ .

{ = = }

Claudio Rocchini - Wikimedia Commons

Corollary ￿.￿. If is any secant line of (i.e. with ), there exist with Corollary ￿.￿. If L is any secant line of C (i.e. L pq with p, q C), there exist µ, ν with Qµ Qν C L. Qµ Qν C L. = ∈ Proof. For any r pq, r p, r q, the space Proof. For∩ any=r ∪pq, r p, r q, the space F F r ￿ Fλ Fλ r ∈ ￿ ≠ ≠ is ￿-dimensional. is ￿-dimensional.￿ ∶ ( ) = ￿ Let F , F be a basis. Since F , F vanish at the three points p, q, r on L, we have L F , F , Let Fµ, Fν be a basis. Since Fµ, Fν vanish at the three points p, q, r on L, we have L Fµ, Fν , hence ⊂ V( ) Q Q F , F C L Qµ Qν Fµ, Fν C L by the exercise above. by the exercise∩ = above.V( ) = ∪ ￿ The rational normal curve The rational normal curveThe rational normal curve

The rational normal curve in Pd is the Veronese embedding of P￿ of degree d. It is the image The rational normal curve in d is the Veronese embedding of ￿ of degree d. It is the image of the map P P of the map ￿ d d d ￿ d ￿ d vd P￿ Pd, X￿, X￿ X￿d, X￿d ￿X￿,...,X￿X￿d ￿, X￿d . vd P P , X￿, X￿ X￿ , X￿ X￿,...,X￿X￿ , X￿ . − − ∶ → [ ] ￿ [ − − ] Any d ∶ ￿ points→ on[ a rational] ￿ [ normal curve are linearly independent.] For given distinct points Any d ￿ points￿ on a rational normal curve are linearly independent. For given distinct points p￿,...,pd P￿, we may assume pi ￿, ￿ for all i, say pi Yi, ￿ . The matrix p￿,...,+pd P , we may assume pi ￿, ￿ for all i, say pi Yi, ￿ . The matrix + d d ￿ d ￿ Y Y Y Y￿ ￿ Z￿d ∈Z￿d ￿ Z￿d ￿ Z￿ ￿ ≠ [ ] = [ ] ￿d ∈ ￿d ￿ ￿ ￿ ￿ ≠ [ ] = [ ] ￿ Y￿d Y￿d−￿ − ￿ ￿ ￿ Z￿ Z￿ − − ⋅ ￿ ￿ ￿ − ⋅ ￿ ￿ − ￿ ￿ ⋅⋅⋅￿ ￿ ￿ ⋅⋅⋅￿ ￿ ￿ ⋅⋅⋅⋅⋅⋅d ￿ ￿ ￿ Y⋅⋅⋅⋅⋅⋅d Yd ￿ ￿ ￿ Zd Zd ￿ ￿ ￿ ⋅⋅⋅⋅⋅d d ￿ is a Vandermonde￿ ⋅⋅⋅⋅⋅ matrix with determinant￿ , showing that is a Vandermonde￿ ⋅⋅⋅ matrix with determinant￿ i j Yi Yj ￿, showing that vd p￿ ,...,vd pd is a Vandermonde￿ ⋅⋅⋅ matrix with determinant￿ i j Zi Z j ￿, showing that vd p￿ ,...,vd pd are independent.￿ ￿ are independent. < Any curve projectively equivalent to the rational∏ < ( normal− ) curve≠ is also a rational( normal) curve.( ) Any curve projectively equivalent to the rational∏ < ( normal− ) curve≠ is also a rational( normal) curve.( ) In particular, if H￿,...,Hd is any basis of K X￿, X￿ d, then In particular, if H￿,...,Hd is any basis of K X￿, X￿ d, then vd X￿, X￿ H￿ X￿, X￿ ,...,Hd X[￿, X￿ ] vd X￿, X￿ H￿ X￿, X￿ ,...,Hd X[￿, X￿ ] is a rational normal curve. is a rational∶ [ normal] ￿ curve.[ ( ) ( )] is a rational∶ [ normal] ￿ curve.[ ( ) ( )]

d Theorem ￿.￿. Through any d ￿ points in general position in Pd, there passes a unique rational Theorem ￿.￿. Through any d ￿ points in general position in Pd, there passes a unique rational normalTheorem curve.￿.￿. Through any d ￿ points in general position in P , there passes a unique rational normal curve. + Construction. Let µ ,...,µ , ν +,...,ν K with µ , ν µ , ν for all i j and consider Construction. Let µ￿,...,µd, ν￿,...,νd K and consideri i j j Construction. Let µ￿,...,µd, ν￿,...,νd K and consider d × d ∈ × [ ] ≠ [ ] ≠ G d µ X ν X K X , X∈ G µi X￿ νi X￿ K X￿, X∈￿ d ￿ G i ￿ µi X￿ νi X￿ K X￿, X￿ d ￿ i ￿ i ￿ G + = ￿( G − ) ∈ [ ] + Hi = ￿= ( G − , i) ∈ ￿,...,[ d. ] Hi µ=i X￿ νi X￿, i ￿,...,d. µi X￿ νi X￿ = = d Then H￿,...,= Hd are a basis of= K X￿, X￿ d. For if d aiHi ￿ is any linear relation, evaluation Then H￿,...,Hd−are a basis of K X￿, X￿ d. For if i ￿ aiHi ￿ is any linear relation, evaluation at H￿,...,givesHd− . K X￿, X￿ d i ￿ aiHi ￿ at µi, νi gives ai ￿. at µi, νi gives ai ￿. [ ] ∑ = = Thus [ ] ∑ = = Thus[ ] = [ ] = vd X￿, X￿ H￿ X￿, X￿ ,...,Hd X￿, X￿ vd X￿, X￿ H￿ X￿, X￿ ,...,Hd X￿, X￿ is a rational∶ [ normal] ￿ curve.[ ( We find) ( )] is a rational∶ [ normal] ￿ curve.[ ( We find) ( )] vd µ￿, ν￿ ￿, ￿,...,￿ ,...,vd µd, νd ￿,...,￿, ￿ vd µ￿, ν￿ ￿, ￿,...,￿ ,...,vd µd, νd ￿,...,￿, ￿ µ￿ µd µ￿ µd µ￿ µd µ￿ µd ￿ ￿ and ￿ ￿ vd([￿, ￿ ]) = [￿ d ,...,] ￿ d ([ µ￿￿,...,]) = [µd￿ and] vd ￿, ￿ ν￿￿,...,νd￿ . vd([￿, ￿ ]) = [µ￿ ,...,] µd ([ µ￿ ,...,]) = [µd and] vd ￿, ￿ ν￿ ,...,νd . µ￿ µd − d− − d− µ￿￿ µ￿d − − − − ([ ]) = ￿ ￿ ￿ ￿ = [ − − ] d ([ ]) = [ − − ] So let p([￿,...,])p=d ￿￿ be any d ￿ points￿ in= general[ position] indP . ([ ]) = [ ] So let p￿,...,pd ￿ be any d ￿ points in general position inP P. d. We can￿ assumedp￿ e for i ￿,...,d by projective equivalence. Then p and p have We can assume pi ei for i ￿,...,d by projective equivalence. Then pd ￿ and pd ￿ have non-zero coordinates.p+i e Wei can+i choose￿,...,d ￿ such that pd ￿ pd ￿ and non-zero coordinates.+ We can+ choose µ￿, ν￿ ,..., µd, νd P￿ such that vd ￿, ￿ pd ￿ and non-zero coordinates.= [ We] can choose= µ￿, ν￿ ,..., µd, νd P such that vd+￿, ￿ p+d ￿ and vd ￿, ￿ pd ￿. Uniqueness= [ ] is left= as an exercise. + + vd ￿, ￿ pd ￿. Uniqueness is left as an[ exercise.] [ ] ∈ [ ] = + [ ] [ ] ∈ [ ] = + [ ] = + ￿ [ ] = + ￿