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Classical Algebraic Geometry

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Classical Algebraic Geometry 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 Pn 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.
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