Commutative Algebra and Algebraic Geometry Lecture Notes Johannes Kepler Universit¨atLinz Summer 2018 Prof. Franz Winkler c Copyright by F. Winkler Contents References 1. Introduction . .1 2. Elimination theory . 9 2.1 Existence and construction of Gr¨obnerbases . .9 2.2 Solving ideal theoretic problems by Gr¨obnerbases . 17 2.3 Basis conversion for 0-dimensional ideals | FGLM . 23 2.4 Resultants . 32 3. Affine algebraic sets and varieties . .35 3.1 Affine space and algebraic sets . 35 3.2 Noetherian rings and Hilbert's Basis Theorem . 41 3.3 Irreducible components of algebraic sets . 44 4. The algebraic-geometric correspondence . 49 4.1 The geometry of elimination . .49 4.2 Hilbert's Nullstellensatz . 53 4.3 Primary decomposition of ideals . 57 5. Projective algebraic sets and varieties . 61 5.1 Projective space . .61 5.2 Homogeneous ideals and projective algebraic sets . 66 6. Functions and mappings on varieties . 75 6.1 Coordinate rings, polynomial functions and mappings . 75 6.2 Rational functions and local rings . 82 7. Plane algebraic curves . .87 7.1 Singularities and tangents . 87 7.2 Intersection of plane curves . .97 7.3 Linear systems and divisors . 106 8. Rational parametrization of curves . 113 8.1 Rational curves and parametrizations . 113 8.2 Proper parametrizations . 121 8.3 Parametrization by lines . 127 8.4 Parametrization by adjoint curves . 134 9. Local parametrization and Puiseux expansion . 143 9.1 Power series, places, and branches . 145 9.2 Puiseux's theorem and the Newton polygon method . 150 10. Dimension and Hilbert function . 157 10.1 An algebraic definition of dimension . .157 10.2 The Hilbert function . 161 Preface Commutative algebra is the theory of polynomial ideals. Problems concerning poly- nomial ideals are representation of ideals, e.g. by Gr¨obnerbases or characteristic sets, decomposition of ideals into prime and primary ideals, determination of radical ideals, di- mension and height of ideals, the structure of polynomial rings modulo ideals, solution of systems of polynomial equations, solution of linear systems with polynomial coefficients, i.e. computation of syzygies, etc. Constructive methods for answering such questions are provided by computer algebra, e.g. algorithms for computation of greatest common divisors, factorization, computation of Gr¨obnerbases, resultants. This theory of polynomial ideals can be generalized to vectors of polynomials, leading to modules over polynomial rings. Algebraic geometry traditionally is the study of sets of solutions of systems of poly- nomial equations, i.e. of algebraic sets. We might be interested in the dimension of an algebraic set, its irreducible components, the tangent space at a point, functions on algebraic sets, different representations such as local or global parametrizations, etc. Algebraic curves and surfaces are an old topic of geometric and algebraic investiga- tion. They appear in biological shapes, in ancient and modern architectural designs, in number theoretic problems, in error-correcting codes, and in cryptographic algorithms. Recently they have gained additional practical importance as central objects in com- puter aided geometric design. Modern airplanes, cars, and household appliances would be unthinkable without the computational manipulation of algebraic curves and surfaces. Algebraic curves and surfaces combine fascinating mathematical beauty with challenging computational complexity and wide spread practical applicability. Obviously commutative algebra and algebraic geometry are closely related. We will investigate some of these relations. For further reading on the topics of this course we suggest the books by Sendra, Winkler, P´erez-D´ıaz[SWP08], Cox, Little, O'Shea [CLO97], [CLO98], Fulton [Ful69], Reid [Rei88], and Walker [Wal50]. As good introductions to commutative algebra we suggest the books by Zariski, Samuel [ZaS58] and Kunz [Kun85]. For computational methods on polynomials we refer to the books by Winkler [Win96] and Kreuzer, Robbiano [KrR00]. Hartshorne [Har77] is a more advanced text on algebraic geometry. The modern language of algebraic geometry is introduced in the book by Eisenbud and Harris [EiH92]. References [Abh90] S.S. Abhyankar, Algebraic Geometry for Scientists and Engineers, Amer.Math.Soc. (1990) [AdL94] W.W. Adams, P. Loustaunau, An Introduction to Gr¨obnerBases, AMS, Provi- dence, RI, Graduate studies in math., vol.3 (1994) [Baj94] C.L. Bajaj, Algebraic Geometry and Applications, Springer (1994) [BeW93] T. Becker, V. Weispfenning, Gr¨obnerBases | A Computational Approach to Commutative Algebra, Springer-Verlag, Berlin, Graduate texts in math., vol.141 (1993) [BrK81] E. Brieskorn, H. Kn¨orrer, Ebene algebraische Kurven, Birkh¨auser(1981) [BCL83] B. Buchberger, G.E. Collins, R. Loos, Computer Algebra | Symbolic and Alge- braic Computation, 2nd ed., Springer-Verlag Wien New York (1983) [BuW98] B. Buchberger, F. Winkler (eds.), Gr¨obnerBases and Applications, London Mathematical Society Lecture Notes Series 251, Cambridge Univ. Press (1998) [CLO97] D. Cox, J. Little, D. O'Shea, Ideals, Varieties, and Algorithms, 2nd edition, Springer (1997) [CLO98] D. Cox, J. Little, D. O'Shea, Using Algebraic Geometry, Springer (1998) [Eis95] D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Springer (1995) [EiH92] D. Eisenbud, J. Harris, Schemes: The Language of Modern Algebraic Geometry, Wadworth & Brooks/Cole (1992) [Ful69] W. Fulton, Algebraic Curves, Benjamin/Cummings (1969) [Gro68] W. Gr¨obner, Algebraische Geometrie I, II, B.I. Hochschultaschenb¨ucher (1968/70) [Har77] R. Hartshorne, Algebraic Geometry, Springer (1977) [KaS95] M. Kalkbrener, B. Sturmfels, \Initial complexes of prime ideals", Advances in Mathematics 116/2, 365{376 (1995) [Kun85] E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry, Birk- h¨auser(1985) [KrW91] H. Kredel, V. Weispfenning, \Computing dimension and independent sets for polynomial ideals", J. Symbolic Computation 12/6, 607{631 (1991) [Lan58] S. Lang, Introduction to Algebraic Geometry, Interscience Publ. (1958) [Rei88] M. Reid, Undergraduate Algebraic Geometry, Cambridge Univ. Press (1988) [Sch72] I.R. Schafarewitsch, Grundz¨ugeder algebraischen Geometrie, Vieweg (1972) [SeW91] J.R. Sendra, F. Winkler, \Symbolic Parametrization of Curves", J. Symbolic Computation 12/6, 607{631 (1991) [SeW97] J.R. Sendra, F. Winkler, \Parametrization of Algebraic Curves over Optimal Field Extensions", J. Symbolic Computation 23/2&3, 191{207 (1997) [SWP08] J.R. Sendra, F. Winkler, S. P´erez-D´ıaz, Rational Algebraic Curves { A Com- puter Algebra Approach, Springer (2008) [SiT92] J.H. Silverman, J. Tate, \Rational Points on Elliptic Curves", Springer-Verlag (1992) [TrW00] Q.-N. Tran, F. Winkler (eds.), \Applications of the Gr¨obnerBasis Method", J. Symbolic Computation 30/4 (2000) [Wae70] B.L. van der Waerden, Algebra I, II, Ungar, New York (1970) [Wal50] R.J. Walker, Algebraic Curves, Springer (1950) [Win96] F. Winkler, Polynomial Algorithms in Computer Algebra, Springer (1996) [Zar69] O. Zariski, An Introduction to the Theory of Algebraic Surfaces, Springer (1969) [ZaS58] O. Zariski, P. Samuel, Commutative Algebra I, II, Springer (1958/60).