Algebraic Geometry - James Milne

1 Algebraic MilneVersion 19, 2017 These notes are an introduction to the theory of Algebraic varieties emphasizing the simi-larities to the theory of manifolds. In contrast to most such accounts they study abstractalgebraic varieties , and not just subvarieties of affine and projective space. This approachleads more naturally into scheme information@misc{milneAG,author={ Milne , James S.},title={ Algebraic Geometry ( )},year={2017},note={Available at },pages={221}} (August 24, 1996). First version on the (June 13, 1998). (October 30, 2003).

2 Fixed errors; many minor revisions; added exercises; added twosections/chapters; 206 (February 20, 2005). Heavily revised; most numbering changed; 227 (March 19, 2008). Minor fixes; TEXstyle changed, so page numbers changed; (September 14, 2009). Minor corrections; revised Chapters 1, 11, 16; 245 (January 13, 2012). Minor fixes; 260 (August 24, 2014). Major revision; 223 (August 23, 2015). Minor fixes; 226 (March 19, 2017). Minor fixes; 221 at send comments and corrections to me at the address on my web curves are a tacnode, a ramphoid cusp, and an ordinary triple 1996 2017 paper copies for noncommercial personal use may be made without explicit permissionfrom the copyright Preliminaries from commutative algebra11a.

3 Rings and ideals, 11 ; b. Rings of fractions, 15 ; c. Unique factorization, 21 ; d. Integraldependence, 24; e. Tensor Products, 30 ; f. Transcendence bases, 33; Exercises, Algebraic Sets35a. Definition of an Algebraic set, 35 ; b. The Hilbert basis theorem, 36; c. The Zariskitopology, 37; d. The Hilbert Nullstellensatz, 38; e. The correspondence between algebraicsets and radical ideals, 39; f. Finding the radical of an ideal, 43; g. Properties of theZariski topology, 43; h. Decomposition of an Algebraic set into irreducible Algebraic sets,44 ; i.

4 Regular functions; the coordinate ring of an Algebraic set, 47; j. Regular maps, 48; ; finite and quasi-finite maps, 48; l. Noether normalization theorem, 50 ; , 52 ; Exercises, Affine Algebraic Varieties57a. Sheaves, 57 ; b. Ringed spaces, 58; c. The ringed space structure on an Algebraic set, 59; d. Morphisms of ringed spaces, 62 ; e. Affine Algebraic varieties , 63; f. The category ofaffine Algebraic varieties , 64; g. Explicit description of morphisms of affine varieties , 65 ;h. Subvarieties, 68; i. Properties of the regular mapSpm.

5 /, 69; j. Affine space withoutcoordinates, 70; k. Birational equivalence, 71; l. Noether Normalization Theorem, 72; , 73 ; Exercises, Local Study79a. Tangent spaces to plane curves, 79 ; b. Tangent cones to plane curves, 81 ; c. The localring at a point on a curve, 83; d. Tangent spaces to Algebraic subsets ofAm, 84 ; e. Thedifferential of a regular map, 86; f. Tangent spaces to affine Algebraic varieties , 87 ; cones, 91; h. Nonsingular points; the singular locus, 92 ; i. Nonsingularity andregularity, 94; j. Examples of tangent spaces, 95; Exercises, Algebraic Varieties97a.

6 Algebraic prevarieties, 97; b. Regular maps, 98; c. Algebraic varieties , 99; d. Maps fromvarieties to affine varieties , 101; e. Subvarieties, 101 ; f. Prevarieties obtained by patching,102; g. Products of varieties , 103 ; h. The separation axiom revisited, 108; i. Fibredproducts, 110 ; j. Dimension, 111; k. Dominant maps, 113; l. Rational maps; birationalequivalence, 113; m. Local study, 114; n. Etale maps, 115 ; o. Etale neighbourhoods,118 ; p. Smooth maps, 120 ; q. Algebraic varieties as a functors, 121 ; r. Rational andunirational varieties , 124 ; Exercises, Projective Varieties1273a.

7 Algebraic subsets ofPn, 127; b. The Zariski topology onPn, 131; c. Closed subsets ofAnandPn, 132 ; d. The hyperplane at infinity, 133; an Algebraic variety, 133; homogeneous coordinate ring of a projective variety, 135; g. Regular functions on aprojective variety, 136; h. Maps from projective varieties , 137; i. Some classical maps ofprojective varieties , 138; j. Maps to projective space, 143; k. Projective space withoutcoordinates, 143; l. The functor defined by projective space, 144; m. Grassmann varieties ,144 ; n. Bezout s theorem, 148; o.

8 Hilbert polynomials (sketch), 149; p. Dimensions, 150;q. Products, 152 ; Exercises, Complete Varieties155a. Definition and basic properties, 155 ; b. Proper maps, 157; c. Projective varieties arecomplete, 158 ; d. Elimination theory, 159 ; e. The rigidity theorem; abelian varieties ,163; f. Chow s Lemma, 165 ; g. Analytic spaces; Chow s theorem, 167; h. Nagata sEmbedding Theorem, 168 ; Exercises, Normal varieties ; (Quasi-)finite maps; Zariski s Main Theorem171a. Normal varieties , 171 ; b. Regular functions on normal varieties , 174 ; c.

9 Finite andquasi-finite maps, 176; d. The fibres of finite maps, 182; e. Zariski s main theorem,184; f. Stein factorization, 189; g. Blow-ups, 190 ; h. Resolution of singularities, 190 ;Exercises, Regular Maps and Their Fibres193a. The constructibility theorem, 193; b. The fibres of morphisms, 196; c. Flat maps andtheir fibres, 199; d. Lines on surfaces, 206; e. Bertini s theorem, 211; f. Birationalclassification, 211; Exercises, to the exercises213 Index2194 NotationsWe use the standard (Bourbaki) notations:NDf0;1;2;:::g,ZDring of integers,RDfieldof real numbers,CDfield of complex numbers,FpDZ=pZDfield ofpelements,paprime number.

10 Given an equivalence relation, denotes the equivalence class containing .A family of elements of a setAindexed by a second setI, , is a functioni7!aiWI!A. We sometimes writejSjfor the number of elements in a finite ,kis an algebraically closed field. Unadorned tensor products are overk. Forak-algebraRandk-moduleM, we often writeMRforR M. The ;k/of a finite-dimensionalk-vector spaceEis denoted rings will be commutative with1, and homomorphisms of rings are required to use Gothic (fraktur) letters for ideals:a b c m n p q A B C M N P Qa b c m n p q A B C M N P QFinallyXdefDY Xis defined to beY, or equalsYby definition;X Y Xis a subset ofY(not necessarily proper, ,Xmay equalY);X Y XandYare isomorphic;X'Y XandYare canonically isomorphic (or there is a given or unique isomorphism).