MIT OpenCourseWare http ocw mit edu 18 727 Topics in Algebraic Geometry Algebraic Surfaces Spring 2008 For information about citing these materials or our Terms of Use visit http ocw mit edu terms ALGEBRAIC SURFACES LECTURE 1 LECTURES ABHINAV KUMAR 1 Introduction This course concerns algebraic surfaces which for our purposes will be projec tive and non singular over a eld k Usually we will assume k is algebraically closed The simplest example of algebraic surfaces arise as hypersurfaces in P3 Let S V f for f an irreducible homogeneous polynomial of degree d ab stractly S Proj k X Y Z W f d 1 We can change coordinates so that f X giving an isomorphism to the rational surface P2 d 2 Changing coordinates we can write f XY ZW and obtain an iso morphism to the rational ruled surface P1 P1 the two rulings are given by X W Y Z or X Z Y W Note that the partic ular surface may have many fewer rational points than P1 P1 e g X 2 Y 2 Z 2 W 2 0 has no rational solutions The smooth quadric in P3 is the Segre embedding of P1 P1 in P3 it is isomorphic to P2 2 1 i e P2 blown up in 2 points with the proper transform of the line joining them blown down d 3 Cubic surfaces in P3 are also rational surfaces realized as P2 blown up at 6 points Each has 27 lines which can be seen explicitly in the case of X 3 Y 3 Z 3 W 3 0 nine of the lines are given by X Y 0 Z W 0 where are cubic roots of unity Similarly we have the lines given by X Z 0 Y W 0 and X W 0 Y Z 0 The con guration of 27 lines and their intersection points is called the Scha i graph It is strongly regular with parameters 27 16 10 8 To see that it is rational choose two non intersecting lines L M taking projections gives L X P1 M X P1 L M X P1 P1 which is birational by Bezout d 4 Quartic surfaces in P3 are examples of K3 surfaces over C these are examples of Calabi Yau manifolds This class includes for instance Kummer surfaces of abelian surfaces which are classically well studied 1 2 LECTURES ABHINAV KUMAR For K3 surfaces the geometry and moduli are well known but the arith metic less so For instance the number of parameters for a quartic surface 4 3 3 in P is 3 dim GL 4 35 16 19 d 5 These are surfaces of general type 2 Rough Plan for the Course The goal will be a full classi cation of surfaces mostly with proofs 1 Preliminaries intersection theory Riemann Roch Hodge index theorem 2 Birational maps minimal models 3 Classi cation In characteristic 0 we have the Enriques Castelnuovo Zariski Kodaira classi cation of minimal models of surfaces based on Kodaira dimen sion S the maximal dimension of the image of S under the linear system nKS if this linear system is always 0 geometric genus pg h0 S KS and irregularity q h1 S OS Here KS is divisor class corresponding to the canonical sheaf 2 1 S These are ruled e g rational surfaces i e there is a map S C onto a curve with generic ber smooth of genus 0 and have geometric genus 0 in fact all their plurigenera are zero S 0 There are four possibilities based on combinations of pg q 0 0 are Enriques surfaces 0 1 are bielliptic or hyperelliptic surfaces 1 0 are K3 surfaces and 1 2 are abelian surfaces S 1 These are honest elliptic surfaces An elliptic surfaces is one with an elliptic bration i e a map S C onto a curve with generic ber smooth of genus 1 S 2 These are surfaces of general type In characteristic 0 we have the Bombieri Mumford classi cation of minimal models which require the adic Betti numbers bi dim Q H i X Q for char k Note that bi is independent of and agrees with the dimension of H i X C for a smooth complex variety X S These are ruled surfaces By Castelnuovo s theorem only ra tional if q p2 0 works in any characteristic S 0 There are four possibilities based on combinations of b1 b2 0 10 are Enriques surfaces classical if pg q 0 0 and non classical if pg q 1 1 2 2 are bielliptic or hyperelliptic surfaces if pg q 0 1 and quasi hyperelliptic surfaces if pg q 1 2 0 22 are K3 surfaces and ALGEBRAIC SURFACES LECTURE 1 3 4 6 are abelian surfaces S 1 These are surfaces with elliptic or quasi elliptic brations in the latter case the generic ber has arithmetic genus 1 but has a cusp only exists in characteristics 2 3 e g y 2 x3 t S 2 These are surfaces of general type 4 We will discuss various aspects of the geometry and arithmetic of surfaces as they arise and some singularity theory and other topics according to interest 2 1 References Beauville Complex Algebraic Surfaces Badescu Algebraic Surfaces Barth et al Compact Complex Surfaces Reid Chapters on Algebraic Surfaces in PCMI vol 3 Hartshone Algebraic Geometry chapter 5 Gri ths and Harris Principles of Algebraic Geometry chapter 4 Kodaira On Compact Complex Analytic Surfaces I II III in the Annals of Mathematics Bombieri and Mumford Enriques Classi cation of Surfaces in Charac teristic p Invent Math 3 Preliminaries Let X S be a nonsingular projective algebraic surface over an algebraically closed eld k We recall the basic notions of intersection theory on surfaces De nition 1 A curve on S is a closed integral subscheme of co dimension 1 A divisor is a formal sum of curves with multiplicity and is e ective if the coe cients are nonnegative The set of divisors form a group Div X For C D distinct curves on X the intersection multiplicity of C and D at p C D is mp C D dim k Op f g where f is an equation for C in Op and g is an equation for D in Op C and D are called transverse if mp C D 1 i e f g span the maximal ideal The intersection product between C D is C D p C D mp C D and extends to divisors in the obvious manner Recall that the ideal sheaf de ning C is OX C Let OC D OX OX C OX D a skyscraper sheaf concentrated on C D At each p C D OC D p OX f g so C D dim H 0 X OC D Recall for short exact sequences of sheaves on X 0 F G H 0 taking right derived functors of global section functor we get an associated long exact sequence 1 0 H 0 X F H 0 X G H 0 X H H 1 X F …