A TASTE OF TWO-DIMENSIONAL COMPLEX ALGEBRAIC GEOMETRY LIVIU I. NICOLAESCU ABSTRACT. These are notes for a talk at a topology seminar at ND.. GENERAL FACTS In the sequel, for simplicity we denote the complex projective space CP N by P N, and we denote the projective coordinates by [ z] = [z 0,..., z N ]. A smooth projective variety is a connected compact complex submanifold of some projective space P N. All the smooth projective varieties are Kähler manifolds, but the converse is not true. An algebraic surface is a smooth projective variety of complex dimension 2. A celebrated theorem of Chow [2, Chap., Sec. 3] states that if X P N is a smooth projective variety, then there exist homogeneous polynomials P,..., P ν C[z 0,..., z N ] such that X = { [ z] P N ; P ( z) = = P ν ( z) = 0 }. In other words, all projective varieties are described by a finite collection of homogeneous polynomial equations. If X is a smooth projective variety of (complex) dimension n then the tanget bundle T X has a complex structure and thus we can speak of Chern classes c k (X) := c k (T X), k =,..., n. We also have an isomorphism of holomorphic vector bundles Λ m T X C = Λ p,q T X. (.) p+q=m If (u,..., u n ) are local holomorphic coordinates on X then the smooth sections of Λ p,q T X are locally described by sums of forms of the type f(u)du i du ip dū j dū jq. For every smooth differential form α of degree m with complex coefficients we denote by α p,q its Λ p,q -component in the Hodge decomposition (.). The holomorphic line bundle Λ n,0 T X is called the canonical line bundle of X and it is denoted by K X. Let us point out that c (X) = c (K X ). From the above equality we deduce that The algebraic variety X is spinnable if and only if c (K X ) = 0 mod 2. (.2) We denote by H m (X) the space of complex differential forms of degree m on X that are harmonic with respect to the Kähler metric, and for any closed form α we denote by [α] its harmonic part in the Hodge decomposition. Then Hodge theory implies that dim C H m (X) = b m (X), Last modified on March 3, 2009.
2 LIVIU I. NICOLAESCU and [α] p,q = [α p,q ], α Ω m (X) C, dα = 0, p + q = m. We denote by H p,q (X) the space of harmonic forms of type (p, q), and we set h p,q (X) := dim C H p,q (X). H m (X) = H p,q (X) and b m (X) = h p,q (X). (.3) p+q=m Moreover, Hodge theory also implies that p+q=m h p,q (X) = h q,p (X) = h n p,n q (X), p, q. (.4) If we denote by Ω p X the sheaf of holomorphic sections of the holomorphic bundle Λp,0 T X then Dolbeault theorem implies that the space H p,q (X) is naturally isomorphic to the q-th Čech cohomology of the sheaf Ω p X, H p,q (X) = H q (X, Ω p X ). Observe that Ω n X coincides with the sheaf of holomorphic sections of the canonical line bundle so that the Hodge number h n,0 (X) is equal to the dimension of the space of global holomorphic sections of K X. This integer is known as the geometric genus of X and it is denoted by p g (X). We define the Euler characteristics χ p (X) == ( ) q h p,q (X). q When X is an algebraic surface, then the Hodge numbers can be organized in a Hodge diamond h 0,0 (X) h,0 (X) h 0, (X) h 2,0 (X) h, (X) h 0,2 (X) h,2 (X) h 2, (X) h 2,2 (X) where the entries symmetric with respect to one of the two axes are equal. From this symmetry we deduce that b (X) = 2h,0 (X) so that the first Betti number of an algebraic surface must be even. Note that in this case χ 0 (X) = h 0, (X) + p g (X) = 2 b (X) + p g (X). This holomorphic Euler characteristic is given by the Noether formula χ 0 (X) = 2( c (X) 2, [X] + c 2 (X), [X] ) = 2 c (X) 2, [X] + χ(x), (.5) 2 where χ(m) is the topological Euler characteristic. The Hodge decomposition (.3) is compatible with the intersection form. This is the content of the Hodge index theorem. For simplicity, we state it only for algebraic surfaces. Theorem. (Hodge index theorem). If X is an algebraic surface, then it is a smooth oriented real 4-manifold. The second Betti number decomposes as b 2 (X) = b + 2 (X) + b 2 (X), where b± 2 (X) is the number of positive/negative eigenvalues of the intersection form of X. Then b + 2 (X) = h2,0 (X) + h 0,2 (X) + = 2p g (X) +, (.6)
and the signature τ(x) of X is A TASTE OF TWO-DIMENSIONAL COMPLEX ALGEBRAIC GEOMETRY 3 τ(x) = b + 2 (X) b 2 (X) = 2p g(x) + 2 h, (X). One consequence of The Hodge index theorem implies that p g (X) and h, (X) are topological invariants of X. From the identity p (X) = c (X) 2 2c 2 (X) and the Hirzebruch signature formula we deduce so that τ(x) = 3 c (X) 2 2c 2 (X), [X], c (X) 2, [X] = 2χ(X) + 3τ(X). Using a bit of Morse theory one can produce some information about the homotopy type of a smooth algebraic variety. The following is a special case of Lefschetz hyperplane theorem. Theorem.2. If X is a compact, complex 2-dimensional submanifold of P 3 or P P 2 then X is connected and simply connected. 2. DIVISORS A hypersurface on a smooth projective variety X is a closed set Y locally described as the zero set of a (nontrivial) holomorphic function. A point p Y is called a smooth point if there exists an open neighborhood U of p in X such that U Y is a smooth submanifold of U. We denote by Y the set of smooth points. Then Y is an open and dense subset of Y. The hypersurface Y is called irreducible if Y is connected. For a general hypersurface Y we define its components to be the closures of the connected components of Y. The hypersurfaces on an algebraic surface are called curves. We denote by Div(X) the free Abelian group generated by the irreducible hypersurfaces in X. The elements in X are called divisors on X. Clearly, every irreducible hypersurface defines a divisor that we denote by [Y ]. Then any divisor D on X can be written as a formal sum ν D = m i [Y i ], i= where Y i are irreducible hypersurfaces on X called the components of D. The integer m i is called the multiplicity of D along Y i. The hypersurface Y Y ν is called the support of D and it is denoted by supp(d). A divisor D on X is called effective if the multiplicities along its components are all nonnegative. We will find convenient use Cartier s descriptions of divisors. Suppose we are given an open cover U = (U α ) α A, nontrivial holomorphic functions p α, q α O(U α ), and nowhere vanishing holomorphic functions g αβ O(U αβ ) such that, if we set then h α = p α q α g αβ = h α h β on U αβ \ {f β = 0}. (2.)
4 LIVIU I. NICOLAESCU This defines a divisor D = D(U, {p α, q α }) whose support is the hypersurface Z = Z 0 Z, Z 0 = α {p α = 0}, Z = α {q α = 0}. If Z 0,i is a component of Z 0. Then we define m 0,i as follows. Choose x Z0,i, U α containing x and then define m 0,i to be the order of vanishing p α along Z0,i U α. The equality (2.) guarantees that the integer is independent of the various choices. We define the integer m j, similarly and then we set D(U, {p α, q α }) = m 0,i [Z 0,i ] m,j [Z,j ] i j This divisor is called the Cartier divisor defined by the open cover U and the meromorphic functions h α on U α. Proposition 2.. On a smooth algebraic variety any divisor can be described as a Cartier divisor. To any Cartier divisor D defined by an open cover U and meromorphic function h α satisfying (2.) we can associate a holomorphic line bundle L D defined by the open cover and gluing cocycle g αβ = hα h β. The functions h α can be interpreted as defining a meromorphic section of L D. Proposition 2.2. On a smooth algebraic variety the above coprrespondence { } { } Cartier divisors pairs (holomorphic line bundle, meromorphic section) is a bijection. Because of the above proposition, the words divisors and holomorphic line bundles are used interchangeably in algebraic geometry. In particular, the canonical line bundle is often called the canonical divisor. A principal (Cartier) divisor is a divisor D such that the associated line bundle is holomorphically trivializable. We denote by PDiv(X) the subgroup of principal divisors. The quotient Div(X)/ PDiv(X) can be identified with the space of holomorphic isomorphism classes of holomorphic line bundles on X. This group is also known as the Picard group of X and it is denoted by Pic(X). Any hypersurface Y on a complex n-dimensional algebraic variety is defines a homology class [Y ] H 2n 2 (X, Z). This extends by linearity to a morphism of groups Div(X) D [D] H 2n 2 (X, Z). For every divisor D Div(X) we let [D] H 2 (X, Z) denote the Poincaré dual of the homology class [D]. Let us observe that the Gauss-Bonnet Chern theorem implies that [D] = c (L D ), D Div(X). Proposition 2.3. The Poincaré dual of the homology class defined by a divisor is a cohomology class of degree 2 and Hodge type (, ). If X is an algebraic surface then any divisor on X defines a (non-torsion) 2-dimensional homology class. The intersection pairing defines an intersection pairing on Div(X). H 2 (X, Z)/Tors H 2 (X, Z)/T ors Z
A TASTE OF TWO-DIMENSIONAL COMPLEX ALGEBRAIC GEOMETRY 5 Suppose C X is a smooth, connected algebraic curve in an algebraic surface X. Then as a topological space C is homeomorphic to a Riemann surface. We denote by g(c) its genus. The normal bundle of C X is a holomorphic line bundle and we have the following topological isomorphism complex vector bundles T X C = T C TC X. Dualizing we deduce T XC = T C TCX Taking the second exterior power in the above equality, and observing that T C = K C we deduce the adjunction formula K X C = KC TCX. (2.2) Hence c (K X ) = c (K C ) + c (TCX) = c (K X ) c (T C X). Integrating over C we deduce c (K X ), [C] = c (K C ), [C] c (T C X), [C] = 2g(C) 2 [C] [C]. We obtain in this fashion the genus formula g(c) = + ( ) [C] [C] + c (K X ), [C] = + 2 2 [C] ( [C] + [K X ] ). 3. EXAMPLES Before we discuss in detail several examples of algebraic surfaces we need to discuss an important construction in algebraic geometry. Example 3. (The blowup construction). We start with the simples case, the blowup of the affine plane C 2 at the origin. Consider the incidence variety I = { (l, p) P C 2 ; the point p belongs to the complex line l C 2 }. The variety I is a complex manifold of dimension 2. The fiber of the natural projection π : I P over the line l P consists of the collection of points in l. Thus, the projection π : I P is none other than the tautological line bundle over P. We denote by E the submanifold P {0} E. Observe that E is exactly the zero section of the tautological line bundle. On the other hand, we have another natural projection This map restrict to a biholomorphic map σ : I C 2 C 2, (l, p) p. σ : I \ E C 2 \ {0}. The manifold I is called the blowup of C 2 at the origin. Equivalently, we say that I is obtained via the σ-process at 0 C 2. For any open set U C 2 we set I(U) = σ (U) U. The map σ defines a surjection I(U) U. Moreover, if U does not contain the origin, then σ : I(U) U is a biholomorphism. Suppose X P N is an algebraic surface and p 0 X. We want to give two descriptions of the blowup of X at p 0. The first construction uses the intuition from the special case discussed above. We choose local holomorphic coordinates (z 0, z ) on a neighborhood U of p 0 so that U can be identified with a neighborhood of 0 in C 2. Set U := U \ {p 0 }. Then the blowup of X at p 0 is obtained by removing the point p 0, and then attaching the manifold I(U) to X \{p 0 } using the biholomorphism σ : I(U ) U. We denote by X p0 the complex surface
6 LIVIU I. NICOLAESCU obtained in this fashion. Its biholomorphism type could depend on U but its diffeomorphism type is independent of U. Note that we have a natural holomorphic map σ : X p0 X. Its fiber over p 0 is the zero section E I. It is a smooth curve in X p0 called the exceptional divisor. It is a rational curve, i.e., it is biholomorphic to the projective line P, and in fact, it can be identified with the projectivization of the complex tangent plane T p0 X. The points in E can thus be identified with the one-dimensional complex subspaces of T p0 X. The self-intersection number of E coincides with the Euler number the tautological line bundle so that [E] [E] =. To see that it is X p0 is algebraic we need to give a more algebraic description of the blowup construction. We follow the approach in [, IV.2]. Recall that X is embedded in a projective space P N. Assume the point p 0 has homogeneous coordinates [0, 0,..., ]. We obtain a map X := X \ {p 0 } The graph of ϕ is a submanifold ϕ P N, X p [z 0 (p),..., z N (p)] P N. Γ ϕ X P N X P N. Then the blowup X p0 of X at p 0 can be identified with the closure of Γ ϕ in P N. The blowdown map σ : X p0 X is induced by the natural projection X P n X. Suppose that C is a curve on X then the total transform of C is the curve σ (C) on X. The proper transform of C is the closure in X p0 of σ (C \ {p 0 }). We denote it by σ (C). If p 0 happens to be a smooth point of C then σ (C) = σ (C \ {p 0 }) [ p 0 ], where [ p 0 ] denotes the point in E corresponding to the one dimensional complex subspace T p0 C T p0 X. In this case σ (C) = E σ (C). The point p 0 is a smooth point on both σ (C) and E, and the components E and σ (C) intersect transversally at p 0. More generally, the operation of proper and strict transforms extend by linearity to maps σ, σ : Div(X) Div( X) The proper transform sends principal divisors to principal divisors and thus induces a morphism of groups σ : Pic(X) Pic( X). This morphism coincides with the pullback operation on holomorphic line bundles. If C 0, C are two curves on X such that p 0 is a smooth point on both, then In particular, [σ (C 0 )] [σ(c )] = [C 0 ] [C ]. For any complex curve C X we have the equality [σ (C 0 )] 2 = [C 0 ] 2. (3.) σ ([C] ) = [σ (C)].
A TASTE OF TWO-DIMENSIONAL COMPLEX ALGEBRAIC GEOMETRY 7 The canonical line bundle of the blowup X is related to the canonical line bundle of X via the equality K bx = σ (K X ) L E. If we think of line bundles as divisors, the last equality can be rewritten as K bx = σ (K X ) + E Pic( X). There is a very simple way of recognizing when an algebraic surface is the blowup of another. More precisely, we have the following remarkable result, [2, Chap 4., Sec ]. Theorem 3.2 (Castelnuovo-Enriques). Suppose Y is an algebraic surface containing an exceptional curve, i.e., a rational curve C with self-intersection [C] 2 =. Then there exists an algebraic surface, a point p 0 X and a biholomorphic map φ : Y X p0 such that φ(c) is the exceptional divisor of the blowdown map σ : Xp0 called the blow down of Y along the exceptional curve C. X. The manifold X is In simpler terms, the Castelnuovo-Enriques theorem states that exceptional curves can be blown down. Definition 3.3. (a) Two algebraic surfaces are said to be birationally equivalent if we can obtain one from the other by a sequence of blowups and blowdowns. A surface is called rational if it is birationally equivalent to the projective plane P 2. (b) A geometric invariant of an algebraic surface is called a birational invariant if it does not change under blowups. Thus, birationally equivalent surfaces have identical birational invariants. Example 3.4. The Hodge numbers h,0 and h 2,0 are birational invariants of an algebraic surface. More generally, if X is an algebraic surface, then we define P n (X) to be the dimension of the space of holomorphic sections of the line bundle K n X. The numbers P n(x) are called the plurigenera of X. All of them are birational invariants of X. Example 3.5 (The projective plane). Consider the projective plane X = P 2, the point p 0 = [, 0, 0], and the line at infinity H given by the equation z 0 = 0. The homology group H 2 (P 2, Z) is generated by the homology class determined by H, and H 2 =. The line bundle determined by the divisor H is the tautological line bundle U P 2. Using Proposition 2.3 we deduce. h, (P 2 ) = = b 2 (X) so thatp g (P 2 ) = h 2,0 (P 2 ) = 0. The canonical line bundle K P 2 coincides with the line bundle associated to the divisor 3H, K X = 3H K X = U 3. In particular, this shows that P 2 is not spinnable. The Hodge diamond of P 2 is 0 0
8 LIVIU I. NICOLAESCU Any projective line l through p 0 is uniquely determined by its intersection with H. For any p H we denote by l p the line going through p 0 and p. The blowup of P 2 at p 0 contains two distinguished 0 L p - E L q p q H FIGURE. Blowing up the projective plane. rational curves: the exceptional divisor E and the proper transform of H which we continue to denote by H. We denote by L p the proper transform of l p. Then (see Figure ) E 2 =, H 2 =, L 2 p = 0, p H. We obtain a holomorphic map P 2 H whose fiber above p H is the rational curve L p ; see Figure. This is an example of ruled surface. Example 3.6 (Quadrics). Consider the ruled surface Q = P P. If we consider the Segre embedding S : P P P 3, ([s 0, s ], [t 0, t ]) [z 0, z, z 2, z 3 ] = [s 0 t 0, s 0 t, s t 0, s t ] we observe that the image S(P P ) coincides with the quadratic hypersurface { [z0, z, z 2, z 3 ] P 2 ; z 0 z 3 z z 2 = 0 }. In fact, any smooth quadratic hypersurface in P 3 is biholomorphic to Q The surface Q is swept by two family of projective lines X(t) = P {t} P P, Y (s) = {s} P P P, forming the pattern in Figure 2. The curves in the family X(t) define the same homology class x H 2 (X, Z), while the curves in the family Y (s) define the same homology class y H 2 (X, Z). The classes x and y form an integral basis of the homology group H 2 (Q, Z). The intersection form is determined by the relations (see Figure 2) x x = y y = 0, x y =. From Proposition 2.3 we deduce that 2 = b 2 (X) h, (X) 2.
A TASTE OF TWO-DIMENSIONAL COMPLEX ALGEBRAIC GEOMETRY 9 p 0 x x y y FIGURE 2. The intersection pattern of the x and y classes on a ruled surface. Hence This shows that h, = 2 and p g = 0 so that the Hodge diamond of Q is 0 2 0 If let l, r : P P P denote the natural projections l(s, t) = s, r(s, t) = r then we deduce that the canonical line bundle of Q satisfies K Q = l K P r K P, c (K Q ) = l c (K P ) + r c (K P ) = 2(x + y ). From (.2) we deduce that Q is spinnable. Consider again the projective plane P 2 and two points p 0, p P 3. Fix a projective line L P 2 that does not contain p 0 and p. For any s L we denote by l x (s) the projective line [p 0 s] and by l y (s) the projective line [p s]. If q denotes the intersection of the line L 0 = [p 0 p ] with L we deduce that L 0 = l x (q) = l y (q). Note that for any p P 2 \ {p 0, p } there exists a unique pair (s, t) = (s(p), t(p)) L L such that p is the intersection of the line l x (t) with the line l y (s). We obtain in this fashion a (rational) map π : P 2 \ {p 0, p } L L, p ( s(p), t(p) ). p 0 L0 l x(t) p p l (s) y q s t L FIGURE 3. Blowing up the projective plane.
0 LIVIU I. NICOLAESCU Let X be the blowup P 2 at the points points p 0, p. We denote by l x(s) the strict transform of l x (s), by l y(s) the strict transform of of l y (s), and by L 0 the strict transform of the line L 0 = [p 0 p ]. Then [l x(s)] 2 = [l y(s)] 2 = [l x (s)] 2 = 0, [L 0] 2 = [L 0 ] 2 2 =. his shows that L 0 is an exceptional curve. After the blowups the points p 0 and p are replaced by exceptional divisors E 0, E. Observe that the points in E 0 can be identified with the lines l x so that we have ( a natural biholomorphic map π 0 : E 0 L such that the point q 0 E corresponds to the line l x π0 (q 0 ) ) we define in a similar way a biholomorphic map π : E L We can now define as follows. π : X L L On the region X \ (E 0 E ) = P 2 \ {p 0, p } we set π = π. If q 0 E 0 then we set π(q 0 ) = (π 0 (x), q), q = L L 0 If q E then we set π(q ) = (q, π (q )). L 0 The resulting map π : P 2 p 0,p L L is holomorphic, the fiber over (q, q) is the exceptional curve, and the induced map P2 p 0,p \ L 0 (L L) \ {(q, q)} is a biholomorphism. This proves that the ruled surface L L is obtained by blowing down the exceptional curve L 0. Thus the quadric Q can be obtained from the projective plane P2 by performing two blowups followed by a blowdown. Thus any quadric is a rational surface. Example 3.7 (Hirzebruch surfaces). For every n 0 we consider holomorphic line bundle L n P of degree n, i.e., c (P ), [P ] = n. We form the rank 2 complex vector bundle E n = C L n, and denote by F n its projectivization. π In other words, the bundle F n P is obtained by projectivizing the fibers of E n, so that the fiber F n (p) of F n over p P is the projectivization of the vector space C L n (p), where L n (p) is the fiber of L n over p. Equivalently, F n (p) can be viewed as the one point compactification of the line L n (p). The zero section of L n determines a section of F n that associates to each p P the the line C 0 C L n (p). This determines a rational curve S 0 in F n with selfintersection number [S 0 ] 2 = n because the self intersection number of the zero section of L n equals the degree of this line bundle. We denote by x 0 the homology class determined by this curve and f the homology class determined by a fiber of F n. They satisfy the intersection equalities x 2 0 = n, f 2 = 0, x 0 f =.
A TASTE OF TWO-DIMENSIONAL COMPLEX ALGEBRAIC GEOMETRY Using these equalities and the Leray-Hirsch theorem we deduce that x 0 and f span H 2 (F n, Z). The Hodge diamond of F n is 0 2 0 The section at is the section of the bundle F n P that associates to each p P b the line L n (p) C L n (p). It defines a rational curve S which determines a homology class x. We can write x = ax 0 + bf, a, b Z. Using the equalities we deduce x x 0 = 0, x f = x = x 0 nf, x 2 = n. Denote by y H 2 (F n, Z) the homology class whose Poincaré dual is c (K Fn ). Again we can write y = ax 0 + bf, a, b Z. Using the genus formula we deduce 0 = g(zero section) = + 2 x 0 (x 0 + y) = + (n + na + b), 2 Hence a = 2, b = n 2 so that 0 = g(fiber) = + 2 f (f + y) = + 2 a. y = 2x 0 + (n 2)f, c (K Fn ) = 2x 0 + (n 2)f. (3.2) This shows that F n is spinnable if n is even and non-spinnable if n is odd. Let us point out that F is biholomorphic to the blowup of P 2 at a point, while F 0 can be identified with the ruled surface P P. Using the genus formula coupled with (3.2) one can prove that if m, n > and m n, then the Hirzebruch surface F n does not contain a smooth complex rational curve with self-intersection number m. This shows that F n and F m are not biholomorphic if n m. On the other hand F n is diffeomorphic to F m if n m mod 2. Moreover, any Hirzebruch surface is rational. Example 3.8 (Hypersurfaces in P 3 ). The zero locus X d of of generic degree d homogeneous polynomial P d C[z 0, z, z 2, z 3 ] is a smooth hypersurface in P 3. By Lefschetz theorem we deduce that X d is simply connected. Using a higher dimensional version of the adjunction formula (2.2) coupled with Noether s formula and basic properties of the Chern classes we deduce that the Hodge diamond of X d is (see [3]) ( d ) 3 h(d) ( d ) 3, where ( ) d = 3 (d )(d 2)(d 3), h(d) = d(d 2 4d + 6) 2 2 6 ( d 3 ).
2 LIVIU I. NICOLAESCU The surface X d is spinnable if and only if d is even. The signature of the intersection form is negative, τ(x d ) = d(d 2)(d + 2). 3 The hypersufaces of degree d 5 are said to have general type. Using M. Freedman s classification of simply connected topological 4-manifolds we deduce that the above data completely determine the homeomorphism type of X d. In fact, they are all diffeomorphic. Example 3.9 (K3 surfaces). These are simply connected surfaces X such that the canonical line bundle K X is holomorphically trivial. For example smooth degree 4 hypersurface in P 3 is a K3- surface. In this case p g (X) = because the space of holomorphic sections of K X is one dimensional. Since b (X) = 0 we deduce that h 0, (X) = 0. Noether s formula then implies that Hence + p g (X) = χ 0 (X) = 2 χ(x). 24 = 2 + b 2 (X) = 22 = b 2 (X) = 2p g + h, (X) = h, (X) = 20. Hence, the Hodge diamond of a K3-surface is 20 Since c (K X ) = 0 we deduce that a K3 surface is spinnable. This implies that all K3-surfaces are homeomorphic. It is much harder to prove that in fact they are all diffeomorphic. Example 3.0 (Elliptic surfaces). An algebraic surface X is called elliptic if there exists a holomorphic map π : X P such that the generic fiber of π is a smooth algebraic curve of genus. The theory of elliptic surfaces is very rich so we limit ourselves to a special class of elliptic surfaces. Fix two degree 3 homogeneous polynomials A 0, A C[z 0, z, z 2 ] and consider the hypersurface E(n) P P 2 described by the equation ([t 0, t ], [z 0, z, z 2 ]) P P 2, t n 0 A 0 (z 0, z, z 2 ) = t n A (z 0, z, z 2 ) = 0. For generic choices of A 0 and A this is a smooth hypersurface in P P 2. We set A t0,t = t n 0 A 0 (z 0, z, z 2 ) = t n A (z 0, z, z 2 ) The natural projection P P 2 P induces a holomorphic map π : E(n) P whose fiber over [t 0, t ] is the elliptic curve {A t0,t (z 0, z, z 2 ) = 0} P 2.
The Hodge diamond of this surface is A TASTE OF TWO-DIMENSIONAL COMPLEX ALGEBRAIC GEOMETRY 3 (n ) 0n (n ) The biholomorphism type of E(n) depends on the choice of polynomials A 0 and A. The Euler characteristic of E(n) is 2n, and E(n) is spinnable if and only if n is even. The elliptic surface E(2) is a K3 surface, but not all K3-surfaces are elliptic. Not all the fibers of the map π : E(n) P are smooth. For generic A 0 and A there are 2n singular fibers. 4. THE KODAIRA CLASSIFICATION The Kodaira classification has as starting point the plurigenera of P n (X) of an algebraic surface. The sequence ( P n (X) ) displays one of the following types of behavior. n ( ) The sequence of plurigenera is identically zero, P n (X) = 0, n. In this case we say that the Kodaira dimension of X is and we write this kod(x) =. (0) The sequence of plurigenera is bounded, but not identically zero. In this case, we say that the Kodaira dimension of X is 0, kod(x) = 0. () The sequence of plurigenera grows linearly, i.e., there exists a constant C > 0 such that C n P n(x) Cn, n. In this case, we say that the Kodaira dimension of X is, kod(x) =. (2) The sequence of plurigenera grows quadratically, i.e., there exists a constant C > 0 such that C n2 P n (X) Cn 2, n. In this case, we say that the Kodaira dimension of X is 2, kod(x) = 2. Example 4.. The rational surfaces have Kodaira dimension, the K3 surfaces have Kodaira dimension 0 the elliptic surfaces E(n), n 3, have Kodaira dimension and the hypersurfaces X d, d 5 have Kodaira dimension 2. REFERENCES [] D. Eisenbud, J. Harris: The Geometry of Schemes, Grad. Texts in Math., vol. 97, Springer Verlag, 2000. [2] P. Griffiths, J. Harris: Principles of Algebraic Geometry, John Wiley, 978. [3] L.I. Nicolaescu: Hodge numbers of projective hypersurfaces, Notre Dame, 2005. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF NOTRE DAME, NOTRE DAME, IN 46556-468. E-mail address: nicolaescu.@nd.edu URL: http://www.nd.edu/ lnicolae/