Page 396 -

§8. Line Bundles and Divisors: Picard and N´ eron–Severi Groups 371

In dimension one elliptic curves arise as cubic curves in the plane or as complete
intersections of two quadrics in 3-space. Elliptic curves also arise either as quartic
4
curves in P 2 (1, 1, 2) and sextic curves in P 2 (1, 2, 3). We can take for example w +
2
6
3
2
4
x + y = 0and w + x + y = 0 respectively.
(7.4) General Toric Varieties. Toric varieties are varieties with an action of a torus
such that there is a dense orbit. They include weighted projective spaces, hence also
projective spaces. The affine pieces of a toric variety are defined by monomial equa-
tions, and the affine pieces are organized by a combinatorial configuration relating
the toric actions on the affine open sets. For a general reference we recommend the
book of Fulton [1993]. We give a guide to some of the sections in Fulton.
From the combinatorial description it is possible to know when a toric variety is
proper and nonsingular. For this, see p. 39 and p. 29 respectively of Fulton [1933]
and in section 2.6 the resolution of singularities of a toric variety can be prescribed
from the combinatorial data needed to prescribe a toric structure.

(7.5) Complete Intersections in Toric Varieties. In Fulton, chapter 3, divisors and
line bundles on a toric variety are studied. Of special importance are the T -invariant
divisors on a tori variety X where T is the torus acting on X. In section 4.3, the
canonical bundle is described using the T -invariant divisors. Then the complete in-
tersections with trivial canonical bundle can be determined. Hence there is a combi-
natorial description of which complete intersections are Calabi–Yau manifolds.

(7.6) Remark. For applications to string theory there is the notion of the mirror
Calabi–Yau manifold, and it can be very concretely determined in cases where the
Calabi–Yau is a complete intersection in a toric variety. A reference for this is Voisin
[1996, chapter 4].

§8. Line Bundles and Divisors: Picard and N´ eron–Severi Groups

In (5.2) we introduced the first Chern class of a line bundle in the setting of ho-
motopy classes [X, P ∞ (C)] of mappings X → P ∞ (C). Basic to this is the double
interpretation of P ∞ (C) leading to the bijection
2
c 1 : {isomorphism classes of complex line bundles/X}→ H (X, Z) = [X, P ∞ (C)]
carrying the tensor product to the sum in the cohomology groups

c 1 (L ⊗ L ) = c 1 (L ) + c 1 (L ).
(8.1) Analytic/Algebraic Line Bundles and Divisors. Let X/k be a proper scheme.
Let M X denote the sheaf of total rings of fractions of O X on the scheme X. The mul-
tiplicative structure of sheaves of rings leads to the following two diagrams relating
line bundles to closed subschemes of codimension one.