3.6 Chows Lemma 171
graph 1,(U) is a closed subscheme of U x P, since P is separated by Pro
p0sition 3.4.(i). The closure scheme of 1,(U) in X x P (cf. 1.27.e), which is
denoted by X, is a closed subscheme of X x P. Composing the closed
immersion i: X+X x
4.2 Fundamental Theorems 175
any open subset U of X. It is easy to see tuat V is a11uX111odu
that for any g,
Hommm y) g H Homm'xm, 1x).
xeX
To show this, let pg: V(U)a Ix be the projection, whenever x e U. The
direct limit of P111; where x E U is denoted
170 3 Projective Schemes
A and 3. Put C0 = R, C1 = A1R 31,., Cd = Ad R Bd, and dene
C to be @1910 C j, which is also a graded R-algebra. Then if a0, ., a, and
B0, ., [is are generators of the R-modules A and B, respectively, a0 69 B0,
., 0:, ,6, become th
196 5 Regular Forms and Rational Forms on a Variety
Such an open cover cfw_Uri/t e A is said to be a coordinate cover of V. In
this case, leu, g 02, for all [t e A; thus (2, is a locally free (Dy-module of
rank n.
Denition. If V is a nonsingular variety,
172 3 Projective Schemes
Corollary. Let X be a complete variety. Then there exist a projective variety
X and a proper birational morphism X> X.
Remarks (1). Any complete algebraic curve is projective (see 6.4.b).
(2) Any complete nonsingular surface is pr
5.1 Modules of Regular Forms and Canonical Derivations 189
(iii)d(b)d(c)=([email protected])=\iwt Cl)
=bc1bccb+lbc
=d(bc)+2bc1 b(d(c)+c1)A c(d(b)+b 1)
= ~(be) baa) dub). E!
b. JB/Jf, is a B A B/JB-module. But since 869 A B/JB ; B, JB/J can be
regarded as a B-module.
Chapter 4
Cohomology of Sheaves
4.1 Injective Sheaves
a. Throughout this section, let (X, (9X) be a ringed space, and let 3;, g, 9?
denote (OX-modules.
Denition. An (x-module V is said to be injective if whenever any sequence
of (Dx-modules of the form
Oa
212 6 Theory of Curves
Whenever i(K D) = 0, i(D) = 1 g + deg D. It is rather difficult to
know l(D) when both l(D) and i(D) = l(K D) are positive. Such an effective
divisor D is called a special divisor, because i(D) is said to be the index of
speciality
5.4 Birational Invariance of Genera 199
Denition. In this case, dimk TM( V) is denoted by PM( V), and is called the
M -genus Of V.
e. If M = (0, ., 0, l, 0, ., 0) (the i-th component is 1), then TM(V) is
denoted by 71V) and q,-(V) is used to denote PM(V)
5.7 Generalized Adjunction Formula and Conductors 205
On the other hand, dening A1, jby w, 235:1A' jwj, one denes a matrix
A1: [(1)1Ai' By denition, one has clearly 1J1, cfw_4114:qu 1,];
hence J =ju, 1",]. Since 21:1 (p', w: 2721 q) w, it follows that
[(P
4.5 Finiteness Theorem 181
HIX, .
u
('D
nee
0
for all j, we have _ ,(X 07): 07,) = 0. Thus the SP
A(X) = (9(X)> A(X) = (9X(X)> H1(X, 9")
is exact. Hence, (DX is surjective, i.e., A(X) = 2;:1 A(X)fj.
(b)=>(a) Let I: X a Spec A(X) be the canonical morphism
5.6 Ramication Formula 203
PROOF. Take a nonzero rational n-form w on W. Then wlwg = (p, dw, and
f*wlU1 =f*(wa)f*(dwa) =f*(pa) . wall (121. By denitions K(W) ~ le(CU) :
divcfw_<p. and KM ~ diV(f*w) = divcfw_f*(<p.) ' mi =f* divcfw_w. + Rf ~
f*K(W) + Rf- E
3.5 Some Properties of Projective Schemes 169
generated graded A(U,1)-algebra, then f is said to be a locally projective
morphism.
By Proposition 3.5, a locally projective morphism is a closed morphism.
Furthermore, by Proposition 3.4.(iv), the property o
5.3 Sheaves of Regular Forms 195
Theorem 2.12. Conversely, for a point p 6 Reg( V), one has x1, x, such
that (x1, ., x,)(9y,p = mp, where r = dim (9,51,. One can choose an affine
open subset W such that p e W andf1,.,f, e A(W) such that x1 =f1/1,
x, =f,/1
4.3 Flabby Sheaves 177
PROOF. We refer the reader to [H, pp. 204205]. [3
To explain the meaning of this theorem, we sh0w some of its easy
consequences.
Corollary
) .
') I V is injective, then H"(U, V) = 0f0r all q > 0.
PROOF. (i) By denition, H(U, 37) = F
4.8 RiemanniRoch Theorem (Weak Form) on a Curve 187
I D ~ A, then deg D = deg A.
deg D 2 deg (9(D) for any divisor D on C.
PROOF. (i) This follows from 0(D) ; 0(A).
(ii) Since deg (0(D) = xc XC(O(D) 1), the assertion follows from the last
theorem. [3
Coro
4.6 Lerays Spectral Sequence 183
We have a complex of (OX-modules cfw_-af*(y1_1)>f*(yq)a - - - . The
q-th cohomology .sheaf Ker( f*(5q)/Im( mar-1) is denoted by qu*(3'7),
wheref*(6") is dened byf*(6")u = 63-1w), for any q.
The next result is a direct cons
222 6 Theory of Curves
(iv) 1'f(p1, ., (p, are kiinearly independent, then W(p1,., (p,) it 0.
(v) For any 1/; e R,
W(WPI, " WW) 2 WWW/91a "'9 (pl)
PROOF. (i) This follows immediately from the denition.
(ii) By (i), one can assume (p1 = 0.
(iii) This is le
Chapter 5
Regular Forms and Rational Forms
on a Variety
5.1 Modules of Regular Forms and
Canonical Derivations
a. Let A be a ring and let B be an A- algebra Dene #3: B 69 A B> B by
[B(Zn:i biOC-):Zr= ibicl' andsetJE:Ker/13.9ega.d.ugBABasa
B-module by b1 b
200 5 Regular Forms and Rational Forms on a Variety
Whenever Pm( V) > 0, one has a rational map (I): V+ P(Q")") =
P(mK(V) associated with the invertible sheaf (Q")", which is denoted by
(Dy, m .
Denition. (Dy, m is said to be the m-th canonical rational m
204 5 Regular Forms and Rational Forms on a Variety
5 7 Generalized Adjunction , For orum Ula,
and Conductors
a. Any variety V of dimension n is birationally equivalent to a hypersurface
of P2, if k has characteristic zero. Thus, it is desirable to establ
5.2 Lemmas 191
Y") for 1 g i s n, one has
PROOF. Letting d : Ba QB be the canonical derivation, dene sd:
318 3 193 by sd(b/s) = d(b)/s bd(s)/s2 for any b/s 6 $43. Clearly,
sd is well-dened and is an A-derivation of 813. We shall prove that
(Slg/A, 5d) is
178 4 Cohomology of Sheaves
be an exact sequence of (RXmodules. If 3 isabby, then the sequence
X 19X
0 $(X) $(X) $(X) 0
is exact.
PROOF. By Lemma 1.19, it sutces to Show that [x is surjective. Given any
section s E $(X), dene the set M to be cfw_(
194 5 Regular Forms and Rational Forms on a Variety
5.3 Sheaves of Regular Forms
a. In 5.l, we discussed the A-module QA/R associated with any R-algebra A.
We dene the sheaf (of germs) of regular forms on Spec A over R to be
(QA,R)~. For any a e A, one ha
6.2 Fujita's Invariant A(C, D) 211
6,2 anmq
J." u
Invariant A(C, D)
a. Denition. For a divisor D with deg D > 0 on a complete nonsingular
curve C, dene A(C, D) to be 1 + deg D l(D).
The invariant A(C, D)
A
cUlllpllele Val lCly V] auu
deciency by T. Fujita
192 5 Regular Forms and Rational Forms on a Variety
there exist x1, ., x,l e R osuch that (i)L :k(x1, x") is a purely trans-
cendental extension, i.e., x1, ., x, are algebraically independent, and (ii)
R/L is a nite separable extension. There exists 6,- E
198 5 Regular Forms and Rational Forms on a Variety
22. Hence, by Theorem 5.2.(ii), F( V, (2%,) = I(dom(f), (2%,). COmposing this
with (p*, one obtains the linear map F(W, (2%V)+ F(V, (2%,), denoted byf*.
If a strictly rational map f is dominating, one ha
180 4 Cohomology of Sheaves
PROOF. Let (37 #, n, X) denote the etale space associated with 37 (cf. 1.10.d).
Then 3;" = L1, a", ; Hg, 37,. Hence, F(X, 37!) g (95:, 37,], by de-
nition. [3
b. Let X be a scheme.
Theorem 4.4. If X is separated and quasicompac