Table Of ContentErrata for
“Higher-Dimensional Algebraic Geometry”
by Olivier Debarre
I thank A. Alzati, A. Chambert-Loir, S. Druel, A. H¨oring, M. Lahyane and F.
Loeser for suggesting some of the corrections below.
page 22
• line -10, for
“the left-hand side of this equality is nonnegative”
read
“the left-hand side of this equality is nonpositive”
page 27
• line -11, for
“nine points”
read
“nine general points”
page 31
• line 10, as of
“Recall also...”
replace the end of the proof of Proposition 1.43 by
“ Each component of E has codimension 1 (1.40) hence, by shrinking Y
again, we may assume that Y and E are smooth and irreducible. Set U =
0
X Sing(π(E)),sothattheclosureinU oftheimageofE∩π−1(U )issmooth,
0 0
of codimension at least 2. Let ε : X → U its blow-up; by the universal
1 1 0
property of blow-ups ([H1], II, prop. 7.14), there exist an open subset V of
1
π−1(U )whosecomplementinY hascodimensionatleast2andafactorization
0
π| :V −π→1 X −ε→1 U ⊂X
V1 1 1 0
and π (E∩V ) is contained in the support of the exceptional divisor of ε . If
1 1 1
thecodimensionofπ (E∩V )inX isatleast2,thedivisorE∩V iscontained
1 1 1 1
in the exceptional locus of π and, upon replacing V by the complement V
1 1 2
of a closed subset of codimension at least 2 and X by an open subset U , we
1 1
may repeat the construction. After i steps, we get a factorization
π :V −π→i X −ε→i U ⊂X −εi→−1 ···−ε→2 U ⊂X −ε→1 U ⊂X
i i i−1 i−1 1 1 0
as long as the codimension of π (E∩V ) in X is at least 2, where V is
i−1 i−1 i−1 i
the complementinY ofaclosedsubsetofcodimensionatleast2. LetE ⊂X
i i
be the exceptional divisor of ε . We have
i
K =ε∗K +c E
Xi i Ui−1 i i
=(ε ◦···◦ε )∗K +c E +c E +···+c E
1 i X i i i−1 i,i−1 1 i,1
where E is the inverse image of E in X and
i,j j i
c =codim (π (E∩V ))−1>0
i Xi−1 i−1 i−1
([H1],II,Ex.8.5). Sinceπ isbirational,π∗O (K )isasubsheafofO (K ).
i i Xi Xi Vi Vi
Moreover,sinceπ (E∩V )iscontainedinthesupportofE ,thedivisorπ∗E −
i i i i i
E| is effective.
Vi
It follows that O (π∗K +(c +···+c )E)| is a subsheaf of O (K )=
Y X i 1 Vi Vi Vi
O (K )| . SinceY isnormalandthecomplementofV inY hascodimension
Y Y Vi i
at least 2, O (π∗K +(c +···+c )E) is also a subsheaf of O (K ). Since
Y X i 1 Y Y
there are no infinite ascending sequences of subsheaves of a coherent sheaf on
a Noetherian scheme, the process must terminate at some point: π (E∩V )
i i
is a divisor in X for some i, hence E∩V is not contained in the exceptional
i i
locusofπ (by1.40again). Themorphismπ theninducesabirationalisomor-
i i
phism between E∩V and E , and the latter is ruled: more precisely, through
i i
every point of E there is a rational curve contracted by ε . This proves the
i i
proposition.”
page 45
• line -3, for
“where g(C)=1−χ(C,O ), and”
C
read
“where, if d is the degree of f onto its image, f C is the 1-cycle df(C) (§1.1)
∗
and g(C)=1−χ(C,O ). We obtain from there”
C
page 47
after formula (2.4), add
“where again, if d is the degree of f onto its image, f C is the 1-cycle df(C).”
∗
page 48
• line 14, in formula (2.5), for
“O (1)n−1”
‘
read
“O (1)N−1”
‘
page 49
• line 9, for
“α(λ ,...,λ )”
2 n
read
“α(λ ,...,λ )”
2 N
page 50
• line 9, for
X
xd =0 for all j
i
i∈Ij
read
X
x 6=0 and xd =0 for all j
Ij i
i∈Ij
page 51
• line -1, for
“and the characteristic of k is ≥d.”
read
“and the characteristic of k is 0 or ≥d.”
page 53
• line -9, read
h0(‘,N (−1))=N−1−min(r,d) h0(‘,N )=2N−2−min(2r,d+1)
‘/Xd ‘/Xd
N N
page 54
In Exercise 3.c), add a footnote:
“It was shown by Green (Some Picard theorems for holomorphic maps to al-
gebraic varieties, Amer. J. Math. 97 (1975), 43–75) that over the complex
numbers, for d>n2−1, the image of any holomorphic map Cm →Xd is con-
n
tained in a linear subspace contained in Xd of the type described page 64. In
n
particular, anyrationalorellipticcurveinXd, ormoregenerally, anyimagein
n
Xd ofaprojectivespaceoracomplextorus, arecontainedinalinearsubspace
n
of this form.”
page 56
• line -9, for
“1-cycle Pr (degf| )f(C ).”
i=1 Ci i
read
“1-cyclePr d f(C ),whered isthedegreeoff| ontoitsimage. Notethat
i=1 i i i Ci
for any Cartier divisor D on X, one has D·f C =deg(f∗D).”
∗
page 57
• line 5, delete
“connected”
• line 8, delete
“We may assume that C is irrational”
• After the illustration, read
“ Ifevisdefinedateverypointof{c}×T,therigiditylemma1.15(a)implies
that there exist a neighborhood V of c in C and a factorization
ev| :V ×T −p→1 V −g→X
V×T
Themorphismg mustbeequaltof| . Itfollowsthatevandf◦p coincideon
V 1
V ×T,henceonC×T. ButthismeansthattheimageofT inMor(C,X;f| )
{c}
is just the point [f], and this is absurd.
Hence there exists a point...”
page 59
• After the illustration, for
“ The rational cycle f(C) bends and breaks.
The surface S might not be smooth. On the other hand, we know that
no component of a fiber of π is contracted by e (because it would then be
contracted by F).
Since T is a smooth curve and S is integral, π is flat ([H1], III, prop. 9.7);
hence each fiber C is a 1-dimensional projective scheme without embedded
component, whose genus is constant hence equal to 0 ([H1], III, cor. 9.10). In
particular, any component C of C is smooth rational, because O is a
1 red C1
quotient of O ; hence H1(C ,O ) is a quotient of H1(C,O ), and therefore
C 1 C1 C
vanishes. In particular, if C is integral, it is a smooth rational curve.
Assume all fibers of π are integral...”
read
“ The rational cycle f C bends and breaks.
∗
Alternatively,theconstructioncanbemadeasfollows. Theindeterminacies
oftherationalmapP1×T 99KX×T inducedbyF canberesolvedbyblowing
up points to get a morphism
F0 :S0 →P1×T 99KX ×T
whose Stein factorization is
F0 :S0 →S −F→X×T
Inotherwords,thesurfaceSisobtainedfromS0bycontractingthecomponents
of fibers of S0 → T that are contracted by F0. In particular, the (reduced)
components of the fibers of π : S → T are smooth rational curves, which are
not contracted by e:S →X (because they would then be contracted by F).
Assume that all fibers of π are integral...”
page 60
• line -7, for
“if D is a divisor associated with the line bundle O (1)”
Y
read
“if D is a divisor associated with the line bundle O (1)”
X
• line -5, for
−K =(r+1)D+π∗(−K −D −···−D )
X Y 1 r
read
−K =rD+π∗(−K −D −···−D )
X Y 1 r
page 61
• line 2, for
“It follows that −K is ample”
Y
read
“It follows that −K is ample”
X
page 64
• line 7, for
2H ·C
H ·Γ≤
Card(B)
read
2H ·f C
H ·Γ≤ ∗
Card(B)
page 65
• line -5, for
“N1(X) ”
R
read
“N1(S) ”
R
page 67
• line 11, for
H ·C
H ·Γ≤2dim(X)
−K ·C
X
read
H ·f C
H ·Γ≤2dim(X) ∗
−K ·f C
X ∗
• lines -7 and -5, for
−pmK ·C
X
read
−pmK ·f C
X ∗
• line -3, for
“a rational curve Γ ”
m
read
“a rational curve Γ on X”
m
• line -1, for
2H ·C 2pm
H ·Γ ≤ m = (H ·C)
m b b
m m
read
2H ·(f ) C 2pm
H ·Γ ≤ m ∗ m = H ·f C
m b b ∗
m m
page 68
• line 1, for
“−dim(X)/(−K ·C).”
X
read
“−dim(X)/(−K ·f C).”
X ∗
• line 3, for
2dim(X)
H ·Γ ≤ (H ·C)
m −K ·C
X
read
2dim(X)
H ·Γ ≤ H ·f C
m −K ·f C ∗
X ∗
• line 13, for
“of degree at most 2dim(X) H·C ”
−KX·C
read
“of degree at most 2dim(X) H·f∗C ”
−KX·f∗C
page 71
• line 1, delete
“connected”
page 77
• line 2, for
“any such morphism factors through α .”
X
read
“any such morphism factors uniquely through α .”
X
page 79
• line -4, for
“π−1(π(x))”
read
“π−1(π(f(c)))”
page 81
• line -12, delete
“connected”
page 83
• line -10, delete
“connected”
page 89
• lines 1 and 16, for
“for some d”
read
“for some positive d”
page 94
• line -3, for
“[Ha], III, prop. 4.6”
read
“[Ha], III, prop. 10.6”
page 98
• line 11, for
“the injection ι:{0}×M ,→X”
read
“the injection ι:{0}×M ,→P1×M”
• line 13, for
π (ι)◦π (ev)=0
1 1
read
π (ev)◦π (ι)=0
1 1
• line -10, for
“Xfree”
read
“Xfree”
x
page 114
• line -15, for
“an irreducible constructible subset”
read
“a constructible subset”
• from line -12 on, read
“The right-hand side is contained in the left-hand side. Let us prove the other
inclusion. WemayassumebyLemma5.1thatV isirreducible. By[H1],II,ex.
3.18(b),V containsasubsetU thatisdenseopeninV. LetW beacomponent
of V U. By a theorem of Chevalley ([H1], II, ex. 3.22(e)), a general fiber of
W →π(V) is either empty or everywhere of dimension
dim(W)−dim(π(V))
whichislessthanthedimensiondim(V)−dim(π(V))ofthefiberofV →π(V)
at the same point. It follows that for y general in π(V), the closed subset
(V U)∩π−1(y) of V ∩π−1(y) is nowhere dense, hence that U ∩π−1(y) is
dense in V ∩π−1(y).”
page 116
• line 8, for
“separable closure of K and let”
read
“separable closure of K, and let”
page 117
• line -5, for
“of length m”
read
“of length 1”
page 118
• line 5, for
“F(Wi,j)”
m
read
“F(Wi,j)”
m
• line -12 on, read
“ Call Vi stable if F(W˜i,j) = Vi for all j, unstable otherwise. Note the
m m m
following:
• if all components of V (x) are stable, V (x)=V (x);
m m+1 m
• if Vi is unstable, it is strictly contained in an irreducible component of
m
V (x);
m+1
• if Vi is stable and an irreducible component of V (x), it is stable as a
m m+1
component of V (x).
m+1
LetV0 =F(W˜i,j )beanunstablecomponentofV (x). Ifthecorrespond-
m m−1 m
ing Vi is a stable component of V (x), it is equal to V0 by definition,
m−1 m−1 m
which contradicts the third item. Hence Vi is unstable, and is strictly con-
m−1
tained in V0 by the second item.
m
It follows that if V (x) has an unstable component V0, then V (x) also
m m m−1
has an unstable component of smaller dimension. In particular, the dimension
ofV0 isatleastm. Bytheseconditem,thisimpliesdim(V (x))>m,which
m m+1
is impossible since m ≥ δ(x). Therefore, V (x) has only stable components,
m
and V (x)=V (x) by the first item. This proves the first step.”
m+1 m
page 119
• line 5, for
[(cid:0) (cid:1)
V = {x}×V (x)
n
x∈X
read
[(cid:0) (cid:1)
V = V (x)×{x}
n
x∈X
page 123
• line -2, for
“hence factors as X0∗ →X∗ →ρ Y,”
read
“hence factors as X0∗ →X∗ →ρ Y∗,”
page 133
• line 7, delete
“Examples in Section 5.11 show that k may grow as logn.”
page 135
• line 11, for
“87 other families with Picard number >1 ([MM]).”
read
“88 other families with Picard number > 1 (Mori and Mukai recently noticed
that the family of blow-ups of P1×P1×P1 along a curve of tridegree (1,1,3)
is missing from the list in [MM]).”
page 136
• line -18, for
“the curve C =f−1(C).”
1
read
“the curve C =π−1(C).”
1
pages 136 and 137
• delete footnote 15 and replace footnote 16 with
“ In this direction, complex Q-Fano threefolds with canonical singularities
(see Definition 7.13) and Picard number 1 are known to form a limited family
([Ka4] for the case of terminal singularities; the canonical case is in Koll´ar,
J., Miyaoka, Y., Mori, S. and Takagi, H., Boundedness of canonical Q-Fano
3-folds, Proc. Japaan Acad. Ser. A Math. Sci. 76 (2000), 73–77). The point
isagaintobound(−K )n andanintegerj suchthatjK isaCartierdivisor.
X X
Work of Ran and Clemens ([RC]) shows that Q-Fano n-folds with canonical
singularitiesandPicardnumber1formalimitedfamilyifonemoreoverbounds
the smallest number m such that −mK is very ample (in the smooth case,
X
one can take m=n(n+1)(n+3) as in the proof of Theorem 5.19).
More generally, it is conjectured that given ε > 0, Fano varieties X such
that, for any desingularization f : X˜ → X, one has K = f∗K +Pa E
X˜ X i i
with a > −1+ε, where the E are the exceptional divisors of f, still form a
i i
limited family. Our surfaces fall in this class for g =0 and d<2/ε.”
page 138
• line 2, read,
“or by noting that L and H are nef, and L+H is ample, since its associated
line bundle is the tautological line bundle associated with the description of X
as P(Or (1)⊕Or (a+1)); it is therefore ample).”
Ps Ps
• replace lines 11 through 15 with
Setting n=dim(X)=r+s, we get, since L and H are nef, for a=s
(−K )n ≥((r+1)L)n =(r+1)nasLr·Hs =(r+1)nas
X
• line -1, for
H0(Ps,O (s−r))r
Ps
read,
H0(Ps,O (s))r
Ps
page 141
• line 18, for
“yields the (5.3).”
read
“yields (5.3).”
page 146
• line 10, for
“ the Riemann-Roch theorem yields
h0(X,O (mC))=m+χ(X,O )≥2
X X
and the linear system |mC| has no base-point (the only possible fixed curve
is C, but h0(X,O ((m−1)C)) < h0(X,O (mC)), and there are no isolated
X X
base-points since C2 =0).”
read
“ the Riemann-Roch theorem yields
h0(X,O (mC))≥m+χ(X,O )≥2
X X
