PROBLEMS: ALGEBRAIC GEOMETRY II
The following questions and problems came up naturally during class. They are not
necessarily difficult but they should help understand the material from class.
(1) For a map f : F → G of sheaves show that U 7→ ker(f (U )) is a sheaf (and not
just a presheaf).
(2) Define what it means to be a subsheaf.
(3) If i : p → X is the inclusion of a point show that i∗ Op is the skyscraper sheaf
supported at p ∈ X (here Op is the structure sheaf of p).
(4) If p ∈ X is a point then show that there exists a short exact sequence
i
π
0 → Ip −
→ OX −
→ Op → 0
(1)
where Ip is the ideal sheaf of p.
(5) Show that if p, q ∈ P1 are distinct points the natural map
(π1 ,π2 )
OP1 −−−−→ Op ⊕ Oq
is surjective, where π1 , π2 are the natural surjective maps as in (1).
(6) If f : X → Y and G is a sheaf on Y describe the natural adjunction map
G → f∗ f −1 G.
(7)
(8)
(9)
(10)
Show that the global section functor Γ is left exact.
What map of rings defines the inclusion of the point (a1 , . . . , an ) ֒→ An .
Describe the glueing maps for the standard cover by affines of Pn .
Show that a homogeneous polynomial f (x0 , x1 ) of degree d carves out d points
in P1 (counted with multiplicity).
(11) Show that C× = C \ {0} is an affine scheme. Show that C2 \ {(0, 0)} is not an
affine scheme.
(12) Find an example of OX -modules F and G so that for open U ⊂ X the assignment
U 7→ F(U ) ⊗OX (U ) G(U )
defines a presheaf but not a sheaf.
(13) For a subscheme i : Y ֒→ X show that i∗ i∗ OY ∼
= OY .
(14) If i : Y ֒→ X = Spec(A) is a closed subscheme corresponding to an ideal I ⊂ A
f for an A-module M .
describe i∗ OY as M
(15) For an open j : U ֒→ X define j! (F) as the sheafification of
(
F|V ifV ⊂ U
V 7→
0
ifV 6⊂ U
1
2
PROBLEMS: ALGEBRAIC GEOMETRY II
Show that there exists a short exact sequence
0 → j! j −1 F → F → i∗ i−1 F → 0
where i : Y = X \ U → X.
(16) Explain why j! (OU ) is not a quasi-coherent sheaf in general.
(17) Is j∗ OY coherent where j : Y = C2 \ {(0, 0)} ֒→ C2 ?
(18) Given an exact sequence
(2)
(19)
(20)
(21)
(22)
(23)
(24)
0 → F1 → F2 → F3 → 0
on an affine scheme X show that if F1 and F3 are quasi-coherent then so is F2 .
Hint: you may use that Γ applied to (2) is exact if X is affine.
Show that if U, V ⊂ X are open affine subschemes then U ∩ V is affine. Hint:
you may use that any closed subscheme Y ⊂ Spec(A) is affine.
Give an example of f : X → Y map of schemes and coherent sheaf F on X so
that f∗ F is not coherent (just quasi-coherent).
Show that a morphism of line bundles CX → L (where CX is the trivial line
bundle on X) is equivalent to a section s : X → L.
Show that a line bundle L on X is trivial if and only if there exists a nonvanishing
section s : X → L (i.e. section where s(x) 6= 0 for all x ∈ X).
If L is a line bundle show that there exists another line bundle L−1 such that
L ⊗ L−1 ∼
= CX .
Show that for an OX -module F we have
HomOX (OX , F) ∼
= Γ(F)
as vector spaces.
(25) If Ψ(V ) denotes the coherent sheaf associated to a vector bundle V show that
Ψ(V ⊗ W ) ∼
= Ψ(V ) ⊗OX Ψ(W ).
(26) Show that for a vector bundle V on Y and morphism f : X → Y we have
f ∗ (Ψ(V )) ∼
= Ψ(V ⊗Y X)
where V ⊗Y X is the pullback (as a vector bundle) of V from Y to X.
(27) Show that Γ(OP1 (−1)) = 0.
(28) Compute dim Γ(OPn (m)) for m ≥ 0.
(29) If X is proper and L a line bundle on X show that both Γ(L) 6= 0 and Γ(L−1 ) 6= 0
if and only if L is trivial. Hint: just need to use that Γ(OX ) ∼
= C.
(30) Describe the cokernel of the map xk : OPn → OPn (1) of sheaves on Pn .
(31) Check that if s ∈ Γ(L) then sn ∈ Γ(L⊗n ) where L is a line bundle.
(32) Consider the map
φ : P1 × · · · × P1 → Pr
PROBLEMS: ALGEBRAIC GEOMETRY II
3
given by ([x1 , y1 ], . . . , [xr , yr ]) 7→ [f0 , . . . , fr ] where
X Y 1−e e
fi =
xj j yj j
e1 +···+er =i j
with 0 ≤ ek ≤ 1. For example, if r = 2 then φ is given by
([x1 , y1 ], [x2 , y2 ]) 7→ [x1 x2 , x1 y2 + x2 y1 , y1 y2 ].
(33)
(34)
(35)
(36)
(37)
(a) Show that φ is well defined.
(b) Compute the number of points in φ−1 (p) for a general point p ∈ Pr .
Show that a line bundle L on X is generated by global section s1 , . . . , sn ∈ Γ(L)
if for any point x ∈ X there exists some si with si (x) 6= 0.
Consider a morphisms f : X → Y of schemes and a line bundle L on Y . If
sections {si } generate L show that {f ∗ si } generate f ∗ L on X.
On X = P1 write (x3 ) as a Weil divisor.
Show that D = {x0 = 0} ⊂ Pn defines the line bundle L(D) ∼
= OPn (1) on Pn .
If D ⊂ X is an effective Cartier divisor show that
i∗ OD ⊗OX OX (D) ∼
= i∗ (L|D )
(38)
(39)
(40)
(41)
(42)
where i : D ֒→ X is the inclusion, L is the line bundle OX (D) and LD is its
restriction to D.
If p, q ∈ P1 are distinct points compute Γ(OP1 (p − q)).
If H ⊂ Pn is a hyperplane compute dim Γ(OPn (mH)) for m ∈ Z.
Show that P GLn+1 (C) acts faithfully on Pn .
Show that Aut(An ) is given by the group of affine transformations on An . Hint:
use that Aut(Pn ) = P GLn (C) together with the fact that a birational map
φ : Pn → Pn induces an isomorphism on Pic(Pn ).
Consider a map B → A of rings. This allows us to define
f : A ⊗B A → A (a, a′ ) 7→ aa′ .
Check that d : A → ker(f )/ker(f )2 given by a 7→ a ⊗ 1 − 1 ⊗ a defines a
B-derivation (i.e. d(aa′ ) = ad(a′ ) + a′ d(a) and d(b) = 0 for b ∈ B).
(43) Show that det(F ∨ ) ∼
= det(F)−1 for any locally free sheaf F.
(44) Show that if
0 → F1 → F2 → F3 → 0
is a short exact sequence of locally free sheaves then det(F2 ) ∼
= det(F1 )⊗det(F3 ).
Pn
∼
n
(45) Since Pic(Pn ) ∼
Z
we
know
ω
O
(ℓ))
for
some
ℓ
∈
Z.
Find ℓ.
=
P =
(46) If X ⊂ Y are smooth varieties (X 6= Y ) show that
det(NX/Y ) ∼
= ωX ⊗ ωY−1 |X .
4
PROBLEMS: ALGEBRAIC GEOMETRY II
(47) Consider the commutative diagram
0
/A
/B
g
f
0
/C
/0
h
/ A′
/ B′
/ C′
/0
where the rows are exact, f is injective and g is an isomorphism. Show that h
is surjective and coker(f ) ∼
= ker(h).
∗
(48) Show from basic principles that ΩX×Y ∼
ΩX ⊗OX×Y πY∗ ΩY where πX and πY
= πX
are the projections form X × Y to X and Y .
(49) Suppose X ⊂ Y are smooth varieties and that X is the zero locus of a section
of a vector bundle V whose rank is the codimension of X in Y . Show that
ωX ∼
= ωY |X ⊗O (det V )|X .
X
(50) Find an example which illustrates why the homotopy category Kom(C) of an
abelian category C is not necessarily abelian.
(51) Given an example of an abelian category C and an object A ∈ C so that the
functor Hom(A, −) is not exact.
∗
∗
(52) Recall that Cartier divisors on an irreducible scheme X correspond to H 0 (KX
/OX
).
Use the short exact sequence
∗
∗
∗
∗
0 → OX
→ KX
→ KX
/OX
→0
to show that Cartier divisors on X modulo principal divisors are also parametrized
∗
by H 1 (X, OX
).
(53) Give an example of a scheme X which is not affine but contains f1 , . . . , fn ∈
Γ(OX ) so that Ui := {fi 6= 0} are all open affine and X = ∪i Ui .
(54) Calculate Ȟ ∗ (P1 , OP1 (m)) for all m ∈ Z with respect to the standard cover
U0 ∪ U1 of P1 .
(55) Consider X = A2 \ (0, 0) where A2 = Spec(C[x, y]). Using an appropriate cover
of X show that
Ȟ 1 (X, OX ) = span{xi y j : i, j < 0}.
In particular, this again shows that X is not affine.
(56) Suppose X is projective and F a coherent sheaf on X. Define
X
χ(F) =
(−1)i dim H i (X, F).
i≥0
If 0 → F1 → F2 → F3 → 0 is a short exact sequence show that χ(F1 ) − χ(F2 ) +
χ(F3 ) = 0.
(57) Let E ⊂ P2 be a smooth curve carved out by a (homogeneous) degree 3 polynomial. Use the short exact sequence
0 → IE → OP2 → OE → 0
PROBLEMS: ALGEBRAIC GEOMETRY II
(58)
(59)
(60)
(61)
(62)
(63)
5
where IE is the ideal sheaf of E, to compute H i (E, OE ) for i ≥ 0. Hint: IE ∼
=
OP2 (−3).
For E as above, let p ∈ E be an arbitrary point. Calculate H i (E, OE (np)) for
n ∈ Z, i ≥ 0. Hint: you can use the calculation of H i (E, OE ) from the above
exercise and induction on n.
For E as above, show that OE (p) is ample but not very ample. Hint: first show
that cohomology criterion for ampleness holds when the test sheaf F is locally
free.
Compute Exti (Op , Op ) where p is a point in P2 .
Check that a closed inclusion i : X → Y is not a flat morphism (X 6= Y ).
Check that the flatness criterion from class implies that the fibres of a flat
morphism over a smooth variety all have the same dimension.
Consider the maps
C[t] → C[x, y]/xy → C[x]
where the first map is given by t 7→ x + y and the second by x 7→ x, y 7→ 0. This
induces maps
f
g
X−
→Y −
→Z
where X = SpecC[x], Y = SpecC[x, y]/xy, Z = SpecC[t]. Determine which of
f, g, g ◦ f are flat morphisms.
(64) Do the same for the maps
C[u, v] ∼
= C[x2 , y 2 ] → C[x2 , xy, y 2 ] → C[x, y]
where both maps are inclusions. Aside: what do these maps represent geometrically?