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Question 1

Define: variety, affine

which topologies used? Compare properties under the different topologies

Answer 1

pg 80-82

2.1.1 Y locally closed subset of variety => Closure in analytic = Closure in Zariski. implies: Zariski dense open subvarieties remain dense in analytic topology

2.1.4 Every variety has a smooth dense subvariety

2.1.5 Connected in Zariski <=> Connected in analytic

2.1.6 In the analytic topology, every variety is locally compact, locally contractible, and of finite c-soft dimension.

Question 2

Define: Noetherian topological space, prove every variety is one

Answer 2

dcc

pg 81

Question 3

Prove: If X is a variety and U1, U2 < X are two locally closed affine subvarieties, then U1 \cap U2 is an affine subvariety.

Answer 3

pg 81 Cartesian square

Question 4

What is Nagata’s compactification theorem?

Answer 4

pg 82

Question 5

Define: quasi-finite morphism, finite morphism

Answer 5

morphism finite iff proper and quasi-finite iff closed and quasi-finite (by 3rd characterization of proper on pg 21)

Question 6

Noether Normalization Lemma?

Answer 6

pg 82

Question 7

Prove: If f: X –> Y is a finite morphism, then o^f_*: Sh(X, k) –> Sh(Y, k) is an exact functor.

Answer 7

pg 82

Question 8

Define/discuss: smooth of relative dimension m-n, smooth morphism, etale morphism

Answer 8

pg 83

Question 9

Discuss generic smoothness of morphisms of varieties

Answer 9

If f: X –> Y and X smooth, then exists nonempty Zariski open subset U < Y s.t. f is smooth from f^-1(U) –> U
pg 83

Question 10

What is the relationship between etale morphisms and covering maps?

Answer 10

pg 83 Lemma 2.1.14

Question 11

Define: divisor, divisor with simple normal crossings

Answer 11

In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mumford). Both are derived from the notion of divisibility in the integers and algebraic number fields.

pg 84

Question 12

Discuss resolution of singularities

Answer 12

pg 84-85

Question 13

Define differentiable locally trivial fibration and discuss presence in algebraic geometry

Answer 13

pg 85

Question 14

Define when map is transverse to divisor w/ snc Z. Transverse locally trivial fibration? Generalization of Ehressmann?

Answer 14

pg 85

Question 15

Discuss the relationship between the fundamental group of a smooth, connected variety and a Zariski open subset of that variety. Proof?

Answer 15

pg 86

Question 16

Prove:Prop 2.1.23

Answer 16

pg86-87

Question 17

Show a smooth morphism of relative dim d is a topological submersion of relative dim 2d

Answer 17

pg 87

Question 18

What is smooth base change? Proof?

Answer 18

pg 88

Question 19

Define: Tate module, nth Tate twist

Answer 19

pg 88

Question 20

Examples of Tate twists?

Answer 20

pg 89

Question 21

f:X–>Y a smooth morphism of rel dim d. What can be said about the rel orientation sheaf? Proof

How does this help with figuring out f^! ?

Answer 21

pg 89-91

Question 22

What is the dualizing complex for a smooth variety of dim n?

Answer 22

pg 91

Question 23

How does smooth pullback interact with sheaf operations?

Answer 23

pg 91

Question 24

Define: smooth pair, discuss restriction with supports functor for smooth pairs

Answer 24

pg 92

Question 25

Define: stratification, strata, closure partial order

Answer 25

A stratification of X is a finite collection of disjoint smooth, connected, locally closed subvarieties s.t. their union is X and s.t. the closure of a strata is a union of strata. pg 93-94

Question 26

Define: filtration of X by smooth varieties relation to stratification? What varieties admit filtration? refinement?

Answer 26

Finite collection of disjoint, smooth, connected, locally closed subvarieties whose union is X and s.t. the elements can be ordered s1, s2, ... , sk in such a way that Xs1 U Xs2 U ... U Xsi is always a closed subset of X. (affine paving if these are affine sets) Any stratification is a filtration. Any filtration admits a refinement that is a stratification. All varieties admit a filtration pg 93

Question 27

Define: weakly constructible and constructible (sheaf and derived)

Answer 27

Sheaf weakly constructible if on each stratum F is a local system (finite type = constructible) An object in derived bounded category is (weakly ) constructible if each cohomology shead has the same property 94

Question 28

What is modified dimension of support over a field? What can be said about it w.r.t. s.e.s of sheaves and distinguished triangles in derived cat?

Answer 28

Over a field - no need to modify. just dimension of support pg 94

Question 29

Prove: f* preserves constructibility

Answer 29

pg 96

Question 30

Prove: if h:Y --> X is inclusion of locally closed sub, h_! preserves constructibility

Answer 30

pg 96

Question 31

Prove: If U open in X and F constructible on U and X - U, then F constructible on U

Answer 31

pg 95

Question 32

How do 6 functors interact with constructibility?

Answer 32

All preserve constructibility pg 96

Question 33

Define: good stratification Importance?

Answer 33

If pushforward of the inclusion j of a strata Xs into X always sends a local system of finite type on Xs to a constructible sheaf on X Can ignore here - in algebraic setting always good Allows easier proof that 4 functors associated to inclusion of locally closed subvariety all preserve constructibility pg 97-98

Question 34

Define: normal crossings stratification is this a good stratification?

Answer 34

Given a divisor with simple normal crossings, we get a stratification. pg 99, 100-101

Question 35

Define: normal crossings coordinate chart

Answer 35

Z divisor with simple normal crossings, Z1 ... Zk components. J < {1,...k} say J = {i1, ..., ij}. For any point of X_J, there is an analytic open set V < X_... pg 99

Question 36

Discuss base change and the affine line Define Z finite/surjective/etale over X

Answer 36

The main point: consider the cartesian square on 103 - find some conditions under which base change maps are isomorphisms. 2.5.3 If Z closed and finite over Y and U = Y \Z. Then if F|U is a local system, base change is an iso.

Question 37

Discuss Artin's vanishing theorem proof?

Answer 37

For affine varieties, H^k(X, k) = 0 for all k > n (you would expect it to be 2n) Thm. X affine variety, F constructible sheaf on X. Then H^k(X, F) is a f.g. k-module and it vanishes unless 0 < k < dimm supp F. In particular, = unless k < dim X.

Question 38

Discuss the cohomology of a constructible sheaf on X

Answer 38

Thm 2.7.4, 2.7.5 pg 112 -113

Question 39

Discuss Verdier duality with respect to constructible sheaves

Answer 39

1. Dualizing complex is constructible, so Verdier duality functor restricts to a functor between (derived) constructible sheaves 2. Evaluation map F --> D(D(F)) is an isomorphism - i.e. natural isomorphism D o D = id 3. Relationship between stalk of dual complex and original stalk 4. Fact that DD = id allows us to prove f^! preserves constructibility

Question 40

Discuss how external tensor product and extension of scalars interact with 6 sheaf functors

Answer 40

Commute with all of them and Verdier duality

Question 41

What is Noetherian induction? Why does it work? Examples?

Answer 41

Borcherds & Pramod