Things that need to be defined:

• General type:

A variety X is of general type if there is a positive power of the canonical bundle whose global sections determine a rational map f : X → P^{n} with dim f(X) = dim X. (If X is of general type then there exists some positive power of the canonical bundle such that the corresponding map is birational to its image.)

“It is a moral judgement of geometers that you would be wise to stay away from the bloody things.” – Swinnerton-Dyer

• Waring’s problem:

Given k, find the smallest number g_{k} such that every positive integer is a sum of g_{k} positive kth powers. The “easier” Waring’s problem refers to the analogous problem where the kth powers are permitted to be either positive or negative. Modification: Given k, find the smallest number G_{k} such that every sufficiently large positive integer is a sum of G_{k} positive kth powers.

• Diophantine set:

Let R be a ring. A subset A ⊂ R^{n} is diophantine over R if there exists a polynomial f ∈ R[t_{1}, …, t_{n}, x_{1}, …, x_{m}] such that

A = {t⃗ ∈ R^{n} : ∃x⃗ ∈ R^{m} such that f(t⃗, x⃗) = 0}.

• The Brauer-Manin obstruction and set X(A_{k})^{Br} of a variety: The terminology is utterly aweful! Many families don’t satisfy Hasse Principle. One explanation of Manin (see his paper): a cohomological obstruction using the Brauer group of the variety

If a variety has a local point everywhere then it has an adelic point. Manin defined, using a cohomological condition involving Brauer group, a subset of the adelic points that must contain the global points. Let X(A_{k}) be the adelic points of X. Consider the subset of points P with the property that for every element z ∈ Br(X) the system of elements (z_{v}(P))_{v} has sum of invariants = 0.

The B-M is an interesting construction in English. It is a nounal-phrase defined purely in terms of the sentences in which in which it may occur. There is no such actual object “the Brauer-Manin obstruction”.

Example: A variety that satisfies X(A_{k}) ≠ ∅ and X(A_{k})^{Br} = ∅ is a counterexample to the Hasse principle explained by the Brauer-Manin obstruction.

For a long time people were interested in whether there are counterexamples ot Hasse principle not explained by the Brauer-Manin obstruction.

X(A_{k})^{Br} ≠ ∅ but still has no global point (Skorobogotav found first example).

After one glass of wine, McCallum advocates that “X(A_{k})^{Br} should be called the set of Brauer points”.

• Hasse Principle:

A family of varieties satisfies the Hasse principle if whenever a variety in the family has points everywhere locally it has a point globally.

• Hilbert’s tenth problem:

Let R be a commutative ring. Hilbert’s tenth problem for R is to determine if there is an algorithm that decides whether or not a given system of polynomial equations with coefficients in R has a solution over R.

• Weak approximation:

For a projective variety X over a global field, say weak approximation holds if X(K) is dense in the adelic points X(A_{K}). Simplest example where it holds: P^{0}, also P^{1}. It does not hold for an elliptic curve over K. (For example, if E has rank 0 it clearly doesn’t hold... but more generally could divide all generators by 2 and choose a prime that splits completely.)

Example: “Weak approximation does not hold for cubic surfaces.”

Example: “The theory of abelian descent in some cases reduces the question of whether the Brauer-Manin obstruction is the only obstruction to Hasse on a base variety X to the question of whether weak approximation holds for a universal torsor.”

Example: “Weak approximation on a moduli space of varieties yields the existence of varieties over a global field satisfying certain local conditions. For example, we want to know there is an elliptic curve over Q with certain behavior at 3, 5, 13, as long as can do it over local fields with that behavior, weak approximation on the moduli space gives you a global curve that has those properties (because P^{1} satisfies weak approximation).’

• BSD conjecture—Birch and Swinnerton-Dyer: Let A be an abelian variety over a global field K and let L(A, s) be the associated L-function. The Birch and Swinnerton-Dyer conjecture asserts that L(A, s) extends to an entire function and ord_{s=1}L(A, s) equals the rank of A(K). Moreover, the conjecture provides a formula for the leading coefficient of the Taylor expansions of L(A, s) about s = 1 in terms of invariants of A.

• Selmer group:

Given Galois cohomology definition for any A ⊂ B. Example A = ker(ϕ) where ϕ is an isogeny of abelian variety. Accessible. It’s what we can compute, at least in theory.

• Fano variety—Fano:

Anticanonical divisor ω^{⊗−1} is ample. This class of varieties is “simple” or “close to rational”. For example, one conjectures that Brauer-Manin is only obstruction. Manin-Batyrev conjecture: asymptotic for number of points of bounded height. A Fano variety of dimension two is also called a Del Pezzo surface.

• Del Pezzo surface:

A Del Pezzo surface is a Fano variety of dimension two.

It can be shown that the Del Pezzo surfaces are exactly the surfaces that are geometrically either P^{1} × P^{1} or a blowup of P^{2} at up to 8 points in general position. By general position we mean that no three points lie on a line, no six points lie on a conic, and no eight lie points lie on a singular cubic with one of the eight points on the singularity.

• Enriques Surface:

A quotient of a K3 surface by a fixed-point free involution.

Equivalently, the normalization of the singular surface of degree 6 in P^{3} whose singularities are double lines that form a general tetrahedron.

Over C an Enriques surface can be characterized cohomologically as follows: H^{0}(Ω_{X}^{2}) = 0 and 2K_{X} = 0 but K_{X} ≠ 0.

• K3 surface:

A surface with trivial canonical bundle and trivial fundamental group (i.e., a Calabi-Yau variety of dimension 2).

• Brauer-Severi variety: A twist of projective space P^{n}. Brauer-Severi varieties satisfy the Hasse principle.

• Abelian variety:

A smooth projective geometrically integral group variety over a field. Over the complex numbers abelian varieties are tori.

• Shimura variety:

A variety having a Zariski open subset whose set of complex points is analytically isomorphic to a quotient of a bounded symmetric domain X by a congruence subgroup of an algebraic group G that acts transitively on X. Examples include moduli spaces X_{0}(N) of elliptic curves with extra structure and Shimura curves which parametrize quaternionic multiplication abelian surfaces with extra structure.

• Prym variety:

A Prym variety is an abelian variety constructed in the following way. Let X and Y be curves and suppose f : X → Y is a degree 2 étale (unramified) cover. The associated Prym variety is the connected component of the kernel of the Albanese map Jac(X) → Jac(Y). The Prym variety can also be defined as the connected component of the −1 eigenspace of the involution on Jac(X) induced by f.

• Jacobian:

The Jacobian of a nonsingular projective curve X is an abelian variety whose points are in bijection with the group Pic^{0}(X) of isomorphism classes of invertible sheaves (or divisor classes) of degree 0.

• Calabi-Yau variety:

An algebraic variety X over C is a Calabi-Yau variety if it has trivial canonical sheaf (i.e., the canonical sheaf is isomorphic to the structure sheaf) and X(C) is simply connected as a topological space. For example, an elliptic curve is not a Calabi-Yau variety because X(C) is a torus, which is not simply connected.

• Descent:

1. The process of expressing the rational points on a variety as the union of images of rational points from other varieties.

2. The descent problem is as follows: Given a field extension L/K and a variety X over L, try to find a variety Y over K such that X = Y ×_{K} L.

• Picard group:

The Picard group of a variety is the group of isomorphism classes of invertible sheaves.

• Rationally connected variety: There are three definitions of rationally connected. These are equivalent in characteristic zero but not in characteristic p.

1. For any two points x, y ∈ X there exists a morphism ϕ : P^{1} → X such that ϕ(0) = x and ϕ(∞) = y.

2. For any n points x_{1}, …, x_{n} ∈ X there exists a morphism ϕ : P^{1} → X such that {x_{1}, …, x_{n}} is a subset of ϕ(P^{1}).

3. For any two points x, y ∈ X there exist morphisms ϕ_{i} : P^{1} → X for i = 1, …, r such that ϕ_{1}(0) = x, ϕ_{r}(0) = y, and for each i = 1, …, r − 1 the images of ϕ_{i} and ϕ_{i+1} have nontrivial intersection.

• Torsor:

Let B be a variety over a field k and let G be an algebraic group over k. A left B-torsor under G is a B-scheme X with a B-morphism G ×_{k} X → X such that for some étale covering {U_{i} → B} there is a G-equivariant isomorphism of U_{i}-schemes from X ×U_{i} to G×U_{i}, for all i. If B = Spec(k) these are also called principal homogenous spaces.

• Fermat curves:

Good examples of many phenomenon. Good source of challenge problems. (E.g., FLT.) Lot of symmetry so you can compute a lot with them. Computations are surprising and nontrivial. They’re abelian covers of P^{1} ramified at 3 points, so they occur in the fund. group of...

• Lang’s conjectures:

1. Suppose k is a number field and X is a variety over k of general type. Then X(k) is not Zariski dense in X. (Also there are refinements where we specify which Zariski closed subset is supposed to contain X(k).)

2. Suppose k is a number field and X is a variety over k. All but finitely many k-rational points on X lie in the special set.

3. Let X be a variety over a number field k. Choose an embedding of k into the complex number C, and suppose that X(C) is hyperbolic: this means that every holomorphic map C → X(C) is constant. Then X(k) is finite.

• Special Set:

The (algebraic) special set of a variety X is the Zariski closure of the union of all positive-dimensional images of morphisms from abelian varieties to X. Note that this contains all rational curves (since elliptic curves cover P^{1}).

• Schinzel’s Hypothesis:

Suppose f_{1}, …, f_{r} ∈ Z[x] are irreducible and no prime divides f_{1}(n)f_{2}(n) ⋯ f_{r}(n) for all n ∈ Z. Then there are infinitely many integers n such that |f_{1}(n)|, …, |f_{r}(n)| are simultaneously prime.

• Hardy-Littlewood circle method: An analytic method for obtaining asymptotic formulas for the number of solutions to certain equations satisfying certain bounds.