Definitions for affine and projective varieties
Results and definitions stated below, for affine varieties, can be translated to projective varieties, by replacing An(kalg) with projective space of dimension n − 1 over kalg, and by insisting that all polynomials be homogeneous.
A k-algebraic set is the zero-locus in An(kalg) of a subset of the polynomial ring k[x1, ..., xn]. A k-variety is a k-algebraic set that is irreducible, i.e. is not the union of two strictly smaller k-algebraic sets. A k-morphism is a regular function between k-algebraic sets whose defining polynomials' coefficients belong to k.
One reason for considering the zero-locus in An(kalg) and not An(k) is that, for two distinct k-algebraic sets X1 and X2, the intersections X1∩An(k) and X2∩An(k) can be identical; in fact, the zero-locus in An(k) of any subset of k[x1, ..., xn] is the zero-locus of a single element of k[x1, ..., xn] if k is not algebraically closed.
A k-variety is called a variety if it is absolutely irreducible , i.e. is not the union of two strictly smaller kalg-algebraic sets. A variety V is defined over k if every polynomial in kalg[x1, ..., xn] that vanishes on V is the linear combination (over kalg) of polynomials in k[x1, ..., xn] that vanish on V. A k-algebraic set is also an L-algebraic set for infinitely many subfields L of kalg. A field of definition of a variety V is a subfield L of kalg such that V is an L-variety defined over L.
Equivalently, a k-variety V is a variety defined over k if and only if the function field k(V) of V is a regular extension of k, in the sense of Weil. That means every subset of k(V) that is linearly independent over k is also linearly independent over kalg. In other words those extensions of k are linearly disjoint.
André Weil proved that the intersection of all fields of definition of a variety V is itself a field of definition. This justifies saying that any variety possesses a unique, minimal field of definition.
Scheme-theoretic definitions
One advantage of defining varieties over arbitrary fields through the theory of schemes is that such definitions are intrinsic and free of embeddings into ambient affine n-space.
A k-algebraic set is a separated and reduced scheme of finite type over Spec(k). A k-variety is an irreducible k-algebraic set. A k-morphism is a morphism between k-algebraic sets regarded as schemes over Spec(k).
To every algebraic extension L of k, the L-algebraic set associated to a given k-algebraic set V is the fiber product of schemes V ×Spec(k) Spec(L). A k-variety is absolutely irreducible if the associated kalg-algebraic set is an irreducible scheme; in this case, the k-variety is called a variety. An absolutely irreducible k-variety is defined over k if the associated kalg-algebraic set is a reduced scheme. A field of definition of a variety V is a subfield L of kalg such that there exists a k∩L-variety W such that W ×Spec(k∩L) Spec(k) is isomorphic to V and the final object in the category of reduced schemes over W ×Spec(k∩L) Spec(L) is an L-variety defined over L.
Analogously to the definitions for affine and projective varieties, a k-variety is a variety defined over k if the stalk of the structure sheaf at the generic point is a regular extension of k; furthermore, every variety has a minimal field of definition.
One disadvantage of the scheme-theoretic definition is that a scheme over k cannot have an L-valued point if L is not an extension of k. For example, the rational point (1,1,1) is a solution to the equation x1 + ix2 - (1+i)x3 but the corresponding Q[i]-variety V has no Spec(Q)-valued point. The two definitions of field of definition are also discrepant, e.g. the (scheme-theoretic) minimal field of definition of V is Q, while in the first definition it would have been Q[i]. The reason for this discrepancy is that the scheme-theoretic definitions only keep track of the polynomial set up to change of basis. In this example, one way to avoid these problems is to use the Q-variety Spec(Q[x1,x2,x3]/(x12+ x22+ 2x32- 2x1x3 - 2x2x3)), whose associated Q[i]-algebraic set is the union of the Q[i]-variety Spec(Q[i][x1,x2,x3]/(x1 + ix2 - (1+i)x3)) and its complex conjugate.
Action of the absolute Galois group
The absolute Galois group Gal(kalg/k) of k naturally acts on the zero-locus in An(kalg) of a subset of the polynomial ring k[x1, ..., xn]. In general, if V is a scheme over k (e.g. a k-algebraic set), Gal(kalg/k) naturally acts on V ×Spec(k) Spec(kalg) via its action on Spec(kalg).
When V is a variety defined over a perfect field k, the scheme V can be recovered from the scheme V ×Spec(k) Spec(kalg) together with the action of Gal(kalg/k) on the latter scheme: the sections of the structure sheaf of V on an open subset U are exactly the sections of the structure sheaf of V ×Spec(k) Spec(kalg) on U ×Spec(k) Spec(kalg) whose residues are constant on each Gal(kalg/k)-orbit in U ×Spec(k) Spec(kalg). In the affine case, this means the action of the absolute Galois group on the zero-locus is sufficient to recover the subset of k[x1, ..., xn] consisting of vanishing polynomials.
In general, this information is not sufficient to recover V. In the example of the zero-locus of x1p- t in (Fp(t))alg, the variety consists of a single point and so the action of the absolute Galois group cannot distinguish whether the ideal of vanishing polynomials was generated by x1 - t1/p, by x1p- t, or, indeed, by x1 - t1/p raised to some other power of p.
For any subfield L of kalg and any L-variety V, an automorphism σ of kalg will map V isomorphically onto a σ(L)-variety.
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