INTRO TO ALGEBRAIC GEOMETRY, PROBLEM SET 6

Due Thursday October 28 in class (no lates). Hand in seven of the following questions. You’re strongly encouraged to collaborate (although write up solutions separately), and you’re also strongly encouraged to ask me questions (if you’re stuck, or if the question is vaguely worded, or if you want to try out an argument). Ask someone else about the non-scheme problems you skip.

The “miscellaneous” questions are more straightforward. The problems on the Hilbert polynomial and Veronese embeddings are harder, but there are fast shortcuts (which aren’t necessarily easy to find).

Miscellaneous.

1. (a) Consider two varieties, ℙ^{1} and 𝔸^{1}. Let one of the standard affine opens of ℙ^{1} have coordinate t, and let x be a coordinate on 𝔸^{1}. Find a dominant rational map f : ℙ^{1} −−→ 𝔸^{1} corresponding to the morphism of function fields k(𝔸^{1}) = \overline{k}(x) → k(ℙ^{1}) = \overline{k}(t) given by x → t^{2}.

(b) If X is a prevariety, give a natural identification between elements of k(X) and the rational maps from X to ℙ^{1}.

2. (a) In Problem Set 4, you showed that the function field of the hypersurface X given by wx = yz in 𝔸^{4} was isomorphic to \overline{k}(t_{1}, t_{2}, t_{3}). By comments in class, X should be rational, i.e. birational to 𝔸^{3}. Prove that it is.

(b) If X is the line with the doubled origin, show that the diagonal Δ(X) is not closed in X × X.

3. Consider the morphism f : 𝔸^{2} \ {(0, 0)} → ℙ^{1} given by (x, y) ↦ (x; y). Show that there is no morphism 𝔸^{2} → ℙ^{1} extending f.

Projective varieties: Hilbert functions, dimension, and degree. Suppose V is a projective variety in ℙ^{n} (with projective coordinates x_{0}, …, x_{n}) with projective coordinate ring R = \overline{k}[x_{0}, …, x_{n}]/I = ⊕_{d}R_{d}, where R_{d} is the degree d part of R. It is a fact that there is a polynomial f(d) and an integer d_{0} such that dim R_{d} = f(d) for d ≥ d_{0} (see Lang’s Algebra, the section on the Hilbert polynomial). f(d) is called the Hilbert polynomial of V ⊂ ℙ^{n}. The dimension of V can be defined as the degree of f(d) (although a proof is required to show that is the same as the definition I’ll give in the course). The degree of V can be defined as (dim V)! times the leading coefficient of f(d). In the next five problems, use these definitions of dimension and degree. (Another, more intuitive, definition of degree is: Intersect V with dim V randomly chosen hyperplanes, and count the points of intersection.)

Date: October 21, 1999.

4. Calculate the Hilbert polynomial when V = ℙ^{n}. Verify that dim V = n and deg V = 1. If V is a codimension m linear subspace, i.e. cut out by m linear independent linear forms in the x_{i}, verify that dim V = n − m and deg V = 1.

5. Suppose V is cut out by a single equation q(x_{0}, …, x_{n}) = 0, where q is irreducible and homogeneous of degree e, so R = \overline{k}[x_{0}, …, x_{n}]/(q). Calculate the Hilbert polynomial of V ⊂ ℙ^{n}, and verify that dim V = n−1 and deg V = e.

6. (The beginnings of intersection theory.) Suppose V′ ⊂ ℙ^{n} is a projective variety of dimension D and degree d, with coordinate ring R = \overline{k}[x_{0}, …, x_{n}]/I(V′), and suppose that q (as in problem 5) cuts out an irreducible subvariety W of V′ (distinct from V′), which is equivalent to q being irreducible in the ring R. Prove that dim W = D − 1 and deg W = de.

The degree d Veronese embedding of ℙ^{n}. Consider the morphism

ℙ^{n} → ℙ^{\binom{n+d}{d}−1}

given by (x_{0}^{d}; x_{0}^{d−1}x_{1}; …; x_{n}^{d}), where (x_{0}^{d}; x_{0}^{d−1}x_{1}; …; x_{n}^{d}) includes all degree d monomials in x_{0}, …, x_{n}.

7. Verify that it is a morphism, and an isomorphism onto its image V.

8. If R = ⊕_{e≥0}R_{e} is the projective coordinate ring of V, verify that R_{e} can be identified with the vector space of (homogeneous) degree de polynomials in the x_{i}. Hence verify that dim V = n and deg V = d^{n}.

Schemes: nilpotents.

9. (a) If f ∈ R is nilpotent, describe D(f) in Spec R.

(b) A scheme (X, O_{X}) is reduced if for every open set U ⊂ X, the ring O_{X}(U) has no nilpotent elements. Show that (X, O_{X}) is reduced if and only if for every P in X, the local ring O_{X,P} has no nilpotent elements. (Translation: “reducedness” is a local property, in the sense that you can check it on points.)

10. Explain intuitively how the morphisms from Spec \overline{k}[t]/(t^{2}) (which has only one point) to 𝔸^{2} (over the field \overline{k}) correspond to the data of a point of 𝔸^{2}, and a tangent vector of 𝔸^{2}. Using this “definition”, calculate the dimension of the tangent space to 𝔸^{n} at the origin; and the dimension of the tangent space to the curve y^{2} = x^{2} + x^{3} in 𝔸^{2} at the origin.