Algebraic geometry M 4 P 33

Exercise sheet 3

The asterisk ^{∗} marks questions that are more difficult than others.

1. More on cubic hypersurfaces. All cubic hypersurfaces in this and the next question are non-conical and absolutely irreducible, that is, irreducible over an algebraic closure \overline{k} of k.

Let X be a cubic surface in P_{k}^{3}, and P be a smooth point on X. Consider the lines contained in X and passing through P. Show that there can be 0, 1, 2 or 3 such lines by giving examples. Prove that these are the only possibilities.

Hint: any such line is contained in the tangent space to X at P understood as a plane in P_{k}^{3}.

2. Let X be a cubic hypersurface in A_{k}^{n} with a singular k-point P.

(i) Prove that X is rational. Hint: imitate the stereographic projection construction. (ii) Let Q be another singular k-point of X. Prove that the line P Q is

contained in X.

(iii) Deduce that a cubic curve in A_{k}^{2} has at most one singular point.

3. A very special cubic hypersurface. Let X be the Veronese surface in P_{k}^{5} (=the image of the morphism from P_{k}^{2} given by all monomials of degree 2).

(i) Check that the images of lines on P_{k}^{2} are plane conics in X. (ii) Let Y ⊂ P_{k}^{5} be the union of planes spanned by these conics. Show

that X and Y can be described as follows.

• P_{k}^{5} is identified with the space of non-zero symmetric 3×3-matrices up to multiplication by a scalar.

• Y is the hypersurface given by the cubic equation det(M) = 0 (degenerate matrices).

• X is the set of symmetric 3×3-matrices of rank 1, up to multiplication by a scalar.

(iii)^{∗} Show that X is the set of singular points of Y. (iv) Let k be algebraically closed. The degree of a projective variety

Z ⊂ P_{k}^{n} of dimension r is the maximal number of points in the intersection Z ∩ P_{k}^{n−r} provided this intersection is finite. (This agrees with the usual concept of degree of a hypersurface.) Show that the degree of X is 4.

(v) Show that plane conics depend on five parameters. More precisely, interpret P_{k}^{5}, Y and X as the parameter space of plane conics, reducible conics and double lines, respectively.

4. Let k be algebraically closed of characteristic 0. Let X ⊂ P_{k}^{4} be the closed subset given by

x_{0}^{2} + x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} = a_{0}x_{0}^{2} + a_{1}x_{1}^{2} + a_{2}x_{2}^{2} + a_{3}x_{3}^{2} + a_{4}x_{4}^{2} = 0,

where a_{i} ∈ k, i = 0, 1, 2, 3, 4. Let P ∈ X. Find T_{X,P}. Find the singular points and the dimension of X in the following cases:

(a) a_{0} = a_{1} = 0, a_{2} = 2, a_{3} = 3, a_{4} = 4;

(b) a_{0} = a_{1} = a_{2} = 0, a_{3} = 1, a_{4} = −1;

(c) a_{i} = i for all i. Can you determine the irreducible components in the case (b)?