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The Stacks project

Exercise 111.39.5. Let k be a field. Suppose that X \subset \mathbf{P}^3_ k is a closed subscheme given by a single homogeneous equation F \in k[X_0, X_1, X_2, X_3]. In other words,

X = \text{Proj}(k[X_0, X_1, X_2, X_3]/(F)) \subset \mathbf{P}^3_ k

as explained in the course. Assume that

F = X_0 G + X_1 H

for some homogeneous polynomials G, H \in k[X_0, X_1, X_2, X_3] of positive degree. Show that if X_0, X_1, G, H have no common zeros then there exists a nonconstant morphism

X \longrightarrow \mathbf{P}^1_ k

of schemes over \mathop{\mathrm{Spec}}(k) which on field points (see Exercise 111.39.1) looks like (x_0 : x_1 : x_2 : x_3) \mapsto (x_0 : x_1) whenever x_0 or x_1 is not zero.


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