I am trying to do the second exercise in page 37 in Ernst Kunz's "Introduction Commutative Algebra and Algebraic Geometry" (I leave a sreenshot of the exercise below), and I got stuck trying to prove the second part of line b).
If I write $X:= \overline{V^*}$ = $X_1 \cup X_2$ with $X_1, X_2$ affine subvarieties of $X$. Since we have $V \subset X$ and it is irreducible, we must have $V \subset X_1$ or $V \subset X_2$. One of the $X_i$'s will have to contain infinite points of some line contained in $V^*$ and so the polynomials in its ideal can't have constant terms. I have also thought of considering the decomoposition in irreducible components, which in the case of $\mathbb{L}$ alg. closed, I would use to deduce that the ideals of each irreducible component would be homogeneous, since they will be the minimal prime divisors of a homogeneous ideal, and then any irreducible component that contained $V$ would be equal to $X$.
Can anyone give me hint for the case $\mathbb{L}$ not alg. closed?
Thank you for all the help in advance :)