Weil restriction of a projective variety to a finite extension over $\mathbb{Q}$

Question

My question maybe a bit stupid, but I can't quite grasp what the Weil restriction actually does... In particular:
Given a projective variety $V$ defined over $L$ algebraically closed, of characteristic 0, given a finite extension $K$ of $\mathbb{Q}$, is the Weil restriction of $V$ to $K$ a model of $V$ over K?

Answer 1

I thought I'd answer this, because Weil restriction is confusing. The best treatment I've seen is in Milne's notes on algebraic geometry, available from his website. (I'd be happy to have other references*.) Given field extension $L/K$, one gets a functor from schemes over $K$ to schemes over $L$ given by base change, that I'll denote by $(-)_L$. Given a scheme $V/L$, the Weil restriction -- if it exists -- is the $K$-scheme $R_{L/K}V$ characterized by $$Hom_L(T_L, V)\cong Hom_K(T, R_{L/K}V)$$ So it is sort of a right adjoint to base change. In particular, the $K$-rational points of $R_{L/K}V$ are the $L$-rational points of $V$. Milne shows that $R_{L/K}V$ exists when $L/K$ is finite separable and $V$ a quasiprojective variety. If you look at the construction, you'll see that certain geometric properties such as dimension can change, as was commented above. The construction is probably most often used when $V$ is an algebraic group, in which case so is the restriction.

*Here are a few more:

1. Bosch, Lütkebohmert, Raynaud, Néron Models, section 7.6 (Very complete treatment.)
2. Poonen, Rational Points on Varieties, Section 4.6 (Suggested by RP).