Is there a projective variety that is a one-point compactification of algebraic torus.

Question

Is there a closed subvariety of $\mathbb{CP}^N$ which is a one point compactification of the complex algebraic torus $(\mathbb{C}^*)^n$?

For example, when $n=1$, we could take the nodal curve defined by $x^3=y^2z-xyz$ in $\mathbb{CP}^2$.

Answer 1

For $n > 1$ the answer is negative, because the class group of Weil divisors on a 1-point compactification is equal to the Picard group of a torus, hence trivial, while the restriction of the hyperplane class to any closed subvariety of positive dimension of a projective space is non-trivial.