Degrees of compactifications of affine space Let $k$ be a field and $V\subset \mathbb{P}^n$ a smooth variety over $k$.  (Note: not assuming $k$ is algebraically closed).  Now assume that for some point (I'm willing to assume every point), the variety $V\setminus T_pV$ is isomorphic to an affine space.  What can be said about the degree of $V$?
I know that for $d=2$, this is true.  Can it happen for anything that isn't a quadric, or does this property determine that you have a quadric, automatically?
 A: Let me give some remarks assuming that $k=\mathbb C$.
a. The intersection $T_p V \cap V$ must have codimension one. Otherwise $H^0(\mathbb C^n,\mathcal O_{\mathbb C^n})$ would inject into $H^0 (V,\mathcal O_V)$.
b. If $T_p V \cap V$ is reduced and irreducible then $V$ is a Fano manifold with $Pic(V)=\mathbb Z$ and therefore has bounded degree.
c. If you are not in the cases of dimension one or codimension one, and you assume that your property holds for every $p\in V$  then the second fundamental form of $V$ is everywhere degenerated. Griffiths-Harris' Algebraic geometry and local differential geometry maybe useful in this case. 

The following remarks are not about the original question, but rather to one of Charles' comments:
"or if it becomes true if I say instead of $V∖T_p V, V∖H$ where $H$ is a hyperplane containing $T_pV$. 
d. As already noted by  mdeland in the comments 
the rational normal curves give examples of curves $C$ such that
$C \setminus T_p C \simeq \mathbb A^1$. Moreover, if we consider 
the osculating hyperplane $H_p$ of $C$ at $p$ then $C \setminus H_p$
is also isomorphic to $\mathbb A^1$. 
e. There are $3$-dimensional examples coming from the study of compactifications of $\mathbb C^3$.
Let  $V_5 \subset \mathbb P^6$ be the Fano $3$-fold of index two and degree $5$. 
 This  $3$-fold can be described as the intersection of $Gr(2,5) \hookrightarrow \mathbb P^9$ with $3$ generic hyperplanes (they are all isomorphic). Alternatively, it can be 
described as closure of the $\mathrm{Aut}(\mathbb P^1)$-orbit of the vertices of a
regular octahedron in $\mathrm{Sym}^6\mathbb P^1$.
I think it was Furushima who first presented an explicit hyperplane $H\subset \mathbb P^6$
such that $V_5 \setminus H \simeq \mathbb C^3$.
Later a number of other divisors in $V_5$ with complement isomorphic to $\mathbb C^3$ where
found.  There are also divisors in Fano $3$-folds of index one with complement isomorphic to $\mathbb C^3$.
Anyway,I think  you  find useful information in the literature about compactifications
of $\mathbb C^3$. I would start  looking  at Mukai's  Fano $3$-folds, and  Peternell-Schneider's Compactifications of $\mathbb C^n$: a survey.  
