  Let $U$ be a $({ k-1 \choose 2}-1)$-dimensional subspace of quadratic forms
defined on $\mathrm{PG}(k-1,{\mathbb F})$ with the property that $U$ does not
contain any reducible quadratic form. Let $V(U)$ be the points of
$\mathrm{PG}(k-1,{\mathbb F})$ which are zeros of all quadratic forms in $U$.
We will prove that if there is a group $G$ which fixes $U$ and no line of
$\mathrm{PG}(k-1,{\mathbb F})$ and $V(U)$ spans $\mathrm{PG}(k-1,{\mathbb F})$
then any hyperplane of $\mathrm{PG}(k-1,{\mathbb F})$ is incident with at most
$k$ points of $V(U)$. If ${\mathbb F}$ is a finite field then the linear code
generated by the matrix whose columns are the points of $V(U)$ is a
$k$-dimensional linear code of length $|V(U)|$ and minimum distance at least
$|V(U)|-k$. A linear code with these parameters is an MDS code or an almost MDS
code. We will construct examples of such subspaces $U$ and groups $G$, which
include the normal rational curve, the elliptic curve, Glynn's arc from
\cite{Glynn1986} and other examples found by computer search. We conjecture
that the projection of $V(U)$ from any $k-4$ points is contained in the
intersection of two quadrics, the common zeros of two linearly independent
quadratic forms. This would be a strengthening of a classical theorem of Fano,
which itself is an extension of a theorem of Castelnuovo, for which we include
a proof using only linear algebra.
