  In 1901, Severi proved that if $Z$ is an irreducible hypersurface in
$\mathbb{P}^4(\mathbb{C})$ that contains a three dimensional set of lines, then
$Z$ is either a quadratic hypersurface or a scroll of planes. We prove a
discretized version of this result for hypersurfaces in $\mathbb{R}^4$. As an
application, we prove that at most $\delta^{-2-\varepsilon}$
direction-separated $\delta$-tubes can be contained in the
$\delta$-neighborhood of a low-degree hypersurface in $\mathbb{R}^4$.
  This result leads to improved bounds on the restriction and Kakeya problems
in $\mathbb{R}^4$. Combined with previous work of Guth and the author, this
result implies a Kakeya maximal function estimate at dimension $3+1/28$, which
is an improvement over the previous bound of $3$ due to Wolff. As a
consequence, we prove that every Besicovitch set in $\mathbb{R}^4$ must have
Hausdorff dimension at least $3+1/28$. Recently, Demeter showed that any
improvement over Wolff's bound for the Kakeya maximal function yields new
bounds on the restriction problem for the paraboloid in $\mathbb{R}^4$.
