The obvious thing to do is to choose $H$ a general hypersurface containing $X$. In other words, choose a general global section of $I_X \otimes O_{P^n}(k)$ for $k \gg 0$.

Certainly $H$ is smooth away from $X$ by Bertini. I don't see why it should be smooth along $H$ though (unless of course, $X$ is a complete intersection).

In general, you are still ok locally, in other words a regular ring is always locally a complete intersection, so in a neighborhood of every point there is such a variety which is smooth near that point (they might not glue, or be smooth elsewhere though). This follows from page 171 of Matsumura's commutative algebra. See in particular 21.2(ii).

The obvious thing to do is to choose $H$ a general hypersurface containing $X$. In other words, choose a general global section of $I_X \otimes O_{P^n}(k)$ for $k \gg 0$.

Certainly $H$ is smooth away from $X$ by Bertini. I don't see why it should be smooth along $H$ though (unless of course, $X$ is a complete intersection).

In general, you are still ok locally, in other words a regular ring is always locally a complete intersection, so in a neighborhood of every point there is such a variety which is smooth near that point (they might not glue, or be smooth elsewhere though). This follows from page 171 of Matsumura's Commutative Ring Theory. See in particular 21.2(ii).

The obvious thing to do is to choose $H$ a general hypersurface containing $X$. In other words, choose a general global section of $I_X \otimes O_{P^n}(k)$ for $k \gg 0$.

Certainly $H$ is smooth away from $X$ by Bertini. I don't see why it should be smooth along $H$ though (unless for example, $H$ is a complete intersection).

Even in general, you are still ok locally, in other words a regular ring is always locally a complete intersection, so in a neighborhood of every point there is such a smooth variety (they might not glue, or be smooth elsewhere though). Let me track down a reference for this at least.

The obvious thing to do is to choose $H$ a general hypersurface containing $X$. In other words, choose a general global section of $I_X \otimes O_{P^n}(k)$ for $k \gg 0$.

Certainly $H$ is smooth away from $X$ by Bertini. I don't see why it should be smooth along $H$ though (unless of course, $X$ is a complete intersection).

In general, you are still ok locally, in other words a regular ring is always locally a complete intersection, so in a neighborhood of every point there is such a variety which is smooth near that point (they might not glue, or be smooth elsewhere though). This follows from page 171 of Matsumura's commutative algebra. See in particular 21.2(ii).