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New answer to both questions. The following technique is usually called "retract rationality". The argument can be summarized as "rational varieties, such as the ambient $\mathbb{P}^n$, are retract rational". In fact, if you do not mind that $Y$ is very singular at the generic point of $X$, you can easily adapt the following argument to $X$ a subvariety of a unirational variety (of codimension $>1$). This actually leads to one of the proposed strategies for the open conjecture that there exist rationally connected varieties that are not unirational: find a rationally connected variety $P$ such that for some $X$ of codimension $>1$, there exists no rational $Y$ containing $X$. For $X$ a subvariety of $\mathbb{P}^n$, or any rational variety $P$, the following argument produces a rational subvariety $Y$ that contains $X$ and that is smooth at the generic point of $Y$.

New answer to both questions. The following technique is usually called "retract rationality". The argument can be summarized as "rational varieties, such as the ambient $\mathbb{P}^n$, are retract rational". In fact, if you do not mind that $Y$ is very singular at the generic point of $X$, you can easily adapt the following argument to $X$ a subvariety of a unirational variety (of codimension $>1$). This actually leads to one of the proposed strategies for the open conjecture that there exist rationally connected varieties that are not unirational: find a rationally connected variety $P$ such that for some $X$ of codimension $>1$, there exists no rational $Y$ containing $X$. For $X$ a subvariety of $\mathbb{P}^n$, or any rational variety $P$, the following argument produces a rational subvariety $Y$ that contains $X$ and that is smooth at the generic point of $X$.

Edit. There is a much simpler argument than the original answer below, and it also answers the second question. I will add that argument in a moment. However, I will leave the "rational simple connectedness" argument as well. The point is that the "rational simple connectedness" argument holds for any embedding of a curve $C$ in a rationally simply connected variety, e.g., a general hypersurface of degree $m$ in $\mathbb{P}^n$ so long as $m^2\leq n$.

New answer to both questions. The following technique is usually called "retract rationality". The argument can be summarized as "rational varieties, such as the ambient $\mathbb{P}^n$, are retract rational". Denote by $\ell$ the dimension of $X$. If $\ell+1$ equals $n$, the result is trivially true by choosing $Y=\mathbb{P}^n$. Thus, assume that $n> \ell+1$. By Bertini, for a general linear projection $$\rho:\mathbb{P}^n \dashrightarrow \mathbb{P}^{\ell + 1},$$ the regular locus of $\rho$ intersects $X$ in a dense open subset of $X$, and the induced rational transformation $$\rho_X:X\dashrightarrow \mathbb{P}^{\ell + 1},$$ is birational to its image. Denote by $U\subset \mathbb{P}^{\ell+1}$ a dense, affine open subset such that $\rho_X^{-1}(U)\subset X$ is a dense open subset and such that $$\rho_{X,U}:\rho_X^{-1}(U)\to U,$$ is a closed immersion. Up to shrinking $U$ further, assume that the invertible sheaf $\mathcal{O}_{\mathbb{P}^n}(1)|_X$ is trivialized on $\rho_X^{-1}(U)$. Since $\rho_{X,U}$ is a closed immersion, the pullback $k$-algebra homomorphism is surjective, $$\rho_{X,U}^\#:\mathcal{O}_{\mathbb{P}^{\ell+1}}(U) \twoheadrightarrow \mathcal{O}_X(\rho_X^{-1}(U)).$$ Thus, the restrictions to $\rho_X^{-1}(U)$ of the $n+1$ defining global sections of $\mathcal{O}_{\mathbb{P}^n}(1)$ lift to $n+1$ global sections of $\mathcal{O}_{\mathbb{P}^{\ell+1}}$ on $U$. By a Bertini-type argument, for a general choice of these $n+1$ lifts, after replacing $U$ by a dense open that still intersects $X$, the $n+1$ global sections have empty base locus, and the associated rational transformation $$j:U\to \mathbb{P}^n,$$ is birational to its image, say $Y=\overline{j(U)}\subset \mathbb{P}^n$. Thus $Y$ is a rational subvariety of $\mathbb{P}^n$ of dimension $\ell+1 = \text{dim}(X)+1$. Since the $n+1$ global sections are chosen to extend the tautological sections on $\rho_X^{-1}(U)$, the composition $j\circ \rho$ is a well-defined rational transformation that agrees with the inclusion $i:X\to \mathbb{P}^n$. Thus, $X$ is contained in an $(\ell+1)$-dimensional rational subvariety $Y$ of $\mathbb{P}^n$.

Edit. There is a much simpler argument than the original answer below, and it also answers the second question. I will add that argument in a moment. However, I will leave the "rational simple connectedness" argument as well. The point is that the "rational simple connectedness" argument holds for any embedding of a curve $C$ in a rationally simply connected variety $P$, e.g., embedded in a general hypersurface of degree $m$ in $\mathbb{P}^n$ so long as $m^2\leq n$.

New answer to both questions. The following technique is usually called "retract rationality". The argument can be summarized as "rational varieties, such as the ambient $\mathbb{P}^n$, are retract rational". In fact, if you do not mind that $Y$ is very singular at the generic point of $X$, you can easily adapt the following argument to $X$ a subvariety of a unirational variety (of codimension $>1$). This actually leads to one of the proposed strategies for the open conjecture that there exist rationally connected varieties that are not unirational: find a rationally connected variety $P$ such that for some $X$ of codimension $>1$, there exists no rational $Y$ containing $X$. For $X$ a subvariety of $\mathbb{P}^n$, or any rational variety $P$, the following argument produces a rational subvariety $Y$ that contains $X$ and that is smooth at the generic point of $Y$.

Denote by $\ell$ the dimension of $X$. If $\ell+1$ equals $n$, the result is trivially true by choosing $Y=\mathbb{P}^n$. Thus, assume that $n> \ell+1$. By Bertini, for a general linear projection $$\rho:\mathbb{P}^n \dashrightarrow \mathbb{P}^{\ell + 1},$$ the regular locus of $\rho$ intersects $X$ in a dense open subset of $X$, and the induced rational transformation $$\rho_X:X\dashrightarrow \mathbb{P}^{\ell + 1},$$ is birational to its image. Denote by $U\subset \mathbb{P}^{\ell+1}$ a dense, affine open subset such that $\rho_X^{-1}(U)\subset X$ is a dense open subset and such that $$\rho_{X,U}:\rho_X^{-1}(U)\to U,$$ is a closed immersion. Up to shrinking $U$ further, assume that the invertible sheaf $\mathcal{O}_{\mathbb{P}^n}(1)|_X$ is trivialized on $\rho_X^{-1}(U)$. Since $\rho_{X,U}$ is a closed immersion, the pullback $k$-algebra homomorphism is surjective, $$\rho_{X,U}^\#:\mathcal{O}_{\mathbb{P}^{\ell+1}}(U) \twoheadrightarrow \mathcal{O}_X(\rho_X^{-1}(U)).$$ Thus, the restrictions to $\rho_X^{-1}(U)$ of the $n+1$ defining global sections of $\mathcal{O}_{\mathbb{P}^n}(1)$ lift to $n+1$ global sections of $\mathcal{O}_{\mathbb{P}^{\ell+1}}$ on $U$. By a Bertini-type argument, for a general choice of these $n+1$ lifts, after replacing $U$ by a dense open that still intersects $X$, the $n+1$ global sections have empty base locus, and the associated rational transformation $$j:U\to \mathbb{P}^n,$$ is birational to its image, say $Y=\overline{j(U)}\subset \mathbb{P}^n$. Thus $Y$ is a rational subvariety of $\mathbb{P}^n$ of dimension $\ell+1 = \text{dim}(X)+1$. Since the $n+1$ global sections are chosen to extend the tautological sections on $\rho_X^{-1}(U)$, the composition $j\circ \rho$ is a well-defined rational transformation that agrees with the inclusion $i:X\to \mathbb{P}^n$. Thus, $X$ is contained in an $(\ell+1)$-dimensional rational subvariety $Y$ of $\mathbb{P}^n$.

New answer to both questions. Denote by $\ell$ the dimension of $X$. If $\ell+1$ equals $n$, the result is trivially true by choosing $Y=\mathbb{P}^n$. Thus, assume that $n> \ell+1$. By Bertini, for a general linear projection $$\rho:\mathbb{P}^n \dashrightarrow \mathbb{P}^{\ell + 1},$$ the regular locus of $\rho$ intersects $X$ in a dense open subset of $X$, and the induced rational transformation $$\rho_X:X\dashrightarrow \mathbb{P}^{\ell + 1},$$ is birational to its image. Denote by $U\subset \mathbb{P}^{\ell+1}$ a dense, affine open subset such that $\rho_X^{-1}(U)\subset X$ is a dense open subset and such that $$\rho_{X,U}:\rho_X^{-1}(U)\to U,$$ is a closed immersion. Up to shrinking $U$ further, assume that the invertible sheaf $\mathcal{O}_{\mathbb{P}^n}(1)|_X$ is trivialized on $\rho_X^{-1}(U)$. Since $\rho_{X,U}$ is a closed immersion, the pullback $k$-algebra homomorphism is surjective, $$\rho_{X,U}^\#:\mathcal{O}_{\mathbb{P}^{\ell+1}}(U) \twoheadrightarrow \mathcal{O}_X(\rho_X^{-1}(U)).$$ Thus, the restrictions to $\rho_X^{-1}(U)$ of the $n+1$ defining global sections of $\mathcal{O}_{\mathbb{P}^n}(1)$ lift to $n+1$ global sections of $\mathcal{O}_{\mathbb{P}^{\ell+1}}$ on $U$. By a Bertini-type argument, for a general choice of these $n+1$ lifts, after replacing $U$ by a dense open that still intersects $X$, the $n+1$ global sections have empty base locus, and the associated rational transformation $$j:U\to \mathbb{P}^n,$$ is birational to its image, say $Y=\overline{j(U)}\subset \mathbb{P}^n$. Thus $Y$ is a rational subvariety of $\mathbb{P}^n$ of dimension $\ell+1 = \text{dim}(X)+1$. Since the $n+1$ global sections are chosen to extend the tautological sections on $\rho_X^{-1}(U)$, the composition $j\circ \rho$ is a well-defined rational transformation that agrees with the inclusion $i:X\to \mathbb{P}^n$. Thus, $X$ is contained in an $(\ell+1)$-dimensional rational subvariety $Y$ of $\mathbb{P}^n$.

New answer to both questions. The following technique is usually called "retract rationality". The argument can be summarized as "rational varieties, such as the ambient $\mathbb{P}^n$, are retract rational". Denote by $\ell$ the dimension of $X$. If $\ell+1$ equals $n$, the result is trivially true by choosing $Y=\mathbb{P}^n$. Thus, assume that $n> \ell+1$. By Bertini, for a general linear projection $$\rho:\mathbb{P}^n \dashrightarrow \mathbb{P}^{\ell + 1},$$ the regular locus of $\rho$ intersects $X$ in a dense open subset of $X$, and the induced rational transformation $$\rho_X:X\dashrightarrow \mathbb{P}^{\ell + 1},$$ is birational to its image. Denote by $U\subset \mathbb{P}^{\ell+1}$ a dense, affine open subset such that $\rho_X^{-1}(U)\subset X$ is a dense open subset and such that $$\rho_{X,U}:\rho_X^{-1}(U)\to U,$$ is a closed immersion. Up to shrinking $U$ further, assume that the invertible sheaf $\mathcal{O}_{\mathbb{P}^n}(1)|_X$ is trivialized on $\rho_X^{-1}(U)$. Since $\rho_{X,U}$ is a closed immersion, the pullback $k$-algebra homomorphism is surjective, $$\rho_{X,U}^\#:\mathcal{O}_{\mathbb{P}^{\ell+1}}(U) \twoheadrightarrow \mathcal{O}_X(\rho_X^{-1}(U)).$$ Thus, the restrictions to $\rho_X^{-1}(U)$ of the $n+1$ defining global sections of $\mathcal{O}_{\mathbb{P}^n}(1)$ lift to $n+1$ global sections of $\mathcal{O}_{\mathbb{P}^{\ell+1}}$ on $U$. By a Bertini-type argument, for a general choice of these $n+1$ lifts, after replacing $U$ by a dense open that still intersects $X$, the $n+1$ global sections have empty base locus, and the associated rational transformation $$j:U\to \mathbb{P}^n,$$ is birational to its image, say $Y=\overline{j(U)}\subset \mathbb{P}^n$. Thus $Y$ is a rational subvariety of $\mathbb{P}^n$ of dimension $\ell+1 = \text{dim}(X)+1$. Since the $n+1$ global sections are chosen to extend the tautological sections on $\rho_X^{-1}(U)$, the composition $j\circ \rho$ is a well-defined rational transformation that agrees with the inclusion $i:X\to \mathbb{P}^n$. Thus, $X$ is contained in an $(\ell+1)$-dimensional rational subvariety $Y$ of $\mathbb{P}^n$.