Line bundles over an affine-space parameter open come from the smooth factor

Represent $M$ by a Cartier divisor and then by a Weil divisor on $X\times V$ using [F1]. Close each of its finitely many prime components in $X\times {\mathbf{A}}^d$ with the same coefficient. The closures have codimension one because this is an open extension of varieties. The product is smooth and integral, so [F1] makes this extended Weil divisor a Cartier divisor $E$. Its restriction represents $M$.

Write $K=k(X)$. The components of $E$ which dominate $X$ restrict to codimension-one primes of $\mathrm{Spec}K[t_1,\dots ,t_d]$, by [F2]. This is a UFD, so choose their irreducible defining polynomials and form their product, with integer exponents given by the coefficients of $E$. View that product as a nonzero rational function $h$ on $X\times {\mathbf{A}}^d$. The divisor $E-\mathrm{div}(h)$ has no component dominating $X$. A remaining codimension-one component has image closure of codimension at most one in $X$, since its fibre dimension is at most $d$ and [F2] gives the total dimension as $\dim X+d-1$. Its image is therefore a prime divisor $T$ on $X$, and its generic fibre over $T$ has dimension $d$. The affine-space fibre is integral; a closed subset of full dimension is the whole fibre. Consequently this component is precisely $T\times {\mathbf{A}}^d$. Its multiplicity as a pullback divisor is one: at the generic point the base DVR uniformizer remains a uniformizer after the purely transcendental residue-field extension.

Thus $E-\mathrm{div}(h)={\mathrm{pr}}_X^{\ast }D$ for a Weil divisor $D$ on $X$. By [F1], $D$ is Cartier and defines $N={\mathcal{O}}_X(D)$. The principal divisor does not change the invertible-sheaf class, so $\mathcal{O}(E)\cong {\mathrm{pr}}_X^{\ast }N$. Restricting to $X\times V$ proves the assertion. Base extension and then restriction to a rational parameter point returns $N$ after that base extension, proving parameter constancy. This argument neither assumes a $k$-rational point of $V$ nor assumes $k$ infinite or perfect.