The variety of complete flags of a finite-dimensional vector space is smooth projective

For $n=1$, the empty product is the one-point variety, there is a unique flag, and all incidence conditions are vacuous. For $n>1$, by [F1] each $\mathrm{Gr}(i,V)$ is a projective variety, and by [F2] the product $P={\prod }_{i=1}^{n-1}\mathrm{Gr}(i,V)$ is a projective variety. By [F3] each incidence condition $V_i\subseteq V_{i+1}$ defines a closed subvariety, so their intersection $\mathrm{F}\mathrm{l}(V)\subseteq P$ is a closed subvariety; by [F4] it is projective, hence complete, and its Plücker embedding in the product of projective spaces is the restriction of the Plücker embeddings of the factors.

Fix a basis $e_1,\dots ,e_n$ and its opposite coordinate flag $E_j=⟨e_{n-j+1},\dots ,e_n⟩$. The locus $U(E)$ of flags $W_{\bullet }$ satisfying $W_i\oplus E_{n-i}=V$ is open: projection of $W_i$ onto $⟨e_1,\dots ,e_i⟩$ is invertible exactly when its leading Plücker coordinate is nonzero. Each such flag is uniquely $W_i=⟨l_1,\dots ,l_i⟩$, where $l_j=e_j+{\sum }_{a>j}{c}_{aj}{e}_a$ are the columns of a lower unitriangular matrix. To construct $l_j$, use the unique vector of $W_j$ projecting to $e_j$ in the first $j$ coordinates; it has the displayed form. Nesting shows the earlier $l_i$ belong to $W_j$, and their distinct leading coordinates make them a basis. Uniqueness follows from that same projection. The coefficients $c_{aj}$ are regular on the standard Grassmannian charts of [F1], obtained by matrix inversion with the nonzero leading minors as denominators. Conversely every lower unitriangular matrix yields such a flag, with polynomial Plücker coordinates. Thus these mutually inverse regular maps identify $U(E)$ with ${\mathbf{A}}_k^{\left(\genfrac{}{}{0px}{}{n}{2}\right)}$. Every flag admits an adapted basis and is transverse to its reverse coordinate flag, so these affine-space charts cover the flag variety. These inversions and polynomial formulas also work over arbitrary $k$-algebras when the leading minors are units, so the charts parameterize locally direct-summand flags after base change. They give smoothness; for $n=1$ the chart is ${\mathbf{A}}_k^0$.

I claim that the stabilizer of a complete flag is solvable. Fix the flag $F$ given by $V_i=⟨e_1,\dots ,e_i⟩$ for a basis $e_1,\dots ,e_n$. For every $k$-algebra $R$, an $R$-point $g\in {\mathrm{G}\mathrm{L}}_n(R)$ stabilizes each $V_i{\otimes }_{k}R$ if and only if its matrix is upper triangular, since the image of $V_i\otimes R$ is spanned by the images of the first $i$ basis vectors; hence the stabilizer of $F$ is exactly the upper triangular closed subgroup scheme $T_n=D_n⋉U_n$ of [F5]. The series $T_n\supseteq U_n=U_n^{(0)}\supseteq U_n^{(1)}\supseteq \cdots \supseteq U_n^{(m)}=1$ is a normal series whose successive quotients are, in order, $D_n$ (commutative, being diagonalizable) and the quotients $U_n^{(r)}\mathrm{/}{U}_n^{(r+1)}\cong {\mathbf{G}}_a$ of the central series of [F5], which are commutative. By [F6] a group scheme with such a series is solvable, so the stabilizer of the complete flag $F$ is solvable.

I claim that $\mathrm{G}\mathrm{L}(V)$ acts transitively on $\mathrm{F}\mathrm{l}(V)$, compatibly with its action on the Grassmannians, and that the stabilizer of a flag is a closed subgroup scheme. Given two flags $V_{\bullet }$, $V_{\bullet }^{\mathrm{\prime }}$, choose bases $v_1,\dots ,v_n$ and $v_1^{\mathrm{\prime }},\dots ,v_n^{\mathrm{\prime }}$ adapted to them; the linear map $g{v}_i=v_i^{\mathrm{\prime }}$ lies in $\mathrm{G}\mathrm{L}(V)(k)$ and carries $V_i$ onto $V_i^{\mathrm{\prime }}$ for every $i$. The action of $\mathrm{G}\mathrm{L}(V)$ on $P$ preserves the incidence conditions, so it restricts to a morphism $\mathrm{G}\mathrm{L}(V)\times \mathrm{F}\mathrm{l}(V)\to \mathrm{F}\mathrm{l}(V)$, an action of the group scheme $\mathrm{G}\mathrm{L}(V)$ by [F5]; the scheme-theoretic stabilizer of a point is then a closed subgroup scheme of $\mathrm{G}\mathrm{L}(V)$. Since $\mathrm{G}\mathrm{L}(V)(k)$ acts transitively, the stabilizer of any complete flag is a $\mathrm{G}\mathrm{L}(V)(k)$-conjugate of the stabilizer $T_n$ of the standard flag, so by [step 1.3] the stabilizer of every complete flag is solvable. Moreover $\mathrm{G}\mathrm{L}(V)$ is the determinant-open integral subscheme of matrix affine space. The closure of its flag orbit is irreducible and contains every closed point by transitivity, hence is all of $\mathrm{F}\mathrm{l}(V)$; thus the flag variety is irreducible.

Collecting: [step 1.1] gives the projective closed-subvariety model of $\mathrm{F}\mathrm{l}(V)$ via incidence, [step 1.2] its smoothness, [step 2.1] the transitive action and closed stabilizer subgroups, and [step 1.3] the solvability of the stabilizer of a complete flag. This proves the statement.