2.1 Spaces of rational curves

Let $X$ be a projective variety. The scheme of morphisms ${{\rm Hom}}({{\mathbb {P}}}^{1},X)$ is equipped with an action of ${{\rm Aut}}({{\mathbb {P}}}^{1})$ that stabilizes the open subscheme ${{\rm Hom}}_{{{\rm bir}}}({{\mathbb {P}}}^{1},X)$ consisting of morphisms which are birational to their image. Moreover, this action lifts uniquely to an action on the normalization

\[ \eta : {{\rm Hom}}_{{{\rm bir}}}^{{{\rm n}}}({{\mathbb{P}}}^{1},X) \longrightarrow {{\rm Hom}}_{{{\rm bir}}}({{\mathbb{P}}}^{1},X) \]

By [Reference KollárKol99, Theorem II.2.15], there is a natural commutative diagram of normal schemes

(1)

where the horizontal arrows are principal ${{\rm Aut}}({{\mathbb {P}}}^{1})$-bundles. As a consequence, the above diagram is cartesian; moreover, $\rho$ is a ${{\mathbb {P}}}^{1}$-bundle.

In view of [Reference KollárKol99, Definition II.2.11], there is another natural commutative diagram

(2)

where ${{\rm Chow}}(X)$ denotes the Chow scheme. Moreover, $\gamma$ is finite over its image, which is the locally closed subscheme of ${{\rm Chow}}(X)$ parameterizing irreducible, geometrically rational $1$-cycles; also, $\delta$ is finite over its image, which is the universal Chow family over the above subscheme. By composing $\delta$ with the second projection, we obtain a morphism

(3)\begin{equation} \mu : {{\rm Univ}}(X) \longrightarrow X \end{equation}

such that the morphism $\rho \times \mu : {{\rm Univ}}(X) \to {{\rm RatCurves}}(X) \times X$ is finite.

The morphism (3) can be constructed alternatively as follows: composing the evaluation map

(4)\begin{equation} {{\mathbb{P}}}^{1} \times {{\rm Hom}}_{{{\rm bir}}}({{\mathbb{P}}}^{1},X) \longrightarrow X, \quad (t,f) \longmapsto f(t) \end{equation}

with the map ${{\rm id}} \times \eta : {{\mathbb {P}}}^{1} \times {{\rm Hom}}_{{{\rm bir}}}^{{{\rm n}}}({{\mathbb {P}}}^{1},X) \longrightarrow {{\mathbb {P}}}^{1} \times {{\rm Hom}}_{{{\rm bir}}}({{\mathbb {P}}}^{1},X)$, we obtain a morphism

\[ \varepsilon : {{\mathbb{P}}}^{1} \times {{\rm Hom}}_{{{\rm bir}}}^{{{\rm n}}}({{\mathbb{P}}}^{1},X) \longrightarrow X. \]

One may check that $\varepsilon$ is the composition of the quotient map

\[ \lambda : {{\mathbb{P}}}^{1} \times {{\rm Hom}}_{{{\rm bir}}}^{{{\rm n}}}({{\mathbb{P}}}^{1},X) \to {{\rm Univ}}(X) \]

with $\mu$. In particular, for any $x \in X$, we have a principal ${{\rm Aut}}({{\mathbb {P}}}^{1})$-bundle $\varepsilon ^{-1}(x) \to \mu ^{-1}(x)$ between (scheme-theoretic) fibers. The first projection ${{\rm pr}}_1 : \varepsilon ^{-1}(x) \to {{\mathbb {P}}}^{1}$ is ${{\rm Aut}}({{\mathbb {P}}}^{1})$-equivariant, and ${{\mathbb {P}}}^{1}$ may be identified with the homogeneous space ${{\rm Aut}}({{\mathbb {P}}}^{1})/{{\rm Aut}}({{\mathbb {P}}}^{1},0)$. This identifies $\varepsilon ^{-1}(x)$ with the associated fiber bundle ${{\rm Aut}}({{\mathbb {P}}}^{1})\times ^{{{\rm Aut}}({{\mathbb {P}}}^{1},0)} {{\rm pr}}_1^{-1}(0)$. Also, note that

\[ {{\rm pr}}_1^{-1}(0) \simeq \eta^{-1}({{\rm Hom}}_{{{\rm bir}}}({{\mathbb{P}}}^{1},X; 0 \mapsto x)) \]

equivariantly for ${{\rm Aut}}({{\mathbb {P}}}^{1},0)$, where ${{\rm Hom}}_{{{\rm bir}}}({{\mathbb {P}}}^{1},X; 0 \mapsto x)$ is the closed subscheme of ${{\rm Hom}}_{{{\rm bir}}}({{\mathbb {P}}}^{1},X)$ consisting of those morphisms $f$ such that $f(0) = x$. Putting all of this together, we obtain a principal ${{\rm Aut}}({{\mathbb {P}}}^{1},0)$-bundle

(5)\begin{equation} \eta^{-1}({{\rm Hom}}_{{{\rm bir}}}({{\mathbb{P}}}^{1},X; 0 \mapsto x)) \to \mu^{-1}(x). \end{equation}

We may view $\mu ^{-1}(x)$ as the space of rational curves on $X$ through $x$.

Another space of rational curves on $X$ through $x$ is constructed in [Reference KollárKol99, Theorem II.2.16]. More specifically, there is a natural commutative diagram of normal schemes

where the horizontal arrows are principal ${{\rm Aut}}({{\mathbb {P}}}^{1},0)$-bundles and the vertical arrows are ${{\mathbb {P}}}^{1}$-bundles. (Here ${{\rm Hom}}_{{{\rm bir}}}^{{{\rm n}}}({{\mathbb {P}}}^{1},X; 0 \mapsto x)$ denotes the normalization of ${{\rm Hom}}_{{{\rm bir}}}({{\mathbb {P}}}^{1},X; 0 \mapsto x)$.) In general, the relation between $\mu ^{-1}(x)$ and ${{\rm RatCurves}}(x,X)$ is not clear to us. But we will see that both share a common smooth open subscheme, consisting of the free curves through $x$; these may be defined as follows.

Let $f \in {{\rm Hom}}_{{{\rm bir}}}({{\mathbb {P}}}^{1},X)$, and $C$ its image. We say that $f$ is free if $C$ is contained in the smooth locus $X_{{{\rm sm}}}$ and the vector bundle $f^{*}(T_{X_{{{\rm sm}}}})$ is generated by its global sections, where $T_{X_{{{\rm sm}}}}$ denotes the tangent bundle of $X_{{{\rm sm}}}$.

Every free morphism $f$ satisfies $H^{1}({{\mathbb {P}}}^{1},f^{*}(T_{X_{{{\rm sm}}}})) = 0$. Thus, the above notion of freeness coincides with that of [Reference KollárKol99, Definition II.3.1] when $X$ is smooth. By [Reference KollárKol99, II.1, II.3], the free morphisms form a smooth open subscheme ${{\rm Hom}}_{{{\rm fr}}}({{\mathbb {P}}}^{1},X)$ of ${{\rm Hom}}({{\mathbb {P}}}^{1},X)$; its dimension at the point $C$ is $- K_{X_{{{\rm sm}}}} \cdot C +\dim (X)$, where $K_{X_{{{\rm sm}}}}$ stands for the canonical class of the smooth locus. We denote by ${{\rm RatCurves}}_{{{\rm fr}}}(X)$ the corresponding smooth open subscheme of ${{\rm RatCurves}}(X)$.

We also have $H^{1}({{\mathbb {P}}}^{1},f^{*}(T_{X_{{{\rm sm}}}})(-1)) = 0$ when $f$ is free. As a consequence, the free morphisms that send $0$ to $x$ form a smooth open subscheme ${{\rm Hom}}_{{{\rm fr}}}({{\mathbb {P}}}^{1},X; 0 \to x)$ of ${{\rm Hom}}({{\mathbb {P}}}^{1},X; 0 \to x)$, with dimension at $C$ being $- K_{X_{{{\rm sm}}}} \cdot C$ (see [Reference KollárKol99, Theorem II.1.7 and Proposition II.3.2] for these results).

Thus, any free morphism $f$ yields a smooth point $C$ of ${{\rm RatCurves}}(X)$. Since the evaluation map (4) is smooth along ${{\mathbb {P}}}^{1} \times f$ (see [Reference KollárKol99, Corollary II.3.5.4]), $\mu ^{-1}(x)$ is smooth at $C$ as well. As a consequence, $\mu ^{-1}(x)$ and ${{\rm RatCurves}}(x,X)$ share a common smooth open subscheme ${{\rm RatCurves}}_{{{\rm fr}}}(x,X)$, the quotient of ${{\rm Hom}}_{{{\rm fr}}}({{\mathbb {P}}}^{1},X; 0 \to x)$ by ${{\rm Aut}}({{\mathbb {P}}}^{1},0)$. The dimension of ${{\rm RatCurves}}_{{{\rm fr}}}(x,X)$ at $C$ equals $-K_{X_{{{\rm sm}}}} \cdot C - 2$.

Consider again $f \in {{\rm Hom}}_{{{\rm bir}}}({{\mathbb {P}}}^{1},X)$ with image $C$. We say that $C$ is embedded if $f$ is an immersion into $X_{{{\rm sm}}}$. This is an open property in view of [Reference KollárKol99, Lemma I.1.10.1], and hence the embedded free curves form an open subscheme ${{\rm RatCurves}}_{{{\rm emfr}}}(X)$ of ${{\rm RatCurves}}(X)$.

Proof. By the above discussion, we may view $\rho _x$ as the natural map

\[ \pi : {{\rm Hom}}_{{{\rm emfr}}}({{\mathbb{P}}}^{1},X; 0 \to x)/{{\rm Aut}}({{\mathbb{P}}}^{1},0) \longrightarrow {{\rm Hom}}_{{{\rm emfr}}}({{\mathbb{P}}}^{1},X)/{{\rm Aut}}({{\mathbb{P}}}^{1}) \]

with an obvious notation.

We check that $\pi$ is injective on $k$-rational points. For any $f_1,f_2$ in ${{\rm Hom}}_{{{\rm emfr}}}({{\mathbb {P}}}^{1},X; 0 \to x)$ such that $f_2 = f_1 \circ \varphi$ for some $\varphi \in {{\rm Aut}}({{\mathbb {P}}}^{1})$, we have $x = f_2(0) = f_1(\varphi (0)) = f_1(0)$, and hence $\varphi (0) = 0$ as desired.

Next, we check that the differential of $\pi$ at any $k$-rational point $f \in {{\rm Hom}}_{{{\rm emfr}}}({{\mathbb {P}}}^{1},X; 0 \to x)$ is injective. We have a commutative diagram (with an obvious notation again)

where $df_0$ is injective. So the induced map

\[ H^{0}({{\mathbb{P}}}^{1}, f^{*}(T_{X_{{{\rm sm}}}})(-1)))/ {{\rm Lie}} {{\rm Aut}}({{\mathbb{P}}}^{1},0) \longrightarrow H^{0}({{\mathbb{P}}}^{1}, f^{*}(T_{X_{{{\rm sm}}}}))/ {{\rm Lie}} {{\rm Aut}}({{\mathbb{P}}}^{1}) \]

is injective as well, as desired.

2.2 Families of rational curves

The normal scheme ${{\rm RatCurves}}(X)$ is the disjoint union of open and closed normal quasi-projective varieties. These (connected or irreducible) components are called the families of rational curves on $X$. Every such family ${{\mathcal {K}}}$ comes with a universal family ${{\mathcal {U}}} := \rho ^{-1}({{\mathcal {K}}}) \to {{\mathcal {K}}}$, where ${{\mathcal {U}}}$ is a component of ${{\rm Univ}}(X)$. For any $x \in X$, we denote the fiber of $\mu : {{\mathcal {U}}} \to X$ by ${{\mathcal {U}}}_x$, and let ${{\mathcal {K}}}_x := \rho ({{\mathcal {U}}}_x)$; then the induced morphism $\rho _x : {{\mathcal {U}}}_x \to {{\mathcal {K}}}_x$ is finite. The family ${{\mathcal {K}}}$ is covering if $\mu$ is dominant, i.e. ${{\mathcal {K}}}_x$ (or equivalently ${{\mathcal {U}}}_x$) is non-empty for a general point $x$. If in addition ${{\mathcal {K}}}_x$ (or equivalently ${{\mathcal {U}}}_x$) is projective for $x$ general, then ${{\mathcal {K}}}$ is called a family of minimal rational curves, or just a minimal family for simplicity. Examples of minimal families include the covering families of lines in some projective embedding of $X$; also, note that lines contained in the smooth locus yield examples of embedded curves.

For any family of rational curves ${{\mathcal {K}}}$ as above, there exists a unique irreducible component $Z$ of ${{\rm Hom}}_{{{\rm bir}}}({{\mathbb {P}}}^{1},X)$ together with a principal ${{\rm Aut}}({{\mathbb {P}}}^{1})$-bundle $Z^{{{\rm n}}} \to {{\mathcal {K}}}$, where $Z^{{{\rm n}}}$ denotes the normalization. Let $Z_{{{\rm fr}}}$ denote the smooth open subscheme of $Z$ consisting of free morphisms; then $Z_{{{\rm fr}}}$ is stable under ${{\rm Aut}}({{\mathbb {P}}}^{1})$, and hence we obtain a principal ${{\rm Aut}}({{\mathbb {P}}}^{1})$-bundle $Z_{{{\rm fr}}} \to {{\mathcal {K}}}_{{{\rm fr}}}$, where ${{\mathcal {K}}}_{{{\rm fr}}} \subset {{\mathcal {K}}}$ denotes the smooth open subscheme of free curves. We may view the points of ${{\mathcal {K}}}_{{{\rm fr}}}$ as (possibly singular) rational curves in $X$. For any such curve $C$, the fiber of $\rho$ at $C$ is a projective line, and the restriction of $\mu$ to this fiber yields the normalization map of $C$. Also, given $x \in X_{{{\rm sm}}}$, the morphism $\rho _x : {{\mathcal {U}}}_x \to {{\mathcal {K}}}_x$ restricts to an isomorphism on the open subscheme ${{\mathcal {K}}}_{{{\rm emfr}},x}$ consisting of embedded free curves (Lemma 2.1). Moreover, by assigning to each such curve its tangent direction at $x$, we obtain a morphism

(6)\begin{equation} \tau = \tau_{X,x} : {{\mathcal{K}}}_{{{\rm emfr}},x} \longrightarrow {{\mathbb{P}}}(T_x X), \end{equation}

where ${{\mathbb {P}}}(T_x X)$ denotes the projectivization of the tangent space. If ${{\rm char}}(k) = 0$ and the point $x$ is general, then $\tau$ extends to a finite morphism from the normalization of ${{\mathcal {K}}}_x$ (see [Reference KebekusKeb02, Theorem 3.4]).

Remark 2.2 Assume that $X$ is smooth. If ${{\rm char}}(k) = 0$ then every covering family contains a free curve, as follows from [Reference KollárKol99, Proposition II.3.10] together with generic smoothness. But this fails whenever ${{\rm char}}(k) = p > 0$, as shown by the following example adapted from [Reference KollárKol99, Example V.1.4.3]. Consider the hypersurface $X$ in ${{\mathbb {P}}}^{2} \times {{\mathbb {P}}}^{2}$ with homogeneous equation

\[ x_0 y_0^{p} + x_1 y_1^{p} + x_2 y_2^{p} = 0, \]

where $x_0, x_1,x_2$ (respectively $y_0, y_1, y_2$) are homogeneous coordinates on the first (respectively second) copy of ${{\mathbb {P}}}^{2}$. Then $X$ is smooth, the geometric fibers of the first projection ${{\rm pr}}_1 : X \to {{\mathbb {P}}}^{2}$ are all non-reduced, and their reduced subschemes are lines. For the corresponding family of rational curves, ${{\mathcal {U}}}$ is the hypersurface in ${{\mathbb {P}}}^{2} \times {{\mathbb {P}}}^{2}$ with equation $x_0 y_0 + x_1 y_1 + x_2 y_2 = 0$, and ${{\mathcal {K}}} = {{\mathbb {P}}}^{2}$; in particular, ${{\mathcal {K}}}$ is minimal. Also, the morphism $\rho$ is the first projection, and

\[ \mu\big( [x_0: x_1: x_2], [y_0: y_1: y_2]\big) := \big([x_0^{p} : x_1^{p} : x_2^{p}], [y_0: y_1: y_2]\big). \]

In particular, all the geometric fibers of $\mu : {{\mathcal {U}}} \to {{\mathcal {K}}}$ are fat points of multiplicity $p$, and hence ${{\mathcal {K}}}$ contains no free curve. Note that $X$ is homogeneous under an appropriate action of ${{\rm Aut}}({{\mathbb {P}}}^{2})$, and the stabilizer of any $x \in X$ is not reduced (or equivalently, not smooth). So $X$ is a variety of unseparated flags in the sense of [Reference Haboush and LauritzenHL93]; one can show that any such variety admits a minimal family which contains no free curve.

We now discuss covariance properties under a morphism $\pi : X \to Y$, where $Y$ is a projective variety. Let ${{\mathcal {K}}}$ be a family of rational curves on $X$, and $Z$ the corresponding irreducible component of ${{\rm Hom}}_{{{\rm bir}}}({{\mathbb {P}}}^{1},X)$. Assume that there exists $f \in Z$ such that the composition $\pi \circ f: {{\mathbb {P}}}^{1} \to Y$ is free. Then $\pi \circ f$ is a smooth point of a unique irreducible component $W$ of ${{\rm Hom}}_{{{\rm bir}}}({{\mathbb {P}}}^{1},Y)$, which defines in turn a family of rational curves ${{\mathcal {L}}}$ on $Y$. The composition of natural morphisms $Z^{{{\rm n}}} \to Z \to {{\rm Hom}}({{\mathbb {P}}}^{1},X) \to {{\rm Hom}}({{\mathbb {P}}}^{1},Y)$ is ${{\rm Aut}}({{\mathbb {P}}}^{1})$-equivariant and its image contains a smooth point of $W$. This yields a rational map

\[ \pi_* : {{\mathcal{K}}} -- \to {{\mathcal{L}}}, \]

which is defined on the open subset consisting of those free curves that are sent to free curves in $Y$.

In the opposite direction, given $\pi$ and ${{\mathcal {K}}}$ as above, we say that $\pi$ contracts some $C_0 \in {{\mathcal {K}}}$ if the composition $\rho ^{-1}(C_0) \stackrel {\mu }{\longrightarrow } X \stackrel {\pi }{\longrightarrow } Y$ is constant. Then $\pi$ contracts all $C \in {{\mathcal {K}}}$: indeed, choose an ample line bundle $M$ on $Y$ and let $L := \mu ^{*} \pi ^{*}(M)$. Then the degree of $L$ on $\rho ^{-1}(C)$ is independent of $C$ (as follows e.g. from [Reference FultonFul98, Proposition 10.3]), and this degree is $0$ if and only if $C$ is contracted by $\pi$.

Based on this observation, we now describe the minimal families in a product of varieties.

Proof. One may easily check that the ‘constant’ morphism

\[ c : Z \longrightarrow {{\rm Hom}}({{\mathbb{P}}}^{1},Z), \quad z \longmapsto (t \mapsto z) \]

is a closed immersion; moreover, the diagram

is cartesian, where the top horizontal arrow is the projection, and the bottom horizontal arrow is the composition with $\varphi$. Thus, $\pi ^{*}$ is a closed immersion as well. Also, $\pi ^{*}$ sends ${{\rm Hom}}_{{{\rm bir}}}({{\mathbb {P}}}^{1},Y) \times Z$ to ${{\rm Hom}}_{{{\rm bir}}}({{\mathbb {P}}}^{1},X)$, and its image (considered with its reduced scheme structure) consists of those morphisms $f$ such that $\deg \, \varphi ^{*} f^{*}(M) = 0$ for a given ample line bundle $M$ on $Z$. This readily yields the assertions (i), (ii) and (iii).

(iv) Let ${{\mathcal {K}}}$ be a minimal family on $X$, and $x \in X$ such that ${{\mathcal {U}}}_x$ is non-empty and projective. Choose $f \in {{\rm Hom}}_{{{\rm bir}}}({{\mathbb {P}}}^{1}, X; 0 \mapsto x)$ with image $C$ such that $C$ is also the image of a curve in ${{\mathcal {U}}}_x$. Write $x = (y,z)$ and $f = (g,h)$, where $g \in {{\rm Hom}}({{\mathbb {P}}}^{1}, Y; 0 \mapsto y)$ and $h \in {{\rm Hom}}({{\mathbb {P}}}^{1}, Z; 0 \mapsto z)$. We may view $f$ as the composition

\[ {{\mathbb{P}}}^{1} \stackrel{\delta}{\longrightarrow} {{\mathbb{P}}}^{1} \times {{\mathbb{P}}}^{1} \stackrel{g \times h}{\longrightarrow} Y \times Z, \]

where $\delta$ denotes the diagonal embedding. If both $g$ and $h$ are non-constant, then $g \times h$ is finite. Since the image of $\delta$ degenerates in ${{\mathbb {P}}}^{1} \times {{\mathbb {P}}}^{1}$ to $({{\mathbb {P}}}^{1} \times \{ 0 \}) \cup (\{ 0 \} \times {{\mathbb {P}}}^{1})$, a reducible curve through $(0,0)$, this yields a degeneration of $C$ in $X$ to a reducible curve through $x$. But this contradicts the minimality of ${{\mathcal {K}}}$. Thus, we may assume that $g$ is constant; then (iii) implies that ${{\mathcal {K}}}$ is the pull-back of a minimal family on $Y$.

2.3 Almost homogeneous varieties

We now assume that the projective variety $X$ is equipped with an action of a connected algebraic group $G$. Then $G$ acts on ${{\rm Hom}}({{\mathbb {P}}}^{1},X)$ via its action on $X$, which commutes with the action of ${{\rm Aut}}({{\mathbb {P}}}^{1})$ and stabilizes the open subscheme ${{\rm Hom}}_{{{\rm bir}}}({{\mathbb {P}}}^{1},X)$. This yields actions of $G$ on ${{\rm Hom}}_{{{\rm bir}}}^{{{\rm n}}}({{\mathbb {P}}}^{1},X)$, ${{\rm RatCurves}}(X)$, ${{\rm Univ}}(X)$ such that the diagram (1) is equivariant. Also, $G$ acts on the Chow scheme and the diagram (2) is equivariant as well. Thus, so is the morphism (3). Since $G$ is connected, every family of rational curves on $X$ is stable by $G$.

Next, assume that there exists a point $x \in X$ such that the orbit $X^{0} = G \cdot x$ is open in $X$, i.e. $X$ is almost homogeneous under $G$. Then there exist covering families of rational curves on $X$, since $G$ is a rational variety. We assume in addition that the (scheme-theoretic) stabilizer $H = G_x$ is smooth and connected.

Consider a family ${{\mathcal {K}}}$ of rational curves on $X$. Let ${{\mathcal {U}}}^{0} := \mu ^{-1}(X^{0})$; this is an open $G$-stable subset of ${{\mathcal {U}}}$. Since $\rho : {{\mathcal {U}}} \to {{\mathcal {K}}}$ is flat, $\rho ({{\mathcal {U}}}^{0}) =: {{\mathcal {K}}}^{0}$ is a $G$-stable open subset of ${{\rm RatCurves}}(X)$; it consists of those curves in ${{\mathcal {K}}}$ that meet $X^{0}$. We also have a smaller $G$-stable open subset ${{\mathcal {K}}}(X^{0})$, consisting of those curves that are contained in $X^{0}$. This yields a commutative diagram of $G$-varieties

(7)

where the horizontal arrows are open immersions, the left and right vertical arrows are ${{\mathbb {P}}}^{1}$-bundles, and the middle vertical arrow is smooth.

Proof. The morphism $\mu$ restricts to a $G$-equivariant morphism $\mu ^{0}: {{\mathcal {U}}}^{0} \longrightarrow X^{0}$ with fiber at $x$ being ${{\mathcal {U}}}_x$. By identifying $X^{0}$ with the homogeneous space $G/H$, this yields a $G$-equivariant isomorphism ${{\mathcal {U}}}^{0} \simeq G \times ^{H} {{\mathcal {U}}}_x$, and in turn a cartesian square

where the vertical arrows are principal $H$-bundles. Since ${{\mathcal {U}}}^{0}$ is a normal variety and $H$ is smooth and connected, it follows that ${{\mathcal {U}}}_x$ is a normal variety as well. Moreover, ${{\mathcal {U}}}_x$ is non-empty if and only if so is ${{\mathcal {U}}}^{0}$, or equivalently ${{\mathcal {K}}}^{0}$; this completes the proof.

Proof. (i) The smooth locus $X_{{{\rm sm}}}$ is $G$-stable and contains the open orbit $X^{0} \simeq G/H$. Thus, the tangent bundle $T_{X_{{{\rm sm}}}}$ is equipped with a space of global sections: the image of the Lie algebra $\mathfrak {g}$ of $G$. Since $H$ is smooth, this space generates $T_{X^{0}}$. Thus, the induced map ${{\mathcal {O}}}_{{{\mathbb {P}}}^{1}} \otimes \mathfrak {g} \to f^{*}(T_{X_{{{\rm sm}}}})$ is generically surjective, where $f: {{\mathbb {P}}}^{1} \to C$ denotes the normalization. Using the fact that every vector bundle on ${{\mathbb {P}}}^{1}$ is a direct sum of line bundles, it follows easily that $f^{*}(T_{X_{{{\rm sm}}}})$ is globally generated.

(ii) This is a consequence of Lemma 2.4.

(iii) This follows from the properties of free morphisms recalled in § 2.1.

(iv) By (ii) and (5), there is a principal ${{\rm Aut}}({{\mathbb {P}}}^{1},0)$-bundle $\eta ^{-1}(Y) \to {{\mathcal {U}}}_x$ for some irreducible component $Y$ of ${{\rm Hom}}_{{{\rm bir}}}({{\mathbb {P}}}^{1},X; 0 \mapsto x)$; moreover, $\eta ^{-1}(Y)$ is a normal variety. By the universal property of the normalization, this yields a finite morphism

\[ \varphi : \eta^{-1}(Y) \to {{\rm Hom}}_{{{\rm bir}}}^{{{\rm n}}}({{\mathbb{P}}}^{1},X; 0 \mapsto x), \]

which restricts to an isomorphism on the open subset of free morphisms. Thus, $\varphi$ yields an isomorphism to a component of ${{\rm Hom}}_{{{\rm bir}}}^{{{\rm n}}}({{\mathbb {P}}}^{1},X; 0 \mapsto x)$. As $\varphi$ is ${{\rm Aut}}({{\mathbb {P}}}^{1},0)$-equivariant, it descends to the desired isomorphism.

Next, we consider covariance properties of covering families, building on the observations after Remark 2.2. Let $X$ be as above, and $\pi : X \to Y$ a surjective morphism, where $Y$ is a projective variety. Assume that $Y$ is equipped with a $G$-action such that $\pi$ is equivariant (this assumption holds if $\pi _* {{\mathcal {O}}}_X = {{\mathcal {O}}}_Y$ in view of Blanchard's lemma, see e.g. [Reference Brion, Samuel and UmaBSU13, § 4.2]). Let $y:= \pi (x)$ and $Y^{0} := Gy$; then $Y^{0}= \pi (X^{0})$ is open in $Y$. Finally, let ${{\mathcal {K}}}$ be a covering family of rational curves on $X$.

Remark 2.7 Under the assumptions of the above lemma, $\pi _*$ restricts to an $H$-equivariant rational map $\pi _* = \pi _{*,x} : {{\mathcal {K}}}_x -- \to {{\mathcal {L}}}_y$. (Indeed, replacing $C$ with the translate $gC$ for some $g \in G$, we may assume that $x \in C$, and hence $y \in D$.) Moreover, we have a commutative diagram of rational maps

(8)

where the horizontal arrows arise from the tangent maps (6).

The assumptions of Lemma 2.6 hold if $\pi$ is birational and ${{\mathcal {K}}}(X^{0})$ is non-empty; then $\pi _*$ restricts to an isomorphism ${{\mathcal {K}}}(X^{0}) \to {{\mathcal {L}}}(X^{0})$, and hence is birational. The assumptions also hold when $\pi$ is birational and $X,Y$ are smooth; then $\pi _*$ is an injective morphism.

3.1 Flag varieties

Let $X$ be a projective variety, homogeneous under the action of a connected linear algebraic group $G$. Choose $x \in X$ and assume that the stabilizer $G_x$ is smooth. Then $G_x$ is a parabolic subgroup of $G$, and hence is connected. Moreover, replacing $G$ with its largest semi-simple quotient and then with its simply-connected cover, we may and will assume that $G$ is semi-simple and simply-connected. We identify $X$ with the homogeneous space $G/P$, where $P := G_x$. The Lie algebras of $G, G_x, P, \ldots$ will be denoted by $\mathfrak {g},\mathfrak {g}_x,\mathfrak {p}, \ldots\,$.

Choose a Borel subgroup $B \subset P$ and a maximal torus $T \subset B$. Let $R$ denote the root system of $(G,T)$, and $R^{+}$ the subset of roots of $(B,T)$; then $R^{+}$ is a set of positive roots of $R$. Denote by $R^{-}$ the corresponding set of negative roots, and by $S = \{ \alpha _1, \ldots , \alpha _n \} \subset R^{+}$ the set of simple roots. The Weyl group $W = N_G(T)/T$ is generated by the associated simple reflections $s_1, \ldots ,s_n$. For any $w \in W$, we denote by $\dot w \in N_G(T)$ a representative. Also, for any $\beta \in R$, we denote by $U_{\beta } \subset G$ the corresponding root subgroup. Let $G_{\beta } \subset G$ denote the subgroup generated by $U_{\beta }$ and $U_{- \beta }$; then $G_{\beta }$ is a closed semi-simple subgroup of rank $1$, normalized by $T$. For any $w \in W$, the conjugation by $\dot w$ sends $U_{\beta }$ to $U_{w(\beta )}$, and $G_{\beta }$ to $G_{w(\beta )}$.

We also have the coroot system $R^{\vee }$ with simple roots $\alpha _1^{\vee }, \ldots ,\alpha _n^{\vee }$; these form a basis of the cocharacter lattice $X_*(T)$. The dual basis of the character lattice $X^{*}(T)$ consists of the fundamental weights $\varpi _1,\ldots , \varpi _n$. More intrinsically, for any simple root $\alpha$, we will denote by $\varpi _{\alpha }$ the fundamental weight with value $1$ at $\alpha ^{\vee }$, and $0$ at all other simple coroots. Let $\rho := \varpi _1 + \cdots + \varpi _n$; then $\rho = ({1}/{2}) \sum _{\alpha \in R^{+}} \alpha$. The height of any $\beta \in R^{\vee }$ is ${{\rm ht}}(\beta ) := \langle \rho , \beta \rangle$; this equals the sum of the coordinates of $\beta$ in the basis of simple coroots.

Consider the Levi decomposition $P = R_u(P) L$, where $L$ is a connected reductive subgroup of $G$ containing $T$; then $B_L := B \cap L$ is a Borel subgroup of $L$. Denote by $R_L \subset R$ the root system of $(L,T)$, with subset of positive roots $R^{+}_L = R_L \cap R^{+}$ and subset of simple roots $I := R_L \cap S$. Then $P$ is generated by $B$ and the $\dot {s}_{\alpha }$, where $\alpha \in I$; we write $P = P_I$ and $L = L_I$.

The character group $X^{*}(P)$ is identified via restriction to the subgroup of $X^{*}(T)$ with basis the $\varpi _{\alpha }$, where $\alpha \in S \setminus I$. Also, every $\lambda \in X^{*}(P)$ defines the associated line bundle ${{\mathcal {L}}}_{G/P}(\lambda )$ on $G/P$; moreover, ${{\mathcal {L}}}_{G/P}(\lambda )$ is ample if and only if $\lambda$ has positive coordinates in the above basis. The assignment $\lambda \mapsto {{\mathcal {L}}}_{G/P}(\lambda )$ yields an isomorphism $X^{*}(P) \simeq {{\rm Pic}}(G/P)$. In particular, $G/P$ has a smallest ample line bundle, namely ${{\mathcal {L}}}_{G/P}(\varpi )$, where $\varpi = \varpi _I := \sum _{\alpha \in S \setminus I} \varpi _{\alpha }$. Every ample line bundle on $G/P$ is very ample, and hence defines a projective embedding $G/P \hookrightarrow {{\mathbb {P}}}(V(\lambda ))$, where $V(\lambda ) := H^{0}(G/P,{{\mathcal {L}}}_{G/P}(\lambda ))^{*}$, and ${{\mathbb {P}}}(V)$ denotes the projective space of lines in a vector space $V$. This embedding is equivariant for the natural action of $G$ on $G/P$, and its linear representation in $V(\lambda )$ (a highest weight module, see e.g. [Reference JantzenJan03, § II.2.13]). In particular, we have a ‘smallest’ projective embedding

(9)\begin{equation} G/P \hookrightarrow {{\mathbb{P}}}(V(\varpi)). \end{equation}

Also, recall that the canonical class of $X$ satisfies

(10)\begin{equation} {{\mathcal{O}}}(K_X) \simeq {{\mathcal{L}}}_{G/P}(- 2 (\rho - \rho_I)). \end{equation}

The parabolic subgroup $P$ is maximal if and only if $I$ is the complement of a unique simple root $\alpha$. We then write $P = P^{\alpha }$. More generally, we will use the notation $P^{S \setminus I}$ for $P_I$ whenever this is convenient.

3.2 Schubert varieties

We keep the notation and assumptions of the previous subsection. The Weyl group $W_L = N_L(T)/T$ is generated by the simple reflections $s_{\alpha }$, where $\alpha \in I$; we also denote this group by $W_I$. Let $W^{I}$ denote the subset of $W$ consisting of those $w$ such that $w(\alpha ) \in R^{+}$ for all $\alpha \in I$; equivalently, $R^{+}_I \subset w^{-1}(R^{+})$. Then $W^{I}$ is a set of representatives of the coset space $W/W_I$, consisting of the elements of minimal length in their right coset (for the length function $\ell$ on $W$ relative to the generators $s_1, \ldots , s_n$). Note that $w \in W^{I}$ has length $1$ if and only if $w = s_{\alpha }$ for some $\alpha \in S \setminus I$. On the other hand, the unique element of maximal length in $W^{I}$ is $w_0 w_{0,I}$, where $w_0$ (respectively $w_{0,I}$) denotes the longest element of $W$ (respectively $W_I$).

For any $w \in W$, the point ${\dot w} x \in G/P$ is independent of the choice of the representative $\dot w$; we thus denote this point by $wx$. Recall that the $wx$, where $w \in W^{I}$, are exactly the $T$-fixed points in $G/P$; moreover, $G/P$ is the disjoint union of the $B$-orbits $B wx$. The stabilizer $B_{wx}$ is generated by $T$ and the root subgroups $U_{\beta }$, where $\beta \in R^{+} \cap w(R^{+})$; in particular, $B_{wx}$ is smooth and connected. The closure of $Bwx$ in $G/P$ is the Schubert variety $X(w)$; we have $\dim (X(w)) = \dim (Bwx) = \ell (w)$. In particular, the Schubert varieties of dimension $1$ are exactly the $X(s_{\alpha })$, where $\alpha \in S \setminus I$. Note that $X(s_{\alpha }) = P_{\alpha } x = L_{\alpha } x = G_{\alpha } x = \overline {U_{-\alpha } x}$ is a $T$-stable curve in $G/P$ with fixed points $x, s_{\alpha }x$.

We now collect some basic properties of $T$-stable curves, which are essentially known (see [Reference Fulton and WoodwardFW04, §§ 3 and 4]); we will provide proofs for completeness.

Proof. Using the action of $N_G(T)$ on $G/P$ which yields a transitive action of $W$ on $T$-fixed points, we may reduce to the case where $w = 1$.

By the Bruhat decomposition, we have a $T$-equivariant open immersion

\[ \prod_{\beta \in R^{+} \setminus R^{+}_I} U_{- \beta} \longrightarrow G/P, \quad (g_{\beta}) \longmapsto \biggl(\prod_{\beta} g_{\beta}\biggr) x, \]

where the product is taken in any order. Thus, $x$ has an open $T$-stable neighborhood in $G/P$, isomorphic to an affine space on which $T$ acts linearly with weights being the roots $-\beta$, where $\beta \in R^{+} \setminus R^{+}_I$; moreover, each such weight has multiplicity $1$. It follows that the $T$-stable curves in $G/P$ through $x$ are exactly the closures of the $T$-stable lines in this neighborhood, i.e. of the orbits $U_{-\beta } x$; moreover, the orbit maps $U_{- \beta } \to U_{- \beta } x$ are isomorphisms. Since $x$ is fixed by $G_{\beta } \cap B$, a Borel subgroup of $G_{\beta }$, we have

\[ C_{1,\beta} = \overline{U_{-\beta} x} = G_{\beta} x \simeq G_{\beta}/(B \cap G_{\beta}). \]

This implies (i), (ii) and (iii).

(iv) The restriction to $C_{1,\beta }$ of the $G$-linearized line bundle ${{\mathcal {L}}}_{G/P}(\lambda )$ is the $G_{\beta }$-linearized line bundle ${{\mathcal {L}}}_{G_{\beta }/(B \cap G_{\beta })}(\lambda )=: {{\mathcal {L}}}$. Moreover, $T \cap G_{\beta }$ is a maximal torus of $G_{\beta }$, the image of the coroot $\beta ^{\vee } : {{\mathbb {G}}}_m \to T$; the scheme-theoretic center of $G_{\beta } \simeq {{\rm SL}}(2)$ is the image of the $2$-torsion subgroup scheme $\mu _2 \subset {{\mathbb {G}}}_m$ under $\beta ^{\vee }$. Also, $\beta ^{\vee }$ acts on the fiber of ${{\mathcal {L}}}$ at $x$ (respectively $s_{\beta } x$) via the weight $\langle \lambda , \beta ^{\vee } \rangle$ (respectively $\langle s_{\beta }(\lambda ), \beta ^{\vee } \rangle = - \langle \lambda , \beta ^{\vee } \rangle$). Identifying $C_{1,\beta }$ with ${{\mathbb {P}}}^{1}$, it follows easily that ${{\mathcal {L}}} \simeq {{\mathcal {O}}}_{{{\mathbb {P}}}^{1}}(n)$, where $n := \langle \lambda , \beta ^{\vee } \rangle$.

(v) The first equality follows readily from (iv) in view of the isomorphism (10). As $\rho - w_{0,I}(\rho ) = 2 \rho _I$, we obtain $2(\rho - \rho _I) = \rho + w_{0,I}(\rho )$; this implies the second equality.

By Lemma 3.1, every one-dimensional Schubert variety $X(s_{\alpha })$ satisfies

\[ {{\mathcal{L}}}_{G/P}(\varpi) \cdot X(s_{\alpha}) = \langle \varpi, \alpha^{\vee} \rangle = 1, \]

that is, $X(s_{\alpha })$ is a line in ${{\mathbb {P}}}(V(\varpi ))$. We thus say that $X(s_{\alpha })$ is a Schubert line.

Also, note that every $T$-stable curve in $G/P$ is a line if and only if we have $\langle \varpi , \beta ^{\vee } \rangle \leq 1$ for all $\beta \in R^{+}$, i.e. the dominant weight $\varpi$ is minuscule. Then the parabolic subgroup $P$ is also called minuscule; it is maximal if $G$ is simple. The projective homogeneous space $G/P$ is called minuscule as well. Moreover, the weights of $T$ in $V(\varpi )$ are exactly the $w(\varpi )$, where $w \in W$; as a consequence, the $G$-module $V(\varpi )$ is simple.

3.3 Lines on flag varieties

We still keep the notation and assumptions of § 3.1, and consider a minimal family ${{\mathcal {K}}}$ on $X = G/P$.

With the notation of Lemma 3.2, we have

\[ P_{I \cup \{ \alpha \} } x \simeq P_{I \cup \{ \alpha \} }/P_I \simeq L_{I \cup \{ \alpha \} }/(P_I \cap L_{I \cup \{ \alpha \} } ). \]

Moreover, the scheme-theoretic intersection $P_I \cap L_{I \cup \{ \alpha \} }$ is smooth (see [Reference BorelBor91, Corollary 13.21]), and hence is a maximal parabolic subgroup of the connected reductive group $L_{I \cup \{ \alpha \} }$. So this lemma reduces the description of minimal families on $G/P$ to the case where $P$ is maximal; then there is a unique such family, and it consists of the lines in $G/P \subset {{\mathbb {P}}}(V(\varpi _{\alpha }))$, where $P = P^{\alpha }$. We may further assume that $G$ is simple; if in addition the simple root $\alpha$ is long, then we have the following result, which is known over the field of complex numbers (see [Reference Hwang and MokHM02, Proposition 1] and [Reference Landsberg and ManivelLM03, Theorem 4.8]).

Proof. By § 2.1 and Lemma 3.2(i), we have $\dim ({{\mathcal {L}}}_x) = - K_X \cdot X(s_{\alpha }) - 2$. Using Lemma 3.1(v), this yields

\[ \dim({{\mathcal{L}}}_x) = 2 \langle \rho - \rho_I, \alpha^{\vee} \rangle -2 = -2 \langle \rho_I, \alpha^{\vee} \rangle = - \sum_{\beta \in R^{+}_I} \langle \beta,\alpha^{\vee} \rangle. \]

Note that $0 \leq - \langle \beta , \alpha ^{\vee } \rangle \leq 1$ for all $\beta \in R^{+}_I$, since $\alpha$ is a long simple root and differs from all the simple roots occurring in $\beta$. As a consequence,

(11)\begin{equation} \dim({{\mathcal{L}}}_x) = \#(R^{+}_I \setminus R^{+}_{I \cap \alpha^{\perp}}). \end{equation}

Next, observe that the tangent map yields a morphism $\tau : {{\mathcal {L}}}_x \to {{\mathbb {P}}}(T_x X)$, since ${{\mathcal {L}}}_x$ consists of embedded free curves. Also, $T_x X \simeq \mathfrak {g}/\mathfrak {p}$ and this identifies $\tau (X(s_{\alpha }))$ with $[\mathfrak {g}_{-\alpha }]$, the image of the root subspace $\mathfrak {g}_{-\alpha } \subset \mathfrak {g}$ in ${{\mathbb {P}}}(\mathfrak {g}/\mathfrak {p})$. Since $\tau$ is $P$-equivariant, we have the inclusion of stabilizers $L_{X(s_{\alpha })} \subset L_{[\mathfrak {g}_{-\alpha }]}$. We now show that

(12)\begin{equation} L_{X(s_{\alpha})} = L_{[\mathfrak{g}_{-\alpha}]} = L \cap P_{\alpha^{\perp}}. \end{equation}

The Lie algebra $\mathfrak {l}_{[\mathfrak {g}_{-\alpha }]}$ of $L_{[\mathfrak {g}_{-\alpha }]}$ is a subalgebra of $\mathfrak {l}$ containing the Borel subalgebra $\mathfrak {b} \cap \mathfrak {l}$, and hence is generated by $\mathfrak {b} \cap \mathfrak {l}$ and the $\mathfrak {g}_{-\beta }$, where $\beta \in I$ and $\mathfrak {g}_{-\beta }$ stabilizes $[\mathfrak {g}_{-\alpha }]$. The latter condition is equivalent to $[ \mathfrak {g}_{- \beta }, \mathfrak {g}_{-\alpha } ] \subset \mathfrak {g}_{-\alpha } + \mathfrak {p}$. But $[ \mathfrak {g}_{- \beta }, \mathfrak {g}_{-\alpha } ] = \mathfrak {g}_{- \alpha - \beta }$ by a result of Chevalley (see e.g. [Reference HumphreysHum72, § 25.2]), and $\mathfrak {g}_{- \alpha - \beta } = 0$ if $- \alpha - \beta$ is not a root. In any case, $- \alpha - \beta$ is not a root of $P$. Thus, the Lie algebra $\mathfrak {l}_{[\mathfrak {g}_{-\alpha }]}$ is generated by $\mathfrak {b} \cap \mathfrak {l}$ and the $\mathfrak {g}_{- \beta }$, where $\beta \in I$ and $[ \mathfrak {g}_{- \beta }, \mathfrak {g}_{-\alpha } ] = 0$; equivalently, $\beta \in \alpha ^{\perp }$. In other terms, $\mathfrak {l}_{[\mathfrak {g}_{-\alpha }]} = \mathfrak {l} \cap \mathfrak {p}_{\alpha ^{\perp }}$. On the other hand, $L \cap P_{\alpha ^{\perp }}$ stabilizes $X(s_{\alpha })$ and hence $[\mathfrak {g}_{- \alpha }]$. Thus,

\[ L \cap P_{\alpha^{\perp}} \subset L_{X(s_{\alpha})} \subset L_{[\mathfrak{g}_{-\alpha}]} \]

and equality holds for the corresponding Lie algebras. Since $L \cap P_{\alpha ^{\perp }}$ is smooth, this easily implies the equalities (12). In turn, this yields (iii) and also the inequalities

\[ \dim({{\mathcal{L}}}_x) \geq \dim(L X(s_{\alpha})) = \dim(L_I/(L_I \cap P_{\alpha^{\perp}})) = \#(R^{+}_I \setminus R^{+}_{I \cap \alpha^{\perp}}). \]

By (11), it follows that ${{\mathcal {L}}}_x = L X(s_{\alpha })$, proving (i). Finally, (ii) follows from (i) and (12).

Remark 3.4 We still assume that $G$ is simple and $P = P^{\alpha }$.

(i) Assume in addition that $G$ is simply-laced, i.e. all the roots have the same length. Then $\alpha$ is long and $- \alpha$ is a minuscule weight of $L$; we denote by $V_L(- \alpha )$ the corresponding highest weight module. By Proposition 3.3 and its proof, ${{\mathcal {L}}}_x$ is a minuscule homogeneous space under $L$; moreover, the tangent map is an immersion onto the closed $L$-orbit in ${{\mathbb {P}}}(V_L(-{\alpha })) \subset {{\mathbb {P}}}(T_x X)$. One can show that the $T$-weights of $V_L(- \alpha )$ are exactly the roots $-\beta$, where $\beta \in R^{+}$ and $\beta = \alpha + \sum _{\alpha _i \in S, \alpha _i \neq \alpha } n_i \alpha _i$ for some non-negative integers $n_i$; these form a unique orbit of $W_L$. Moreover, $P$ is minuscule if and only if $V_L(- \alpha ) = T_x X$; equivalently, $W_L$ acts transitively on $R^{-} \setminus R_L$.

On the other hand, if $G$ is not simply-laced, then there exists a short root $\alpha$ such that the variety of lines through $x$ in $G/P^{\alpha }$ is not homogeneous under $P^{\alpha }$; see the main theorem of [Reference Cohen and CoopersteinCC98] for a more specific result, valid over an arbitrary field, and [Reference StricklandStr02, Reference Landsberg and ManivelLM03] for further developments.

(ii) When $G$ is simply-laced, the minuscule homogeneous spaces $G/P$ and their varieties of lines are exactly those in the following table:

Here ${{\mathbb {G}}}(m,n+1)$ denotes the Grassmannian of $m$-dimensional linear subspaces of $k^{n+1}$, and ${{\mathbb {Q}}}^{n} \subset {{\mathbb {P}}}^{n+1}$ the $n$-dimensional smooth quadric. Also, ${{\mathbb {S}}}^{n(n-1)/2}$ stands for the spinor variety of the corresponding dimension, and ${{\mathbb {X}}}^{16}, {{\mathbb {X}}}^{27}$ are two exceptional varieties of the corresponding dimensions again. The simple roots are ordered as in [Reference BourbakiBou08, Chapter VI], and the list of minuscule weights is taken from [Reference BourbakiBou08, § 4 and Exercise 15].

When $G$ is not simply-laced, one obtains in addition the pairs $({{\rm B}}_n, \alpha _n)$ and $({{\rm C}}_n,\alpha _1)$. The associated minuscule varieties are isomorphic to those of the pairs $({{\rm D}}_{n+1},\alpha _n)$ and $({{\rm A}}_{2n-1},\alpha _1)$ respectively, that is, ${{\mathbb {S}}}^{n(n+1)/2}$, respectively ${{\mathbb {P}}}^{2n-1}$. Moreover, this identifies the Schubert varieties of the former pairs to those of the latter ones. Thus, we may assume that $G$ is simply-laced when studying Schubert varieties in minuscule $G$-homogeneous spaces.

3.4 Lines on Schubert varieties

We keep the notations and assumptions of §§ 3.1 and 3.2, and start with the following observation.

Proof. The Bruhat decomposition yields a $T$-equivariant open immersion

\[ \prod_{\beta \in R^{+} \cap w(R^{-})} U_{\beta} \longrightarrow X(w), \quad (g_{\beta}) \longmapsto \Biggl(\prod_{\beta} g_{\beta}\Biggr) w x \]

with image $Bwx$, where the product is taken in any order. The assertion follows from this by arguing as in the proof of Lemma 3.1.

Our next result implies that every Schubert variety is covered by translates of Schubert lines (see also [Reference Hong and MokHM13, Proposition 3.1]).

Next, assume that $P = P^{\alpha }$ where $\alpha$ is a long simple root. Let ${{\mathcal {K}}}$ be a family of lines on $X(w)$, and ${{\mathcal {K}}}_{wx}$ the subfamily of lines through $wx$. It will be convenient to consider the translate ${\dot w}^{-1} {{\mathcal {K}}}$, a family of lines in $w^{-1} X(w)$ through the base point $x$. Recall from Proposition 3.3 that the family ${{\mathcal {L}}}_x$ of lines in $G/P$ through $x$ is the minuscule variety $L X(s_{\alpha }) \simeq L/(L \cap P_{\alpha ^{\perp }})$.

Proof. By Lemma 2.4, ${{\mathcal {K}}}_{wx}$ is a projective variety equipped with an action of $B \cap w P w^{-1}$. Thus, $w^{-1} {{\mathcal {K}}}_{wx}$ is a projective variety equipped with an action of $w^{-1} B w \cap P$, and hence of $B_L$. Moreover, there is a quasi-finite equivariant morphism

\[ \gamma : w^{-1} {{\mathcal{K}}}_{wx} \longrightarrow {{\mathcal{L}}}_x, \]

since we may view ${{\mathcal {L}}}_x$ as the Chow variety of lines in $G/P$. Since ${{\mathcal {L}}}_x$ is a flag variety under $L$, it has only finitely many $B_L$-orbits by the Bruhat decomposition; also, the $B_L$-isotropy groups are smooth and connected. As a consequence, $w^{-1} {{\mathcal {K}}}_{wx}$ contains an open orbit of $B_L$; moreover, the stabilizer of a point $C$ of this orbit is contained in the stabilizer $(B_L)_{\gamma (C)}$, and both have the same dimension. Since $(B_L)_{\gamma (C)}$ is smooth and connected, both stabilizers must be equal and hence $\gamma$ is birational. Using the normality of Schubert varieties, it follows that $\gamma$ is an isomorphism. This proves the first assertion.

For the second assertion, recall that every $B_L$-orbit in ${{\mathcal {L}}}_x$ contains a unique $T$-fixed point; moreover, these fixed points are exactly the $v X(s_{\alpha })$, where $v \in W_L$. Also, $v X(s_{\alpha }) = C_{1,v(\alpha )}$ as $v(\alpha ) \in R^{+}$. By Lemma 3.5, it follows that $v X(s_{\alpha }) \subset w^{-1} X(w)$ if and only if $wv(\alpha ) \in R^{-}$.

Example 3.9 We illustrate the results of this subsection in the case where $G := {{\rm SL}}(4)$ (with simple roots $\alpha _1,\alpha _2,\alpha _3$), and $I := \{ \alpha _1, \alpha _3 \}$; then the parabolic subgroup $P = P_I = P^{\alpha _2}$ is minuscule. Let $w := s_1 s_3 s_2$; then $w \in W^{I}$ and $\ell (w) = 3$. The $T$-stable lines through $wx$ in $X(w)$ are exactly

\[ C_1 := C_{w, - \alpha_1} , \quad C_2 := C_{w, - \alpha_1 - \alpha_2 - \alpha_3}, \quad C_3 := C_{w, -\alpha_3}. \]

Moreover, $C_1 = G_{\alpha _1} wx$, $C_2 = s_1 s_2 G_{\alpha _3} wx$ and $C_3 = G_{\alpha _3} wx$. Since $s_1 w, s_3 w < w$, we see that $C_1,C_3$ are contained in the smooth locus of $X(w)$. But $C_2$ contains $x$, which is the unique singular point of $X(w)$.

The family ${{\mathcal {L}}}_x$ of lines in $G/P$ through $x$ satisfies ${{\mathcal {L}}}_x = L X(s_2) \simeq {{\mathbb {P}}}^{1} \times {{\mathbb {P}}}^{1}$; this isomorphism identifies $X(s_2)$ with $(\infty ,\infty )$. Moreover, the Chow variety of lines in $w^{-1} X(w)$ through $x$ is identified with $({{\mathbb {P}}}^{1} \times \{ \infty \}) \cup (\{ \infty \} \times {{\mathbb {P}}}^{1})$, the union of two lines meeting at the point corresponding to $X(s_2) = w^{-1} C_2$. These lines are the $B_L$-orbit closures of $w^{-1} C_1, w^{-1} C_3$. As a consequence, there are exactly two families of minimal rational curves on $X(w)$; those through $wx$ are the $w$-translates of the above lines.

These results can also be obtained by direct geometric arguments, since $X = {{\mathbb {G}}}(2,4)$ is embedded in ${{\mathbb {P}}}(V(\varpi _2)) = {{\mathbb {P}}}(\Lambda ^{2} k^{4}) \simeq {{\mathbb {P}}}^{5}$ as a quadric; moreover, $X(w)$ is the intersection of $X$ with a tangent hyperplane. Thus, $X(w)$ is the projective cone over ${{\mathbb {Q}}}^{2} \simeq {{\mathbb {P}}}^{1} \times {{\mathbb {P}}}^{1}$ with vertex $x$. This cone contains two families of planes (the projective cones over the two families of lines in ${{\mathbb {P}}}^{1} \times {{\mathbb {P}}}^{1}$) and the lines in these planes form the two minimal families.

4.1 Bott–Samelson desingularizations

We keep the notation of §§ 3.1 and 3.2. Let $w \in W^{I}$. If $w \neq 1$, then there exists a decomposition $w = s_{i_1} w'$, where $s_{i_1}$ is a simple reflection, $w' \in W$, and $\ell (w) =\ell (w') + 1$. It follows that the minimal parabolic subgroup $P_{i_1}$ stabilizes $X(w)$, and $w' \in W^{I}$. Consider the Schubert variety $X(w') \subset G/P$ and the associated fiber bundle $P_{i_1} \times ^{B} X(w')$. This is a projective variety equipped with an action of $P_{i_1}$ and an equivariant morphism

\[ f_{i_1,w'} : P_{i_1} \times^{B} X(w') \longrightarrow P_{i_1}/B \simeq {{\mathbb{P}}}^{1}, \]

which is a locally trivial fibration (for the Zariski topology) with fiber $X(w')$. We also have a $P_{i_1}$-equivariant morphism

\[ \pi_{i_1,w'} : P_{i_1} \times^{B} X(w') \longrightarrow X(w), \]

which restricts to an isomorphism above the open orbit $Bwx$.

The above ‘one-step construction’ can be iterated: given a reduced decomposition

\[ {{\tilde{w}}} = (s_{i_1}, s_{i_2}, \ldots, s_{i_{\ell}}) \]

(i.e. a sequence of simple reflections such that $w = s_{i_1} s_{i_2} \cdots s_{i_{\ell }}$ and $\ell (w) = \ell$), this yields a projective variety

\[ {{\tilde{X}}}({{\tilde{w}}}) = P_{i_1} \times^{B} P_{i_2} \times^{B} \cdots \times^{B} P_{i_{\ell}}/B \]

of dimension $\ell$, equipped with an action of $P_{i_1}$ and two equivariant morphisms,

\[ f : {{\tilde{X}}}({{\tilde{w}}}) \longrightarrow P_{i_1}/B, \]

a locally trivial fibration with fiber ${{\tilde {X}}}({{\tilde {w}}}')$, where ${{\tilde {w}}}':= (s_{i_2}, \ldots , s_{i_{\ell }})$, and

\[ \pi : {{\tilde{X}}}({{\tilde{w}}}) \longrightarrow X(w), \]

which restricts to an isomorphism above the open orbit $Bwx$. Also, note that ${{\tilde {X}}}({{\tilde {w}}})$ is smooth, and hence $\pi$ is a desingularization of $X(w)$.

More generally, for $1 \leq j \leq \ell$, we have a fibration

\[ f_j : {{\tilde{X}}}({{\tilde{w}}}) \longrightarrow {{\tilde{X}}}(s_{i_1}, \ldots, s_{i_j}) \]

with fiber ${{\tilde {X}}}(s_{i_{j+1}}, \ldots , s_{i_{\ell }})$. In particular, $f_{\ell - 1}$ is a ${{\mathbb {P}}}^{1}$-bundle; also, $f_1 = f$.

The Bott–Samelson variety ${{\tilde {X}}}({{\tilde {w}}})$ is equipped with a base point ${{\tilde {x}}}$, defined as the image of $(\dot s_{i_1}, \dot s_{i_2}, \ldots , \dot s_{i_{\ell }}) \in P_{i_1} \times P_{i_2} \times \cdots \times P_{i_{\ell }}$. Moreover, $\pi ({{\tilde {x}}}) = wx$ and $B_{{{\tilde {x}}}} = B_{wx}$. In particular, ${{\tilde {x}}}$ is fixed by $T$. Denoting by ${{\tilde {x}}}_j$ the base point of ${{\tilde {X}}}(s_{i_1}, \ldots , s_{i_j})$, we have $f_j({{\tilde {x}}}) = {{\tilde {x}}}_j$. Further, the fiber of $f_j$ at ${{\tilde {x}}}_j$ is $T$-equivariantly isomorphic to ${{\tilde {X}}}(s_{i_{j+1}}, \ldots ,s_{i_{\ell }})$ on which the $T$-action is twisted by the Weyl group element $s_{i_1} \cdots s_{i_j}$.

We now recall the description of line bundles on ${{\tilde {X}}}({{\tilde {w}}})$ obtained in [Reference Lauritzen and ThomsenLT04]. For $1 \leq j \leq \ell$, consider the natural morphism

\[ \pi_j : {{\tilde{X}}}(s_{i_1}, \ldots, s_{i_j}) \longrightarrow G/B \]

with image the Schubert variety $X(s_{i_1} \cdots s_{i_j})$, and let

\[ {{\mathcal{L}}}_j := f_j^{*} \pi_j^{*} {{\mathcal{L}}}_{G/B}(\varpi_{i_j}). \]

Then the isomorphism classes of ${{\mathcal {L}}}_1,\ldots , {{\mathcal {L}}}_{\ell }$ form a basis of ${{\rm Pic}}({{\tilde {X}}}({{\tilde {w}}}))$. Further, for any integers $n_1, \ldots , n_{\ell }$, the line bundle ${{\mathcal {L}}}_1^{\otimes n_1} \otimes \cdots \otimes {{\mathcal {L}}}_{\ell }^{\otimes n_{\ell }}$ is ample if and only if $n_1,\ldots ,n_{\ell } > 0$; also, every ample line bundle on ${{\tilde {X}}}({{\tilde {w}}})$ is very ample. In particular, ${{\mathcal {L}}}_1 \otimes \cdots \otimes {{\mathcal {L}}}_{\ell }$ is the smallest very ample line bundle on ${{\tilde {X}}}({{\tilde {w}}})$.

Next, we describe the $T$-stable curves through ${{\tilde {x}}}$ in ${{\tilde {X}}}({{\tilde {w}}})$. Let

\[ \beta_1 := \alpha_{i_1}, \beta_2 := s_{i_1}(\alpha_{i_2}), \ldots, \beta_{\ell} := s_{i_1} \cdots s_{i_{\ell -1}}(\alpha_{i_{\ell}}). \]

Then we have

(13)\begin{equation} R^{+} \cap w(R^{-}) = \{ \beta_1, \beta_2, \ldots, \beta_{\ell} \}. \end{equation}

We may now state a version of Lemma 3.1 for Bott–Samelson varieties.

Proof. (i) This follows from Lemmas 3.1 and 3.5, since $\pi$ sends $B {{\tilde {x}}}$ (an open $T$-stable neighborhood of ${{\tilde {x}}}$ in ${{\tilde {X}}}({{\tilde {w}}})$) isomorphically to $Bwx$.

For (ii), note that $\pi$ restricts to a birational morphism ${{\tilde {C}}}_j \to C_{w,-\beta _j} \simeq {{\mathbb {P}}}^{1}$.

(iii) If $j > k$ then ${{\tilde {C}}}_j$ is contracted by $f_k$. This implies the first assertion by using the projection formula.

If $j \leq k$ then $f_k$ restricts to an isomorphism of ${{\tilde {C}}}_j$ onto the $j$th $T$-stable curve in ${{\tilde {X}}}(s_{i_1},\ldots ,s_{i_k})$ through ${{\tilde {x}}}_k$. Further, the latter curve is sent isomorphically by $\pi _k$ to the $T$-stable curve $C_{s_{i_1} \cdots s_{i_k},-\beta _j}$ in $G/B$. Using Lemma 3.1, it follows that

\[ {{\mathcal{L}}}_k \cdot {{\tilde{C}}}_j = \langle \varpi_{i_k}, - s_{i_k} \cdots s_{i_1}(\beta_j^{\vee}) \rangle = \langle \varpi_{i_k}, s_{i_k} \cdots s_{i_{j+1}}(\alpha_j^{\vee}) \rangle . \]

This yields the second assertion.

(iv) We first determine $-K_{{{\tilde {X}}}({{\tilde {w}}})} \cdot {{\tilde {C}}}_1$. Note that $G_{\alpha _{i_1}} \subset P_{i_1}$ acts on ${{\tilde {X}}}({{\tilde {w}}})$ and we have ${{\tilde {C}}}_1 = G_{\alpha _{i_1}} {{\tilde {x}}}$; also, the tangent space of ${{\tilde {X}}}({{\tilde {w}}})$ at ${{\tilde {x}}}$ is a direct sum of $T$-stable lines with weights $\beta _1, \ldots , \beta _{\ell }$. Arguing as in the proof of Lemma 3.1(iv), it follows that

\[ - K_{{{\tilde{X}}}({{\tilde{w}}})} \cdot {{\tilde{C}}}_1 = \langle \beta_1 + \cdots + \beta_{\ell}, \alpha_{i_1}^{\vee} \rangle. \]

But $\beta _1 + \cdots + \beta _{\ell } = \rho - w(\rho )$ in view of (13), and hence

\[ - K_{{{\tilde{X}}}({{\tilde{w}}})} \cdot {{\tilde{C}}}_1 = \langle \rho - w(\rho), \alpha_{i_1}^{\vee} \rangle = 1 + {{\rm ht}}( -w^{-1}(\alpha_{i_1}^{\vee}) ). \]

This yields the assertion, since $- w^{-1}(\alpha _{i_1}) = s_{i_{\ell }} \cdots s_{i_2}(\alpha _{i_1})$.

Next, we determine $-K_{{{\tilde {X}}}({{\tilde {w}}})} \cdot {{\tilde {C}}}_j$, where $j \geq 2$. Then ${{\tilde {C}}}_j$ is contracted by $f$, i.e. we have ${{\tilde {C}}}_j \subset F := f^{-1} f({{\tilde {x}}})$. Since $f$ is a locally trivial fibration, it follows that $-K_{{{\tilde {X}}}({{\tilde {w}}})} \cdot {{\tilde {C}}}_j = -K_F \cdot {{\tilde {C}}}_j$. Recall that $F$ is $T$-equivariantly isomorphic to the Bott–Samelson variety ${{\tilde {X}}}({{\tilde {w}}}')$ on which the $T$-action is twisted by $s_{i_1}$, and this isomorphism sends ${{\tilde {x}}}$ to the base point of ${{\tilde {X}}}({{\tilde {w}}}')$. Using an easy induction argument, this completes the proof.

Remark 4.2 By Lemma 4.1, we have ${{\mathcal {L}}}_k \cdot {{\tilde {C}}}_{\ell } = 0$ for $1 \leq k < \ell$, and ${{\mathcal {L}}}_{\ell } \cdot {{\tilde {C}}}_{\ell } = 1$. Thus, ${{\tilde {C}}}_{\ell }$ is a line in the smallest projective embedding of ${{\tilde {X}}}({{\tilde {w}}})$.

Also, ${{\mathcal {L}}}_j \cdot {{\tilde {C}}}_j = 1$ for $1 \leq j \leq \ell$; as a consequence, ${{\tilde {C}}}_j$ is a line if and only if ${{\mathcal {L}}}_k \cdot {{\tilde {C}}}_j = 0$ for all $j < k$. By an easy argument, this is equivalent to the assertion that $s_{i_j}$ commutes with $s_{i_{j+1}},\ldots , s_{i_{\ell }}$. Then there is an isomorphism of resolutions of $X(w)$

\[ {{\tilde{X}}}({{\tilde{w}}}) = {{\tilde{X}}}(s_{i_1}, \ldots, s_{i_{\ell}}) \simeq {{\tilde{X}}}(s_{i_1}, \ldots, s_{i_{j-1}}, s_{i_{j+1}}, \ldots, s_{i_{\ell}}, s_{i_j}) \]

which identifies ${{\tilde {C}}}_j$ with the line in the right-hand side constructed as above.

Next, we determine the minimal rational curves in ${{\tilde {X}}}({{\tilde {w}}})$ (these include of course the lines discussed above).

Proof. We argue as in the proof of Lemma 3.2. By Lemma 2.4, ${{\mathcal {K}}}_{{{\tilde {x}}}}$ is a projective variety; moreover, ${{\mathcal {K}}}$ consists of free curves in view of Lemma 2.5. Since ${{\mathcal {K}}}_{{{\tilde {x}}}}$ is equipped with an action of $T$, it contains a $T$-fixed point, say ${{\tilde {C}}}_j$.

If $j > 1$ then ${{\tilde {C}}}_j$ is contracted by $f$. In view of Lemma 2.8, it follows that ${{\mathcal {K}}}_{{{\tilde {x}}}} = {{\mathcal {L}}}_{{{\tilde {x}}}}$ for a unique minimal family ${{\mathcal {L}}}$ on the fiber of $f$ at ${{\tilde {x}}}$. Since this fiber is a translate of a smaller Bott–Samelson variety, we may conclude by induction on $\ell$.

Thus, we may assume that $j = 1$; then ${{\tilde {C}}}_1$ is the unique $T$-fixed point of ${{\mathcal {K}}}_{{{\tilde {x}}}}$. Also, ${{\mathcal {K}}}_{{{\tilde {x}}}}$ admits an ample $T$-linearized line bundle: indeed, it is equipped with a finite $T$-equivariant morphism to some Chow variety of ${{\tilde {X}}}({{\tilde {w}}})$, which in turn is equipped with a finite $T$-equivariant morphism to the projectivization of a $T$-module in view of its construction in [Reference KollárKol99, § I.3]. As a consequence, ${{\mathcal {K}}}_{{{\tilde {x}}}}$ admits a $T$-equivariant immersion in the projectivization of a $T$-module. Using [Reference BorelBor91, Proposition 13.5], it follows that ${{\mathcal {K}}}_{{{\tilde {x}}}}$ consists of the unique curve ${{\tilde {C}}}_1$.

On the other hand, by Lemma 2.5, ${{\tilde {C}}}_1$ lies in a unique family of rational curves ${{\mathcal {L}}}$ on ${{\tilde {X}}}({{\tilde {w}}})$; moreover, ${{\mathcal {L}}}$ is covering and satisfies

\[ \dim({{\mathcal{L}}}_{{{\tilde{x}}}}) = - K_{{{\tilde{X}}}({{\tilde{w}}})} \cdot {{\tilde{C}}}_1 - 2. \]

In view of Lemma 4.1(iii), this vanishes if and only if the root $s_{i_{\ell }} \cdots s_{i_2}(\alpha _{i_1})$ is simple.

Remark 4.4 (i) Since $s_{i_{\ell }} \cdots s_{i_{j + 1}}(\alpha _{i_j}) =- w^{-1}(\beta _j)$, the simple roots $\alpha _k$ which can be obtained as $s_{i_{\ell }} \cdots s_{i_{j + 1}}(\alpha _{i_j})$ for some $j$ are exactly those such that $w(\alpha _k) \in R^{-}$. Then $\pi$ sends ${{\tilde {C}}}_j$ to $w X(s_k)$, a translate of a Schubert line. This relates the minimal rational curves in ${{\tilde {X}}}({{\tilde {w}}})$ through ${{\tilde {x}}}$ to the lines in $X(w)$ through $wx$ constructed in Lemma 3.6. In particular, we may take $j = \ell$, ie., $\alpha _k = \alpha _{i_{\ell }}$; this just gives back the line ${{\tilde {C}}}_{\ell }$ (Remark 4.2).

If $P = P^{\alpha }$ is maximal, then we must have $\alpha = \alpha _k$ and $j = \ell$. Thus, ${{\tilde {X}}}({{\tilde {w}}})$ has a unique minimal family, consisting of the fibers of the ${{\mathbb {P}}}^{1}$-bundle $f_{\ell - 1} : {{\tilde {X}}}({{\tilde {w}}}) \to {{\tilde {X}}}(s_{i_1}, \ldots , s_{i_{\ell - 1}})$.

(ii) The condition that $s_{i_{\ell }} \cdots s_{i_{j + 1}}(\alpha _{i_j})$ is a simple root, say $\alpha _k$, turns out to be equivalent to the exchange condition

\[ s_{i_1} s_{i_2} \cdots s_{i_{\ell}} = s_{i_1} \cdots \widehat{s_{i_j}} \cdots s_{i_{\ell}} s_k, \]

where both sides are reduced decompositions of $w$.

4.2 Their generalizations à la Perrin

Let $w \in W$. Recall that the set of simple roots $\alpha$ such that $s_{\alpha }$ occurs in a reduced decomposition of $w$ is independent of the reduced decomposition, and called the support of $w$. We denote this set by ${{\rm Supp}}(w)$. The subgroup of $G$ generated by the $U_{\pm \alpha }$, where $\alpha \in {{\rm Supp}}(w)$, will be denoted by $G_w$; this is the derived subgroup of the Levi subgroup $L_{{{\rm Supp}}(w)}$, and hence is a semi-simple subgroup of $G$, normalized by $T$ and containing a representative of $w$.

Denote by $P^{w}$ be the largest parabolic subgroup of $G$ such that $P^{w} \supset B$ and $w \in W^{P^{w}}$; then $P^{w} = P_{I^{w}}$, where $I^{w} := \{ \alpha \in S \vert w(\alpha ) \in R^{+} \}$. Consider the associated Schubert variety $X(w) \subset G/P^{w}$, and denote by $P_w$ the closed reduced subgroup of $G$ consisting of those $g$ such that $g X(w) = X(w)$. Then $P_w$ is a parabolic subgroup of $G$ containing $B$, and hence $P_w = P_{I_w}$, where $I_w := \{ \alpha \in S ~\vert ~ s_{\alpha } w \leq w \}$; here $\leq$ denotes the Bruhat order on $W^{I^{w}}= W/W_{I^{w}}$. Note that $P_w \cap G_w$ is a parabolic subgroup of $G_w$, and we have

(14)\begin{equation} X(w) = \overline{P_w w P^{w}}/P^{w} \simeq \overline{(P_w \cap G_w) w (P^{w} \cap G_w)}/(P^{w} \cap G_w) \subset G_w/(P^{w} \cap G_w). \end{equation}

We say that $w \in W$ is minuscule if so is $G/P^{w}$; then one may readily check that $G_w/(P^{w} \cap G_w)$ is minuscule as well. Also, note that $G_w$ is simply-laced if so is $G$. For any minuscule $w \in W$, we have $X(w)_{{{\rm sm}}} = (P_w \cap G_w) w x$ by [Reference Brion and PoloBP99, Proposition 3.3]. In particular, $X(w)$ is smooth if and only if it is homogeneous under $P_w \cap G_w$.

Next, let $w = w_1 w'$, where $w_1, w' \in W$ satisfy $P^{w_1} \cap G_{w_1} \subset P_{w'}$; equivalently, we have the inclusion $I^{w_1} \cap {{\rm Supp}}(w_1) \subset I_{w'}$. We may then define

\[ {{\tilde{X}}}(w_1,w') := \overline{(P_{w_1} \cap G_{w_1}) w_1 (P^{w_1} \cap G_{w_1})} \times^{P^{w_1} \cap G_{w_1}} X(w'). \]

This is a projective variety equipped with an action of $P_{w_1} \cap G_{w_1}$ and an equivariant morphism

\[ f_{w_1,w'} : {{\tilde{X}}}(w_1,w') \longrightarrow X(w_1), \]

which is a Zariski locally trivial fibration with fiber $X(w')$. If in addition $\ell (w) = \ell (w_1) + \ell (w')$, then $w' \in W^{P^{w}}$ and hence $P^{w'} \supset P^{w}$. Thus, if $P^{w}$ is maximal and $w' \neq 1$, then $P^{w'} = P^{w}$. Under these assumptions, we obtain another equivariant morphism

\[ \pi_{w_1,w'} : {{\tilde{X}}}(w_1,w') \longrightarrow G/P^{w}. \]

One may check that $\pi _{w_1,w'}$ is birational to its image $X(w)$; it restricts to an isomorphism above the open orbit $Bwx$. Also, note that

(15)\begin{equation} P_{w_1} \cap G_{w_1} \subset P_w \cap G_w. \end{equation}

Remark 4.5 Assume that $w_1$ is a simple reflection $s_{\alpha }$. Then $P^{w_1}$ is the maximal parabolic subgroup $P^{\alpha } = P_{S \setminus \{ \alpha \} }$, and $X(w_1)$ is the Schubert line in $G/P^{\alpha }$. Moreover, we have $G_{w_1} =G_{\alpha }$ and $P_{w_1} = P_{\{ \alpha \} \cup \alpha ^{\perp } }$. So $P^{w_1} \cap G_{w_1} = B \cap G_{\alpha }$ is a Borel subgroup of $G_{\alpha }$. Thus, we have

\[ {{\tilde{X}}}(s_{\alpha},w') = G_{\alpha} \times^{B \cap G_{\alpha}} X(w') \simeq P_{\alpha} \times^{B} X(w'), \]

with fibration $f_{s_{\alpha },w'}$ over $P_{\alpha }/B \simeq {{\mathbb {P}}}^{1}$. If in addition $\ell (w) = \ell (w') + 1$, then $\pi _{s_{\alpha },w'}$ yields a birational morphism to $X(w)$. Therefore, the above ‘one-step construction’ generalizes that of Bott–Samelson varieties.

This construction can be iterated, under certain additional assumptions that are discussed in detail in [Reference PerrinPer07, § 5.2]. We now present some notions and results from [Reference PerrinPer07, § 5.2]: a finite sequence $\hat {w} = (w_1,\ldots ,w_m)$ of elements of $W$ is called a generalized reduced decomposition of $w$, if we have $w = w_1 \cdots w_m$ and $\ell (w) = \ell (w_1) + \cdots + \ell (w_m)$. Such a decomposition is called good if in addition $w$ is minuscule and we have

\[ I^{w_i} \cap {{\rm Supp}}(w_i) \subset I_{w_{i+1} \cdots w_m} \subset w_i^{\perp} \cup {{\rm Supp}}(w_i) \quad (1 \leq i \leq m -1), \]

where $w_i^{\perp }$ denotes the set of simple roots $\alpha$ such that $s_{\alpha }$ commutes with $w_i$. Under these assumptions, $(w_{i+1},\ldots ,w_m)$ is a good generalized reduced decomposition of $w_{i+1} \cdots w_m$ for $i = 1, \ldots , m-1$. Moreover, $P^{w_i} \cap G_{w_i} \subset P_{w_{i+1} \cdots w_m}$ for all such $i$.

Given a good generalized reduced decomposition $\hat {w}$ of $w$, we obtain a projective variety $\hat {X}(\hat {w})$ equipped with an action of $P_w$, a locally trivial fibration

\[ \hat{f} : \hat{X}(\hat{w}) \longrightarrow X(w_1) \]

with fiber $\hat {X}(w_2,\ldots ,w_m)$, and a birational morphism

\[ \hat{\pi} : \hat{X}(\hat{w}) \longrightarrow X(w). \]

Also, $\hat {X}(\hat {w})$ has a base point $\hat {x}$ such that $\hat {\pi }(\hat {x}) = wx$ and $\hat {f}(\hat {x}) = w_1 x_1$, where $x_1$ denotes the base point of $G/P^{w_1}$.

More generally, for $1 \leq i \leq m$, we have a fibration

\[ \hat{f}_i : \hat{X}(\hat{w}) \longrightarrow \hat{X}(w_1,\ldots,w_i) \]

with fiber $\hat {X}(w_{i+1},\ldots ,w_m)$. Further, $\hat {f}_i(\hat {x}) = \hat {x}_i$ with an obvious notation, and $\hat {f}_1 = \hat {f}$.

By [Reference PerrinPer07, § 5.1], the morphism $\hat {f}$ is $P_w$-equivariant. As a consequence, we have the equality of stabilizers $P_{w,\hat {x}} = P_{w,wx} = P_w \cap w P^{w} w^{-1}$. Thus, $P_{w,\hat {x}}$ contains the maximal torus $T$. Also, note that $P_{w,\hat {x}}$ is smooth and connected, in view of the following result.