# Quantum $K$-theory for flag varieties

Sybille Rosset

(PhD Thesis defended in July 2020)

## Summary

We present here an overview of the different results presented in this thesis.
Chapters 2 and 3 use some of the results of Chapter 1, but are otherwise
independent and each of them may be read separately.

####  0.0.1 Chapter 1: Gromov-Witten varieties of partial flag varieties.

Let $n\geq 0$, let $J=\\{j_{1},\dots,j_{M}\\}$ be a set of $M$ integers
satisfying $0<j_{1}<\dots<j_{M}<n$. Consider the flag variety $X=Fl_{J}$
parametrizing the flags of vector spaces of type

$V_{j_{1}}\subset V_{j_{2}}\subset\dots\subset
V_{j_{M}}\subset{\mathbb{C}}^{n},$

where $V_{j_{k}}$ is a vector subspace of ${\mathbb{C}}^{n}$ of dimension
$j_{k}$. For an element ${\mathbf{d}}$ in the semi-group $E(X)\subset
H_{2}(X,{\mathbb{Z}})$ of effective classes of curves, we denote by
${\mathcal{M}_{0,r}}(X,{\mathbf{d}})$ the space of maps
$(f:{\mathbb{P}}^{1}\to X,\\{p_{1},\dots,p_{r}\\})$ with $r$ marked points
$p_{i}$ on ${\mathbb{P}}^{1}$ satisfying
$f_{*}[{\mathbb{P}}^{1}]={\mathbf{d}}$. Associating to a map the image of its
marked points yields an evaluation morphism
$ev_{\mathbf{d}}:{\mathcal{M}}_{0,r}(X,{\mathbf{d}})\rightarrow X^{r}$. For
$r=3$ and $X$ a cominuscule variety, and in particular for $X$ a Grassmannian,
Chaput-Perrin proved that the general fiber of the evaluation map is a
rational variety [CP11a]. For $X={\mathbb{P}}^{2}$, Pandharipande computed the
genus of the general fiber of the evaluation map
$ev_{d}:{{\mathcal{M}}_{0,r}}({\mathbb{P}}^{2},d)\rightarrow({\mathbb{P}}^{2})^{r}$
when this fiber has dimension $1$ [Pan97], genus which turns out to be
positive for $d\geq 3$. For $X={\mathbb{P}}^{1}$, examples of values of $d$
such that the general fiber of
$ev_{d}:{{\mathcal{M}}_{0,r}}({\mathbb{P}}^{1},d)\rightarrow({\mathbb{P}}^{1})^{r}$
is not a rationally connected variety have been obtained using techniques from
quantum $K$-theory [IMT14]. One might wonder what happens for a variety of
Picard rank greater than one. Chapter 1 provides a step in that direction by
studying the geometry of the general fiber of the evaluation map
$ev_{\mathbf{d}}:{\mathcal{M}}_{0,r}(X,{\mathbf{d}})\rightarrow X^{r}$ for $X$
a flag variety, and more generally the geometry of Gromov-Witten varieties of
flag varieties.

######  Definition 0.0.1.1.

Let $X_{1}$, $\dots$, $X_{r}$ be Schubert varieties of the flag variety $X$,
let $\mathbf{d}$ be an element in $E(X)$. For any element
$g=(g_{1},\dots,g_{r})$ general in ${\rm GL}_{n}^{r}$, we call Gromov-Witten
variety of degree $\mathbf{d}$ associated with the varieties $X_{i}$ and with
$g$ the subscheme

${\mathcal{W}}_{X;X_{1},\dots,X_{r}}^{g,\mathbf{d}}:=ev_{1}^{-1}(g_{1}X_{1})\cap\dots\cap
ev_{r}^{-1}(g_{r}X_{r})$

of $\overline{\mathcal{M}_{0,r}}(X,\mathbf{d})$.

Note that the Gromov-Witten variety
${\mathcal{W}}_{X;X_{1},\dots,X_{r}}^{g,\mathbf{d}}$ parametrizes morphisms
${\mathbb{P}}^{1}\rightarrow X$ representing the class ${\mathbf{d}}$ whose
image has a non empty intersection with the translates $g_{i}\cdot X_{i}$ of
the Schubert varieties $X_{i}$. When these varieties are zero dimensional,
their number of points is a Gromov-Witten invariant. More generally, the
Euler-Poincaré characteristic of their adherence in the space
$\overline{\mathcal{M}_{0,r}}(X,\mathbf{d})$ is a correlator in quantum
$K$-theory [Giv00].

Let $I=\\{i_{1},\dots,i_{m}\\}$ be a set of $m$ integers satisfying
$0<i_{1}<\dots<i_{m}<n$, such that $I$ is contained in $J$. We denote by
$\pi:X=Fl_{J}\rightarrow X^{\prime}=Fl_{I}$ the forgeftul map. Fix a class
$\lambda=[l]\in H_{2}(X,{\mathbb{Z}})$ of a curve $l$ whose projection by the
forgetful map $\pi$ is a point. We observe in Chapter 1 a relation between the
general fiber of the evaluation map

$ev_{[C]+\lambda}:\overline{\mathcal{M}_{0,r}}(X,[C]+\lambda)\rightarrow
X^{r}$

and the general fiber of the map
$ev_{[\pi(C)]}:\overline{\mathcal{M}_{0,r}}(X^{\prime},[\pi(C)])\to(X^{\prime})^{r}$.

######  Theorem.

(Theorem 1.1.2.1) Suppose the collection $(I,J,[C]+\lambda)$ is a stabilized
collection in the sense of Definition 1.1.1.1. Then:

  * •

For any Schubert varieties $X_{1}$, $\dots$, $X_{r}$ of $X$, for $g$ general
in ${\rm GL}_{n}^{r}$, the Gromov-Witten variety
${\mathcal{W}}_{Fl_{J};X_{1},\dots,X_{r}}^{g,[C]+\lambda}$ of $Fl_{J}$ is a
rationally connected fibration above the Gromov-Witten variety
${\mathcal{W}}_{Fl_{I};\pi(X_{1}),\dots,\pi(X_{r})}^{g,[\pi(C)]}$ of $Fl_{I}$.

  * •

For $x$ general in $(Fl_{J})^{r}$, the fiber $ev_{[C]+\lambda}^{-1}(x)$ is a
tower of unirational fibrations over the fiber $ev_{[\pi(C)]}^{-1}(\pi(x))$.

We called a variety $W$ a rationally connected fibration (respectively
unirational fibration) above $W^{\prime}$ if there exists a dominant map
$W\rightarrow W^{\prime}$ sending each irreducible component of $W$ onto a
different irreducible component of $W^{\prime}$, and whose general fiber is a
rationally connected variety (respectively unirational). A variety $W$ is a
tower of unirational fibrations above a variety $W^{\prime}$ if there exists a
sequence of morphisms

$W_{1}=W\rightarrow W_{2}\rightarrow\dots\rightarrow W_{l}=W^{\prime}$

such that for $1\leq i<l$, the variety $W_{i}$ is a unirational fibration
above $W_{i+1}$.

####  0.0.2 Chapter 2: A comparison formula between genus $0$ correlators of
flag varieties.

Fix integers $1\leq n_{1}<\dots<n_{m}<n$. We consider the flag variety $X$
parametrizing flags of vector spaces

$V_{n_{1}}\subset\dots\subset V_{n_{m}}\subset{\mathbb{C}}^{.}$

satisfying ${\rm dim}V_{n_{i}}=n_{i}$. Let $T$ be a torus in ${\rm GL}_{n}$
acting on $X$ by left multiplication. Let $\mathbf{d}=(d_{1},\dots,d_{m})$ be
an effective class of curve in $H_{2}(X,{\mathbb{Z}})$. We denote by
$\overline{\mathcal{M}_{0,r}}(X,{\mathbf{d}})$ the coarse moduli space
parametrizing genus $0$ stable maps representing the class ${\mathbf{d}}$.
Assocating to a stable map with $r$ marked points the image of its $i$-th
marked point induces an evaluation map
$ev_{i}:\overline{\mathcal{M}_{0,r}}(X,{\mathbf{d}})\rightarrow X$. Let
$E_{1}$, $\dots$, $E_{r}$ be $T$-equivariant fiber bundles over $X$. In
quantum $K$-theory, $T$-equivariant correlators are defined by

$\langle\phi_{1},\dots,\phi_{r}\rangle_{T,(d_{1},\dots,d_{m})}^{X}:=\chi_{T}\left(ev_{1}^{*}E_{1}\otimes\dots\otimes
ev_{r}^{*}E_{r}\otimes{\mathcal{O}}_{\overline{\mathcal{M}_{0,r}}(X,{\mathbf{d}})}\right),$
(1)

where we denote by $\chi_{T}$ the $T$-equivariant Euler-Poincaré
characteristic [Giv00, Lee04]. For $1\leq k\leq m$, we denote by
$X_{\widehat{k}}$ the flag variety obtained from $X$ by forgetting the vector
space $V_{n_{k}}$, and by

$\pi_{\widehat{k}}:X\rightarrow X_{\widehat{k}}$

the forgetful map. In order to facilitate notations, we set $d_{0}=0=d_{m+1}$.
Chapter 2 yields the following equality between correlators-cf. Theorem
2.1.3.1bis.

######  Theorem.

Suppose:

  * •

$\forall\>p<k,\>d_{k}\geq
n_{k}\left\lceil\frac{d_{p}-d_{p-1}}{n_{p}-n_{p-1}}\right\rceil$,

  * •

$d_{k-1}\leq\lfloor\frac{d_{k+1}}{n_{k+1}}\rfloor$,

  * •

$d_{k}\geq
r(n_{k}-n_{k-1})+d_{k-1}+(n_{k}-n_{k-1})(\lfloor\frac{d_{k+1}-d_{k-1}}{n_{k+1}-n_{k-1}}\rfloor+1)$.

Then for any $T$-equivariant fiber bundles $E_{1}$, $\dots$, $E_{r}$ over
$X_{\widehat{k}}$, the correlators of $X$ and $X_{\widehat{k}}$ associated
with the degree $\mathbf{d}$ and the fiber bundles $E_{i}$ are equal:

$\langle(\pi_{\widehat{k}})^{*}E_{1},\dots,(\pi_{\widehat{k}})^{*}E_{r}\rangle_{T,(d_{1},\dots,d_{m})}^{X}=\langle
E_{1},\dots,E_{r}\rangle_{T,(d_{1},\dots,d_{k-1},d_{k+1},\dots,d_{m})}^{X_{\widehat{k}}}.$

Note that for $3$-pointed Gromov-Witten invariants, a result of
Peterson/Woodward allows to deduce all invariants with $3$ marked points of
generalized flag varieties $G/P$ from $3$-pointed Gromov-Witten invariants of
$G/B$ [Woo05].

####  0.0.3 Chapter 3: Schubert calculus for the incidence variety
$Fl_{1,n-1}$.

We study in Chapter 3 different variations of modern Schubert calculus for the
incidence variety $X=Fl_{1,n-1}$ parametrizing pairs $(p,H)$ where $p$ is a
point in ${\mathbb{P}}^{n-1}$ and $H$ is a hyperplane in ${\mathbb{P}}^{n-1}$
containing the point $p$. Note that $X$ can be written as $G/P$, where $G={\rm
GL}_{n}$, $T$ is the set of diagonal matrices, $B$ is the set of upper
triangular matrices and $P$ is the parabolic subgroup of $G$ satisfying
$B\subset P$ associated with the roots $\\{\alpha_{2},\dots,\alpha_{n-1}\\}$.
The smallest length representative $w$ of an element in
$W/W_{P}=\mathfrak{S}_{n}/\mathfrak{S}_{n-2}$ is a permutation $w_{i,j}$
satisfying $w(1)=i$, $w(n)=j$, and such that for $1<k<n-1$ $w(k)<w(k+1)$. For
$1\leq i,j\leq n$ and $i\neq j$, we call $X(i,j):=X(w_{i,j})$ the Schubert
variety of $Fl_{1,n-1}$ associated with the element $w_{i,j}$ in $W^{P}\simeq
W/W_{P}$.

Littlewood-Richardson coefficients in $K(Fl_{1,n-1})$. For $1\leq i,j\leq n$,
$i\neq j$, the natural embedding $i:X(i,j)\hookrightarrow X$ defines a sheaf
$i_{*}{\mathcal{O}}_{X(i,j)}$, which is a coherent sheaf of
${\mathcal{O}}_{Fl_{1,n-1}}$-module. We denote by
${\mathcal{O}}_{i,j}:=[i_{*}{\mathcal{O}}_{X(i,j)}]$ the class of
$i_{*}{\mathcal{O}}_{X(i,j)}$ in the Grothendieck group $K(Fl_{1,n-1})$ of
coherent sheaves of ${\mathcal{O}}_{Fl_{1,n-1}}$-module. Schubert classes
$({\mathcal{O}}_{i,j})_{1\leq i,j\leq n,\>i\neq j}$ form a basis of
$K(Fl_{1,n-1})$, and the product

${\mathcal{O}}_{i,j}\cdot{\mathcal{O}}_{k,p}=\sum_{i=0}^{{\rm
dim}Fl_{1,n-1}}(-1)^{i}[Tor_{i}^{Fl_{1,n-1}}\left({\mathcal{O}}_{X(i,j)},{\mathcal{O}}_{X(k,p)}\right)]$

defines a structure of associative and commutative ring
$(K(Fl_{1,n-1}),\cdot)$ with identity element $[{\mathcal{O}}_{Fl_{1,n-1}}]$.
We set ${\mathcal{O}}_{i,j}=0$ if $i<1$ or $j>n$.

######  Proposition (Littlewood-Richardson coefficients in $K(Fl_{1,n-1})$-cf.
Proposition 3.4.1.1).

Let $1\leq i,j,k,p\leq n$, where $i\neq j$ and $k\neq p$. Then

$\left\\{\begin{aligned}
\mathcal{O}_{k,p}\cdot\mathcal{O}_{i,j}&=\mathcal{O}_{i+k-n,j+p-1}\>&\mathrm{if}\>i+k-n\geq
j+p\>\mathrm{or}\>i<j\>\mathrm{or}\>k<p;\\\
\mathcal{O}_{k,p}\cdot\mathcal{O}_{i,j}&=\mathcal{O}_{i+k-n-1,j+p-1}+\mathcal{O}_{i+k-n,j+p}-\mathcal{O}_{i+k-n-1,j+p}\>&\mathrm{else}.\end{aligned}\right.$

Chevalley formula and positive algorithm in $QK_{s}(Fl_{1,n-1})$. We denote by
$l_{1}:=X(2,1)$ et $l_{2}:=X(1,2)$ the two one-dimensional Schubert varieties
of $Fl_{1,n-1}$. Their classes $[l_{1}]$ and $[l_{2}]$ in the singular
homology of $Fl_{1,n-1}$ generate $H_{2}(Fl_{1,n-1},{\mathbb{Z}})$. The small
quantum $K$-ring
$QK_{s}(Fl_{1,n-1})=(K(Fl_{1,n-1}))\otimes\mathbb{Q}[Q_{1},Q_{2}],\star)$ of
$Fl_{1,n-1}$ is a deformation of the ring $K(Fl_{1,n-1})$ by the Novikov
variables $Q_{1}$ and $Q_{2}$ associated with the classes $[l_{1}]$ and
$[l_{2}]$ [Lee04]. This deformation depends on the correlators
$\langle{\mathcal{O}}_{i,j},{\mathcal{O}}_{k,p},{\mathcal{O}}_{s,t}\rangle_{d_{1}[l_{1}]+d_{2}[l_{2}]}$
defined by (1). Considering the limit $Q_{1},Q_{2}\rightarrow 0$ yields the
$K$-ring $K(Fl_{1,n-1})$.

We name $h_{1}:=X(n-1,1)$ and $h_{2}:=X(n,2)$ the two Schubert varieties of
$Fl_{1,n-1}$ of codimension $1$.

######  Proposition (Chevalley formula in $QK_{s}(Fl_{1,n-1})$-cf. Proposition
3.7.0.1).

$\displaystyle{\mathcal{O}}_{h_{1}}\star{\mathcal{O}}_{k,p}$
$\displaystyle=\left\\{\begin{array}[]{lr}Q_{1}{\mathcal{O}}_{n-1,n}+Q_{1}Q_{2}\left([{\mathcal{O}}_{X}]-{\mathcal{O}}_{h_{1}}\right)&\mathrm{if}\>k=1,p=n\\\
Q_{1}{\mathcal{O}}_{n,p}&\mathrm{if}\>k=1,\>p<n\\\
{\mathcal{O}}_{1,2}+Q_{1}\left([{\mathcal{O}}_{X}]-{\mathcal{O}}_{h_{1}}\right)&\mathrm{if}\>k=2,p=1\\\
{\mathcal{O}}_{k-1,p}&\mathrm{if}\>k>1,\>k\neq p+1\\\
{\mathcal{O}}_{p-1,p}+{\mathcal{O}}_{p,p+1}-{\mathcal{O}}_{p-1,p+1}&\mathrm{if}\>1<p<n-1,\>k=p+1\\\
\end{array}\right.$
$\displaystyle{\mathcal{O}}_{h_{2}}\star{\mathcal{O}}_{k,p}$
$\displaystyle=\left\\{\begin{array}[]{lr}Q_{2}{\mathcal{O}}_{1,2}+Q_{1}Q_{2}\left([{\mathcal{O}}_{X}]-{\mathcal{O}}_{h_{2}}\right)&\mathrm{if}\>k=1,p=n\\\
Q_{2}{\mathcal{O}}_{k,1}&\mathrm{if}\>k>1,\>p=n\\\
{\mathcal{O}}_{n-1,n}+Q_{2}\left([{\mathcal{O}}_{X}]-{\mathcal{O}}_{h_{2}}\right)&\mathrm{if}\>k=n,p=n-1\\\
{\mathcal{O}}_{k,p+1}&\mathrm{if}\>p<n,\>k\neq p+1\\\
{\mathcal{O}}_{p,p+1}+{\mathcal{O}}_{p-1,p}-{\mathcal{O}}_{p-1,p+1}&\mathrm{if}\>1<k<n-1,\>k=p+1\\\
\end{array}\right.$

Note that for $X$ a cominuscule variety, a Chevalley formula in small quantum
$K$-theory is given by Buch-Chaput-Mihalcea-Perrin [BCMP16].

Since $({\mathcal{O}}_{w})_{w\in W^{P}}$ is a basis of $K(Fl_{1,n-1})$, for
any elements $u$ and $v$ in $W^{P}$, there is a unique decomposition

${\mathcal{O}}_{u}\star{\mathcal{O}}_{v}=\sum_{d_{1},\>d_{2}\in\mathbb{N}}\sum_{w\in
W^{P}}a_{u,v}^{w;d_{1},d_{2}}Q_{1}^{d_{1}}Q_{2}^{d_{2}}{\mathcal{O}}_{w}.$

We call the polynomials
$\sum_{d_{1},\>d_{2}\in\mathbb{N}}a_{u,v}^{w;d_{1},d_{2}}Q_{1}^{d_{1}}Q_{2}^{d_{2}}$
Littlewood-Richardson coefficients in $QK_{s}(Fl_{1,n-1})$. We call the ring
$QK_{s}(Fl_{1,n-1})$ positive if for any elements $u$ and $v$ in $W^{P}$,
$d_{1}$, $d_{2}$ in $\mathbb{N}$ the sign of the coefficient
$a_{u,v}^{w;d_{1},d_{2}}$ is given by the following inequality.

$(-1)^{{\rm codim}X(w)-{\rm codim}X(u)-{\rm
codim}X(v)+\int_{d_{1}[l_{1}]+d_{2}[l_{2}]}c_{1}(T_{X})}a_{u,v}^{w;d_{1},d_{2}}\geq
0.$

Note that for $d_{1}=0=d_{2}$, this inequality is satisfied according to M.
Brion’s result on positivity in the Grothendieck groups of generalized flag
varieties [Bri02]. Finally, we call an algorithm positive if at each iteration
the computation performed is a sum of coefficients having the same sign.

######  Proposition.

The algorithm given Subsection 3.8.1 is a positive algorithm computing
Littlewood-Richardson coefficients in $QK_{s}(Fl_{1,n-1})$.

######  Corollary.

(Cf. Proposition 3.8.2.1) The small quantum $K$-theory ring
$QK_{s}(Fl_{1,n-1})$ is positive.

######  Conjecture.

The closed formula described in Subsection 3.8.3 yields Littlewood-Richardson
coefficients in $QK_{s}(Fl_{1,n-1})$.

## Definitions and notations

###  0.1 Space of stable maps $\overline{{\mathcal{M}}_{0,r}}(Y,\gamma)$.

Definitions. Let $Y$ be a complex projective smooth variety. Let $C$ be a
connecetd reduced nodal curve of genus zero equipped with $r$ distinct non
singular marked points. A morphism $C\rightarrow Y$ from $C$ to the variety
$Y$ is called a stable map if each irreducible component of $C$ sent onto a
point contains at least three points that are nodal or marked. Let $\gamma$ be
an element in $H_{2}(Y,{\mathbb{Z}})$ corresponding to an effective class of
curve. We say the morphism $\varphi:C\rightarrow Y$ represents the class
$\gamma$ if $\varphi_{*}[C]=\gamma\in H_{2}(Y,{\mathbb{Z}})$. Let $S$ be a
complex scheme. A family of quasi-stable maps with $r$ marked points above $S$
is given by

  * •

A flat and projective morphism $\pi:\mathcal{C}\rightarrow S$ equipped with
$r$ sections $p_{i}:S\rightarrow\mathcal{C}$, such that each geometrical fiber
is a reduced nodal projective connected curve of genus zero $C_{s}$ and such
that the marked points $p_{i}(s)$ are distinct and non singular.

  * •

A morphism $\mathcal{C}\rightarrow Y$.

The space $\overline{{\mathcal{M}}_{0,r}}(Y,\gamma)$ of genus $0$ stable maps
to $Y$ representing $\gamma$ is the coarse moduli space parametrizing families
of stable maps with $r$ marked points above $S$ from genus $0$ curves to $Y$
representing the class $\gamma$, such that the geometric fiber
$\mathcal{C}_{s}$ above any point$s$ in $S$ defines a stable map
$\mathcal{C}_{s}\rightarrow Y$ [FP96]. For a homogeneous variety $Y$, the
space $\overline{{\mathcal{M}}_{0,r}}(Y,\gamma)$ is an irreducible rational
projective variety, ofe dimension the expected dimension and with quotient
singularities [FP96, KP01]. Furthermore, the subvariety
${{\mathcal{M}}_{0,r}}(Y,\gamma)$ parametrizing maps
${\mathbb{P}}^{1}\rightarrow Y$ is a dense open subset of
$\overline{{\mathcal{M}}_{0,r}}(Y,\gamma)$. A general point in
$\overline{{\mathcal{M}}_{0,r}}(Y,\gamma)$ thus corresponds to a map
${\mathbb{P}}^{1}\rightarrow Y$ representing the class $\gamma$.

Note that a general point in $\overline{{\mathcal{M}}_{0,r}}(Y,\gamma)$ is not
necessarily associated with an injective map ${\mathbb{P}}^{1}\to Y$. This is
immediate if $Y={\mathbb{P}}^{1}$; indeed, a general point in
$\overline{{\mathcal{M}}_{0,r}}({\mathbb{P}}^{1},d)$ is associated with a
degree $d$ map ${\mathbb{P}}^{1}\rightarrow{\mathbb{P}}^{1}$. This phenomenon
may also be observed in higher dimensions. Consider for example the full flag
variety $Y=Fl_{1,2}={\rm GL}_{3}/B$ parametrizing pairs $(p,l)$ where $p$ is a
point in ${\mathbb{P}}^{2}$ and $l$ is a line in ${\mathbb{P}}^{2}$ taht
contains $p$. Fix a line $l$ in ${\mathbb{P}}^{2}$. Denote by $\gamma$ the
class of the curve $C\simeq{\mathbb{P}}^{1}\subset Fl_{1,2}$ containing all
pairs $(p,l)$, where $p\subset l$ and by
$\pi:Fl_{1,2}\rightarrow{\mathbb{P}}^{2}$ the map forgetting the line
$l^{\prime}$ in a pair $(p,l^{\prime})$. Then the restriction of $\pi$ to the
curve $C$ is an isomorphism onto its image, which is the line
$l\subset{\mathbb{P}}^{2}$. Denote by $\pi^{\prime}:Fl_{1,2}\rightarrow{\rm
Gr}(2,3)$ the map induced by forgetting the point $p$ in a pair $(p,l)$. An
irreducible curve $C^{\prime}$ in $Fl_{1,2}$ of class $2\gamma$ satisfies
$\pi_{*}[C^{\prime}]=2$ in $A_{*}({\mathbb{P}}^{2})$. Furthermore its image
$\pi(C^{\prime})$ in ${\mathbb{P}}^{2}$ is included in the line in
${\mathbb{P}}^{2}$ defined by the image $\pi^{\prime}(C^{\prime})\in{\rm
Gr}(2,3)$. Then the projection $\pi$ induces a degree $2$ morphism between the
curve $C^{\prime}$ and its image $\pi(C^{\prime})$, which contradicts the fact
that the restriction of $\pi$ to $C^{\prime}$ is an injection. There thus is
no irreducible curve of class $2\gamma$ in $Fl_{1,2}$, and a general point in
${{\mathcal{M}}_{0,r}}(Fl_{1,2},2\gamma)\hookrightarrow\overline{{\mathcal{M}}_{0,r}}(Fl_{1,2},2\gamma)$
corresponds to a degree $2$ map ${\mathbb{P}}^{1}\rightarrow Fl_{1,2}$.

$G$-action on $\overline{{\mathcal{M}}_{0,r}}(Y,\gamma)$. Let $G$ be a complex
algebraic group, $Y$ be a smooth projective $G$-variety (in this thesis $G$
will be the group ${\rm GL}_{n}$ of invertible matrices and $Y$ will be a flag
variety parametrizing flags of vector spaces included in ${\mathbb{C}}^{n}$).
Let $\gamma$ be an element of $H_{2}(Y,{\mathbb{Z}})$ corresponding to an
effective class of curve. We consider the natural transformation of functors
associating to a morphism $\phi:S\rightarrow G$ and to a family of quasi-
stable maps $(\pi:\mathcal{C}\rightarrow S,\mu:\mathcal{C}\rightarrow Y)$
above $S$ with $r$ marked points representing the class $\gamma$ the following
family:

${\mathcal{C}}$${Y}$${S}$$\scriptstyle{G\cdot\mu}$$\scriptstyle{\pi}$

where we denote by $G\cdot\mu:\mathcal{C}\rightarrow Y$ the morphism
$\mathcal{C}\ni c\rightarrow\Phi\circ\pi(c)\cdot\mu(c)\in Y$ translating
curves $\mu(\mathcal{C}_{s})$ by the element in $G$ image of $s$ by $\Phi$.
This natural transformation of functors induces an action
$\overline{{\mathcal{M}}_{0,r}}(Y,\gamma)\times
G\rightarrow\overline{{M}_{0,r}(Y,\gamma)}$, associating to an element $g$ in
$G$ and to a map $\mu:C\rightarrow Y$ the map

$g\cdot\mu:C\ni p\rightarrow g\cdot\mu(p)\in Y.$

###  0.2 Flag varieties

We recall here some classical definitions and properties of flag varieties.
Let $n>0$. We consider a collection $I=\\{i_{1},\dots,i_{m}\\}$ of $m$ non-
negative integers $i_{k}$ satisfying $i_{0}:=0<i_{1}<\dots<i_{m}<i_{m+1}:=n$.
We denote by $Fl_{I}$ the variety parametrizing flags of vector spaces

$V_{i_{1}}\subset V_{i_{2}}\subset\dots\subset
V_{i_{m}}\subset{\mathbb{C}}^{n},$

where $V_{i_{k}}$ is a vector subspace of ${\mathbb{C}}^{n}$ of dimension
$i_{k}$. Note that the variety $Fl_{I}$ represents the functor $Sch\to Sets$
associating to a scheme $S$ the following set:

$\\{\mathcal{F}_{1}\subset\dots\subset\mathcal{F}_{m}\subset{\mathcal{O}}_{S}^{\oplus
n}\\},$

where the fiber bundles over $S$ $\mathcal{F}_{k}$ have rank $i_{k}$ and the
quotient sheaves $\mathcal{F}_{k}/\mathcal{F}_{k-1}$ are rank
$(i_{k}-i_{k-1})$ fiber bundles. Furthermore, we consider the complex
algebraic group $G={\rm GL}_{n}$, the maximal torus $T$ of $G$ given by the
set of diagonal matrices and the Borel subgroup $B\subset G$ given by the set
of upper triangular matrices. We denote by $W\simeq\mathfrak{S}_{n}$ the Weyl
group of $G={\rm GL}_{n}$, and $\Delta={\alpha_{1},\dots,\alpha_{n}}$ the set
of simple roots positive with respect to $B$. We denote by
$\Delta(I)=\\{\alpha_{1},\dots\alpha_{i_{1}-1},\dots,\alpha_{i_{m}-1}\\}$ the
subset of $\Delta$ associated with $I$, and by
$W_{I}\simeq\prod_{k=1}^{m+1}\mathfrak{S}_{i_{k}-i_{k-1}}$ the subgroup of $W$
generated by reflexions $s_{\alpha}$ associated with elements $\alpha$ in
$\Delta(I)$. Finally, we denote by $P_{I}:=BW_{I}B$ the parabolic subgroup
associated with $\Delta(I)$. Note that the block diagonal matrix whose $k$-th
block is $GL_{i_{k}-i_{k-1}}$ is a Levi sugroup $L_{I}$ associated with
$P_{I}$. We have the following identification: $Fl_{I}\simeq G/P_{I}$.

Let $1\leq k<m$. Composing the map forgetting all vector spaces except from
the $k$-th one with the Plücker embedding ${\rm
Gr}(i_{k},n)\hookrightarrow{\mathbb{P}}^{\binom{n}{i_{k}}-1}$ induces a map
$\varphi:Fl_{I}\rightarrow{\mathbb{P}}^{\binom{n}{i_{k}}-1}$. Let $l_{i}$ be a
curve on $Fl_{I}$ sent onto a line in ${\mathbb{P}}^{\binom{n}{i_{k}}-1}$ by
$\varphi$, and such that the restriction of $\varphi$ to $l_{i}$ is an
isomorphism. The classes $[l_{i}]$ generate the free abelian group
$H_{2}(Fl_{I},{\mathbb{Z}})$, and we may thus identify an effective class of
curve ${\mathbf{d}}=\sum_{k=1}^{m}d_{k}[l_{k}]$ in $E(Fl_{I})\subset
H_{2}(Fl_{I},{\mathbb{Z}})$ with $m$ positive integers $(d_{1},\dots,d_{m})$.

###  0.3 Algebraic $K$-theory

####  0.3.1 Equivariant algebraic $K$-theory.

We recall here some classical definitions and properties of the equivariant
$K$-theory of a variety; complete proofs can be found in [C+09], chapter 5.
Let $H$ be a complex linear algebraic group, let $X$ be an irreducible variety
with an $H$-action. Let $E$ be a vector bundle on $X$. We call $E$ an
$H$-equivariant vector bundle if there exists an $H$-action $\Phi:H\times
E\rightarrow E$ satisfying

  * –

The map $E\rightarrow X$ commutes with $H$-actions.

  * –

For any $x$ in $X$ and $g$ in $G$, the map $\Phi(g,\bullet):E_{x}\rightarrow
E_{g\cdot x}$ is a linear map of vector spaces,

where we denote by $E_{x}$ the fiber over a point $x$ in $X$. Let
$\mathcal{F}$ be a coherent sheaf on $X$. Denote by $a:H\times X\rightarrow X$
the action of $H$ on $X$, by $p:H\times X\rightarrow X$ the projection map and
by $p_{2\>3}:H\times H\times X\rightarrow H\times X$ the projection along the
first factor $H$. We call $\mathcal{F}$ an $H$-equivariant coherent sheaf if
there exists an $H$-action $\Phi:H\times\mathcal{F}\rightarrow\mathcal{F}$
satisfying

  * –

There exists an isomorphism $I:a^{*}\mathcal{F}\rightarrow p^{*}\mathcal{F}$.

  * –

${p_{2\>3}}^{*}I\circ({\rm id}_{H}\times a)^{*}I=(m\times{\rm id}_{X})^{*}I$
of sheaves on $H\times X$.

  * –

There is an identification $I_{1\times X}={\rm
id}:\mathcal{F}=a^{*}\mathcal{F}_{|1\times X}\rightarrow
p^{*}\mathcal{F}_{|1\times X}=\mathcal{F}$.

Denote by $K_{H}(X)$ the Grothendieck group of $H$-equivariant coherent
sheaves on $X$ and by $K^{H}(X)$ the Grothendieck group of $H$-equivariant
vector bundles on $X$. The tensor product of equivariant vector bundles
defines a ring structure on $K^{H}(X)$. Moreover, the map $K^{H}(X)\otimes
K_{H}(X)\rightarrow K_{H}(X)$ defined by the tensor product

$[E]\otimes[\mathcal{F}]\rightarrow[E\otimes_{\mathcal{O}_{X}}\mathcal{F}]$

makes $K_{H}(X)$ a $K^{H}(X)$-module. When $X$ is non singular, the map
$[E]\rightarrow[E\otimes{\mathcal{O}}_{X}]$ yields an isomorphism
$K_{H}(X)\simeq K^{H}(X)$. Indeed, any equivariant sheaf then has a finite
resolution by equivariant vector bundles.
Furthermore, for any equivariant proper morphism of irreducible varieties with
an $H$-action $f:X\rightarrow Y$, there is a pushforward map
$f_{*}:K_{H}(X)\rightarrow K_{H}(Y)$ defined by:

$[\mathcal{F}]\rightarrow\sum_{i}(-1)^{i}[R^{i}f_{*}(\mathcal{F})].$

Indeed, if $\mathcal{F}$ is an $H$-equivariant sheaf on $X$, the Godement
resolution implies that $R^{i}f_{*}(\mathcal{F})$ will also be an
$H$-equivariant sheaf on $Y$.
In particular, if $X$ is an irreducible projective $H$-variety, the
equivariant pushforward $K_{H}(X)\rightarrow K_{H}(point)$ is called
equivariant Euler characteristic, and denoted by $\chi_{H}$.
Moreover, an equivariant morphism $f:X\rightarrow Y$ induces a pullback map
$f^{*}:K^{H}(Y)\rightarrow K^{H}(X)$:

$[E]\rightarrow[f^{*}(E)].$

Consider an equivariant proper morphism $f:X\rightarrow Y$ of irreducible
varieties with an $H$-action. Let $E$ be an equivariant vector bundle on $Y$,
and $\mathcal{F}$ be an equivariant coherent sheaf on $X$. Note that the
following projection formula is satisfied in $K_{H}(Y)$:

$f_{*}(f^{*}[E]\otimes[\mathcal{F}])=[E]\otimes f_{*}[\mathcal{F}].$

####  0.3.2 Algebraic K-theory of $G/P$.

We recall here some classical definitions and properties of the Grothendieck
ring of coherent sheaves on a variety. Denote by $K_{\circ}(X)$ the
Grothendieck group of coherent sheaves on $X$ and by $K^{\circ}(X)$ the
Grothendieck group of vector bundles on $X$. Note that for $H={\rm id}$, the
equivariant groups $K_{H}(X)$ and $K^{H}(X)$ are equal to $K_{\circ}(X)$ and
$K^{\circ}(X)$. In particular, the tensor product of vector bundles defines a
ring structure on $K^{\circ}(X)$, and for a non singular variety $X$ we can
identify $K(X)=K_{\circ}(X)\simeq K^{\circ}(X)$. Furthermore, for any coherent
sheaves $\mathcal{F}$ and $\mathcal{G}$ on $X$ we have

$[\mathcal{F}]\cdot[\mathcal{G}]=\sum_{i=0}^{{\rm
dim}X}(-1)^{i}Tor_{i}^{X}(\mathcal{F},\mathcal{G}).$

Furthermore, note that for an irreducible projective variety $X$, the
pushforward $K_{\circ}(X)\to K_{\circ}({\rm Spec}{\mathbb{C}})$ is given by:

$\displaystyle\chi:K_{\circ}(X)\rightarrow{\mathbb{Z}},$
$\displaystyle[\mathcal{F}]\rightarrow\chi(\mathcal{F})=\sum_{i}(-1)^{i}h^{i}(\mathcal{F}),$

where $h^{i}$ denote the dimension of the $i$-th cohomology group of
$\mathcal{F}$; i.e. $\chi$ is the Euler-Poincaré characteristic of
$\mathcal{F}$. A morphism of schemes $f:X\rightarrow Y$ is called perfect if
there is an $N$ such that

$Tor_{i}^{Y}({\mathcal{O}}_{X},\mathcal{F})=0$

for all $i>N$ and for all coherent sheaves $\mathcal{F}$ on $Y$. For a perfect
morphism $f:X\rightarrow Y$, one can define the pull-back

$f^{*}:K_{\circ}(Y)\rightarrow K_{\circ}(X)$

by

$f^{*}[\mathcal{F}]=\sum_{i=0}^{N}[Tor_{i}^{Y}({\mathcal{O}}_{X},\mathcal{F})].$

Furthermore suppose $f$ is a morphism locally of finite type. Then if $f$ is
flat or $Y$ is regular, $f$ is perfect-cf. for example Stacks Project $37.53$.
We will also be using the following easy results.

######  Lemma 0.3.2.1.

Let $Y$ be an irreducible projective variety and $\Phi$ be an irreducible
projective subvariety of $\mathbb{P}^{1}\times Y$. Denote by
$\pi:\Phi\rightarrow\mathbb{P}^{1}$ the natural projection. Let $p_{1}$,
$p_{2}$ be two points on $\mathbb{P}^{1}$. Then we have the following equality
in $K_{\circ}(Y)$:

$[\mathcal{O}_{\pi^{-1}(p_{1})}]=[\mathcal{O}_{\pi^{-1}(p_{2})}]$

###### Proof.

Since the projection map $\pi$ is flat, we have
$\pi^{*}[{\mathcal{O}}_{p_{i}}]=[{\mathcal{O}}_{\pi^{-1}(p_{i})}]$ in
$K_{\circ}(\Phi)$. Denote by $\Pi:\Phi\rightarrow Y$ the natural projection.
We obtain
$\Pi_{*}\pi^{*}[{\mathcal{O}}_{p_{i}}]=\Pi_{*}[{\mathcal{O}}_{\pi^{-1}(p_{i})}]=[{\mathcal{O}}_{\pi^{-1}(p_{i})}]$
in $K_{\circ}(Y)$. Finally, the equality
$[{\mathcal{O}}_{p_{1}}]=[{\mathcal{O}}_{p_{2}}]$ in
$K_{\circ}({\mathbb{P}}^{1})$ yields the result. ∎

######  Lemma 0.3.2.2.

Let $X$ be a projective ${\rm GL}_{n}$-variety. Consider a subvariety $Y$ of
$X$. Then for all $g$ in ${\rm GL}_{n}$, we have the following equality in
$K_{\circ}(X)$:

$[\mathcal{O}_{g\cdot Y}]=[\mathcal{O}_{Y}].$

###### Proof.

Let $g$ in ${\rm GL}_{n}$. Since ${\rm GL}_{n}$ is a dense open subset of
$\mathcal{M}_{n,n}\simeq\mathbb{A}^{n^{2}}$, there exits an irreducible
rational curve $f:{\mathbb{P}}^{1}\supset U\rightarrow{\rm GL}_{n}$ sending
points $p_{1}$ and $p_{2}$ on ${\mathbb{P}}^{1}$ onto the points $g$ and ${\rm
id}$. Denote by $\Phi$ the irreducible projective subvariety of
${\mathbb{P}}^{1}\times X$ obtained as the adherence of the image of the map
$U\times Y\rightarrow U\times Y\subset{\mathbb{P}}^{1}\times X$ sending
$(x,y)$ to $(x,f(x)\cdot y)$. Denote by $\pi:\Phi\rightarrow{\mathbb{P}}^{1}$
the first projection. Then according to Lemma 0.3.2.1

$[{\mathcal{O}}_{\pi^{-1}(p_{1})}]=[{\mathcal{O}}_{\pi^{-1}(p_{2})}]\>\mathrm{hence}\>[\mathcal{O}_{g\cdot
Y}]=[\mathcal{O}_{Y}].$

∎

###  0.4 Rational singularities.

We recall here known definitions and results on rational singularities. The
treatment in this section is inspired by [Bri02]. An irreducible complex
variety $X$ has rational singularities if there exists a desingularization
$\psi:\widetilde{X}\rightarrow X$ such that:

$\psi_{*}{\mathcal{O}}_{\widetilde{X}}={\mathcal{O}}_{X}\>\mathrm{and}\>\>\forall
i>0,\>R^{i}\pi_{*}{\mathcal{O}}_{\widetilde{X}}=0.$ (2)

Note that, according to Zariski’s main theorem, the equality
$\psi_{*}{\mathcal{O}}_{\widetilde{X}}={\mathcal{O}}_{X}$ is equivalent to the
normality of $X$. The relevance of this notion here comes from the following
results.

######  Theorem 0.4.0.1.

(Viehweg [Vie77]) The quotient of a smooth complex scheme by a finite group
has rational singularities.

Note that, since for $X$ convex the moduli spaces
$\overline{{\mathcal{M}}_{0,r}}(X,{\mathbf{d}})$ have quotient singularities
[FP96], this implies that for $X$ a generalized flag variety the moduli spaces
$\overline{{\mathcal{M}}_{0,r}}(X,{\mathbf{d}})$ have rational singularities.

Let $H$ be a complex linear algebraic group. Recall a variety $X$ is called
unirational if there exists a dominant map ${\mathbb{P}}^{N}\dashrightarrow
X$, and rationally connected if there exists an irreducible rational curve
joining two general points in $X$. Note that since ${\mathbb{P}}^{N}$ is
rationally connected, a unirational variety is rationally connected.

######  Theorem 0.4.0.2.

(Buch-Mihalcea [BM11], Theorem 3.1.) Let $f:X\rightarrow Y$ be a surjective
equivariant morphism of projective varieties with rational singularities
equipped with an $H$-action. Assume that the general fiber of $f$ is
rationally connected. Then $f_{*}[{\mathcal{O}}_{X}]=[{\mathcal{O}}_{Y}]\in
K_{H}(Y)$.

###### Proof.

Buch and Mihalcea prove in [BM11] that this statement is satisfied if a
desingularization $\psi:\widetilde{X_{y}}\rightarrow X_{y}$ of the fiber of
$f$ over $y$ general in $Y$ satisfying
$H^{i}(\widetilde{X_{y}},{\mathcal{O}}_{\widetilde{X_{y}}})=0$ for all $i>0$.
Since $X_{y}$ is rationally connected, and $\widetilde{X_{y}}$ is birational
to $X_{y}$, $\widetilde{X_{y}}$ is also rationally connected. Hence
$\widetilde{X_{y}}$ is a smooth rationally connected variety, hence according
to [Deb13], $\widetilde{X_{y}}$ satisfies
$H^{i}(\widetilde{X_{y}},{\mathcal{O}}_{\widetilde{X_{y}}})=0$ for all $i>0$.
∎

Note that for $X$ a full flag variety, Brion gives a nice proof of the
rational singularities of Schubert varieties in [Bri05].

######  Theorem 0.4.0.3.

(Ramanathan, [Ram85]) Schubert varieties have rational singularities.

##  Chapter 1 Gromov-Witten varieties of partial flag varieties

###  1.1 Introduction

Removing subspaces from a partial flag gives another partial flag composed of
fewer subspaces. This induces a forgetful map $\pi:X\rightarrow X^{\prime}$
between the corresponding flag varieties. The main result of this chapter is
that, for a degree large enough, the variety associated with rational curves
of a given degree in $X$ having their marked points within Schubert varieties
$X_{i}$ of $X$ is a rationally connected fibration over its image, which is
the variety associated with rational curves of a given degree in $X$ having
their marked points within Schubert varieties $\pi(X_{i})$ of $X^{\prime}$.
These subvarieties of the space of genus $0$ stable maps to a homogeneous
space are called Gromov-Witten varieties. The Euler characteristics of the
structure sheaf of these varieties are quantum K-theoretical invariants
[Giv00]. When these varieties are zero dimensional, their number of points are
Gromov-Witten invariants. Our result in particular implies the equality
between (non equivariant) correlators proved in Chapter 2, providing an
alternative proof-cf. chapter 2 Section 2.4.

####  1.1.1 Definitions.

Let $n>0$. Let $I=\\{i_{1},\dots,i_{m}\\}$ be a collection of integers
satisfying $0<i_{1}<i_{2}<\dots<i_{m}<n$. Denote by $Fl_{I}$ the flag variety
parametrizing $m$-tuples of vector spaces ordered by inclusion

$V_{i_{1}}\subset V_{i_{2}}\dots\subset V_{i_{m}}\subset\mathbb{C}^{n}$ (1.1)

such that ${\rm dim}V_{i_{k}}=i_{k}$. Let $B$ be a Borel subgroup of ${\rm
GL}_{n}$. Note that $Fl_{I}$ is the homogeneous variety $Fl_{I}\simeq{\rm
GL}_{n}/P_{I}$, where $P_{I}$ is the parabolic subgroup of ${\rm GL}_{n}$
satisfying $B\subset P_{I}$ with associated reductive subgroup
$L_{I}\simeq\prod_{p=1}^{m+1}{\rm GL}(i_{p}-i_{p-1})$, where we set $i_{0}=0$
and $i_{m+1}=n$. Let $J=\\{j_{1},\dots,j_{M}\\}$ be a collection of integers
satisfying $0<j_{1}<\dots<j_{M}<n$ containing $I$, i.e. for each $k$ there
exists an element $s$ such that $j_{s}=i_{k}$. Consider an integer $p$ such
that $1\leq p\leq M$. We denote by $J_{\geq p}$ the collection $J_{\geq
p}:=\\{i_{1},\dots,i_{p^{\prime}},j_{p},\dots,j_{M}\\}$, where $p^{\prime}$ is
the largest integer such that $i_{p^{\prime}}<j_{p}$. Forgetting all vector
spaces included in $V_{j_{p}}$ except those of the form $V_{i_{k}}$ yields a
forgetful morphism

$\pi_{I/J_{\geq p}}:Fl_{J_{\geq p}}\rightarrow Fl_{I}.$

In the same way, for $1\leq p\leq M$, we name $J_{\leq
p}:=\\{j_{1},\dots,j_{p},i_{p^{\prime}+1}\dots,i_{m}\\}$, where $p^{\prime}$
is the smallest integer such that $j_{p}<i_{p^{\prime}}$. We name
$\pi_{I/J_{\leq p}}:Fl_{J_{\leq p}}\rightarrow Fl_{I}$ the morphism induced by
forgetting all vector spaces containing $V_{j_{p}}$ except those of the form
$V_{i_{k}}$.

Let $W=N_{{\rm GL}_{n}}(T)/T\simeq{\mathfrak{S}}_{n}$ be the Weyl group of the
algebraic group ${\rm GL}_{n}$. We denote by
$Fl_{J}(u)=\overline{BuP_{J}/P_{J}}$ the Schubert variety of $Fl_{J}$
associated with $u\in W$. Note that the image of a Schubert variety
$Fl_{J}(u)$ by the projection $\pi_{I/J}$ is the Schubert variety
$\pi_{I/J}(Fl_{J}(u))=Fl_{I}(u)$.

Let $r>0$. The group $H_{2}(Fl_{I},{\mathbb{Z}})$ is a free abelian group
generated by classes of Schubert varieties of dimension $1$. Denote by
$E(Fl_{I})\subset H_{2}(Fl_{I},\mathbb{Z})$ the semi-group of effective
classes of curves. Note that $H_{2}({\rm
Gr}(k,n),{\mathbb{Z}})\simeq{\mathbb{Z}}$, and the class of a curve on the
Grassmannian ${\rm Gr}(k,n)$ is given by the degree of its image in
${\mathbb{P}}^{\binom{n}{k}}$ by the Plücker embedding. The class
$\mathbf{d}=(d_{1},\dots,d_{m})\in E(Fl_{I})$ of a curve $C$ in $Fl_{I}$ is
determined by $m$ non-negative integers $d_{1}$, $d_{2}$,$\dots$, $d_{m}$,
where $d_{k}$ is the degree of the projection of $C$ to the Grassmannian ${\rm
Gr}(i_{k},n)$. For a class ${\mathbf{d}}$ in $E(Fl_{I})$, we denote by
$\overline{{\mathcal{M}}_{0,r}}(Fl_{I},{\mathbf{d}})$ the moduli space
parametrizing stable maps $C\rightarrow Fl_{I}$ from genus $0$ curves $C$ with
$r$ marked points $x_{i}\in C$ to $Fl_{I}$ representing the class
${\mathbf{d}}$. We name
$ev:\overline{{\mathcal{M}}_{0,r}}(Fl_{I},{\mathbf{d}})\rightarrow(Fl_{I})^{r}$
the evaluation morphism sending the point $p$ associated with a map
$f_{p}:C\rightarrow Fl_{I}$ to the image $\prod_{i=1}^{r}f_{p}(x_{I})$ of the
$r$ marked points $x_{i}\in C$. For any class ${\mathbf{d}}$ in
$E(Fl_{J})\simeq\mathbb{N}^{M}$, the morphism $\pi_{I/J}:Fl_{J}\rightarrow
Fl_{I}$ induces a morphism

$\Pi_{I/J}\times
ev:\overline{{\mathcal{M}}_{0,r}}(Fl_{J},{\mathbf{d}})\rightarrow\overline{{\mathcal{M}}_{0,r}}(Fl_{I},(\pi_{I/J})_{*}{\mathbf{d}})\times_{(Fl_{I})^{r}}(Fl_{J})^{r}.$
(1.2)

Let $1\leq k\leq M$ such that the integer $j_{k}$ is not in $I$. We denote by

$\Pi_{J_{\geq k}}\times ev:\overline{\mathcal{M}_{0,r}}(Fl_{J_{\geq
k}},(\pi_{I/J_{\geq
k}})_{*}{\mathbf{d}})\rightarrow\overline{\mathcal{M}_{0,r}}(Fl_{J_{\geq
k+1}},(\pi_{I/J_{\geq k+1}})_{*}{\mathbf{d}})\times_{(Fl_{J_{\geq
k+1}})^{r}}(Fl_{J_{\geq k}})^{r}$

the morphism induced by forgetting the vector space $V_{j_{k}}$ and by
evaluating the $r$ marked points. Let $1\leq\mu\leq k\leq M$ such that the
element $j_{k}$ is not in $I$. We denote by $J_{\mu\leq k}$ the collection
$\\{i_{1},\dots,i_{p},j_{\mu},\dots,j_{k},i_{p^{\prime}},\dots i_{m}\\}$,
where $p$ is the largest element such that $i_{p}<j_{\mu}$ and $p^{\prime}$ is
the smallest element such that $i_{p^{\prime}}>j_{k}$. We denote by

${\Pi}_{J_{\mu\leq k}}\times ev:\overline{\mathcal{M}_{0,r}}(Fl_{J_{\mu\leq
k}},(\pi_{J_{\mu\leq
k}/J})_{*}{\mathbf{d}})\rightarrow\overline{\mathcal{M}_{0,r}}(Fl_{J_{\mu\leq
k-1}},(\pi_{J_{\mu\leq k-1}/J})_{*}{\mathbf{d}})\times_{(Fl_{J_{\mu\leq
k-1}})^{r}}(Fl_{J_{\mu\leq k}})^{r}$

the morphism induced by forgetting the vector space $V_{j_{k}}$ from
${Fl}_{\mu\leq k}$ and evaluating the $r$ marked points.

######  Definition 1.1.1.1.

Consider a collection $\\{I,J,{\mathbf{d}}\\}$ of a set
$I=\\{i_{1},\dots,i_{m}\\}$ of $m$ positive integers satisfying

$0<i_{1}<\dots<i_{m}<n,$

a set $J=\\{j_{1},\dots,j_{M}\\}$ of $M$ positive integers satisfying

$0<j_{1}<\dots<j_{M}<n$

such that $I$ is contained in $J$, and a set
${\mathbf{d}}=(d_{1},\dots,d_{M})$ in $\mathbb{N}^{M}\simeq E(Fl_{J})$. We
denote by
${\mathbf{d}}^{\prime}=(d^{\prime}_{1},\dots,d^{\prime}_{m})\in\mathbb{N}^{m}\simeq
E(Fl_{I})$ the pushforward of ${\mathbf{d}}$ by the forgetful map $\pi_{I/J}$.
Let $1\leq\mu\leq M$ such that $j_{\mu}$ is contained in $I$. We call the
collection $\\{I,J,{\mathbf{d}}\\}$ stabilized with respect to $\mu$ if the
following conditions are satisfied. First, for all elements $1\leq k<\mu$ such
that the integer $j_{k}$ is not contained in $I$, the two following conditions
are satisfied.

  1. $\bullet$

$\mathrm{The}\>\mathrm{morphism}\>\Pi_{J_{\geq k}}\times
ev\>\mathrm{is}\>\mathrm{surjective.}$ (1.3)

  2. $\bullet$

Let $k^{\prime}$ be the largest integer such that $i_{k^{\prime}}<j_{k}$. For
any $1\leq p<k^{\prime}$ we have

$d_{k}\geq\left\lceil\left(d^{\prime}_{p}-d^{\prime}_{p-1}\right)/(i_{p}-i_{p-1})\right\rceil
j_{k}.$ (1.4)

Futhermore, for all $\mu\leq k\leq M$ such that the integer $j_{k}$ is not
contained in $I$ the two following symmetric conditions are satisfied.

  1. $\bullet$

The morphism ${\Pi}_{J_{\mu\leq k}}\times ev$ is surjective.

  2. $\bullet$

Let $k^{\prime}$ be the smallest integer such that $i_{k^{\prime}}>j_{k}$. For
$k^{\prime}<p\leq m$ we have

$d_{k}\geq\left\lceil\left(d^{\prime}_{p}-d^{\prime}_{p+1}\right)/(i_{p+1}-i_{p})\right\rceil(n-j_{k}).$

A collection $\\{I,J,{\mathbf{d}}\\}$ will be called stabilized if there
exists an integer $\mu$ such that the collection $\\{I,J,{\mathbf{d}}\\}$ is
stabilized with respect to $\mu$. A collection $\\{I,J,{\mathbf{d}}\\}$ will
be called stabilized with respect to $M$ if for all $1\leq k\leq M$ such that
$j_{k}$ is not contained in $I$ the conditions (1.3) and (1.4) are satisfied.

Observe that when $\mu=1$, the conditions (1.3) and (1.4) are empty. For
simplicity we have set $d^{\prime}_{0}=d^{\prime}_{m+1}=0=i_{0}$ and
$i_{m+1}=n$. We have denoted by $\lceil q\rceil$ the ceiling of
$q\in\mathbb{Q}$. We will see in Section 1.6 some conditions to ensure that a
collection is stabilized. The following interesting cases yield examples of
stabilized collections (cf. Subsection 1.6.1).

  1. 1.

Suppose $I=\\{i_{1},\dots,i_{m}\\}$ is a collection of integers satisfying
$1<i_{1}<\dots<i_{m}<n$ and $J=\\{j_{1},i_{1},\dots,i_{m}\\}$ satisfies
$0<j_{1}<i_{1}$. Note that then $Fl_{J}\rightarrow Fl_{I}$ is the morphism
forgetting the first vector space. Then if $d_{1}\geq j_{1}\left(r+1+\lfloor
d_{2}/i_{1}\rfloor\right)$ the collection $(I,J,(d_{1},d_{2},\dots d_{m+1}))$
is stabilized.

  2. 2.

Suppose $J=\\{k\\}$ and $I=\emptyset$; then $Fl_{J}$ is the Grassmannian ${\rm
Gr}(k,n)$ and $Fl_{I}$ is the point ${\rm Spec}({\mathbb{C}})$. Then if $d\geq
rk$ or $d\geq r(n-k)$ the collection $(I,J,d)$ is stabilized.

Note that the dependency on the integer $r$ in the examples here above comes
from the condition asking the surjectivity of the following morphisms.

${\Pi}_{J_{\mu\leq k}}\times ev:\overline{\mathcal{M}_{0,r}}(Fl_{J_{\mu\leq
k}},(\pi_{J_{\mu\leq
k}/J})_{*}{\mathbf{d}})\rightarrow\overline{\mathcal{M}_{0,r}}(Fl_{J_{\mu\leq
k-1}},(\pi_{J_{\mu\leq k-1}/J})_{*}{\mathbf{d}})\times_{(Fl_{J_{\mu\leq
k-1}})^{r}}(Fl_{J_{\mu\leq k}})^{r}$

######  Definition 1.1.1.2.

Let $\mathbf{d}$ be an element in $E(Fl_{I})$, let $u_{1}$, $\dots$, $u_{r}$
be elements in $W$. We call Gromov-Witten variety of degree $\mathbf{d}$
associated with $u_{1}$, $\dots$, $u_{r}$ the subvariety

${\mathcal{W}}_{Fl_{I};u_{1},\dots,u_{r}}^{g,\mathbf{d}}:=ev_{1}^{-1}(g_{1}\cdot
Fl_{I}(u_{1}))\cap\dots\cap ev_{r}^{-1}(g_{r}\cdot Fl_{I}(u_{r}))$

of $\overline{\mathcal{M}_{0,r}}(Fl_{I},\mathbf{d})$.

Note that for $\mathbf{d}$ in $E(Fl_{I})$, $u_{1}$, $\dots$, $u_{r}$ in $W$,
${\mathcal{W}}_{Fl_{I};u_{1},\dots,u_{r}}^{g,\mathbf{d}}$ is the variety
associated with degree ${\mathbf{d}}$ rational curves on $X$ having their
$i$-th marked point within $g_{i}\cdot Fl_{I}(u_{i})$. Denote by $1$ the
identity in $W\simeq\mathfrak{S}_{n}$; note that $Fl_{I}(1):=P_{I}/P_{I}$ a
point in $Fl_{I}$. Furthermore, note that
${\mathcal{W}}_{Fl_{I};1,\dots,1}^{g,\mathbf{d}}$ is the variety associated
with degree ${\mathbf{d}}$ rational curves on $X$ whose $i$-th marked point is
the point $g_{i}P_{I}/P_{I}\in Fl_{I}$.

######  Remark 1.1.1.1.

According to the proof of Kleiman’s transversality theorem [Kle74] for a
general $g\in{\rm GL}_{n}^{r}$ the fiber product
$overline{\mathcal{M}_{0,r}}(Fl_{I},\mathbf{d})\underset{(Fl_{I})^{r}}{\times}(g_{1}\cdot
Fl_{I}(u_{1})\times\dots\times g_{r}\cdot Fl_{I}(u_{r}))$ is a reduced
equidimensional scheme. Hence for $g$ general in ${\rm GL}_{n}^{r}$, the
natural map

$\overline{\mathcal{M}_{0,r}}(Fl_{I},\mathbf{d})\underset{(Fl_{I})^{r}}{\times}(g_{1}\cdot
Fl_{I}(u_{1})\times\dots\times g_{r}\cdot
Fl_{I}(u_{r}))\to\overline{\mathcal{M}_{0,r}}(Fl_{I},\mathbf{d})\supset{\mathcal{W}}_{Fl_{I};u_{1},\dots,u_{r}}^{g,\mathbf{d}}$

is a closed immersion surjecting into
${\mathcal{W}}_{Fl_{I};u_{1},\dots,u_{r}}^{g,\mathbf{d}}$, and we can identify
the two schemes.

####  1.1.2 Main results.

######  Definition 1.1.2.1.

Let $Y$, $Z$ be varieties. We call $Y$ a rationally connected fibration over
$Z$ (respectively unirational fibration) if there exists a morphism
$Y\rightarrow Z$ such that each irreducible component of $Y$ dominates a
different irreducible component of $Z$, $Y$ dominates $Z$, and the general
fiber $Y\to Z$ is a rationally connected (respectively unirational)
irreducible variety. We call $Y$ a tower of unirational fibrations over $Z$ if
there exists a sequence of morphisms

$Y_{1}=Y\rightarrow Y_{2}\rightarrow\dots\rightarrow Y_{n}=Z$

such that, for all $1\leq i<n$, $Y_{i}$ is a unirational fibration over
$Y_{i+1}$.

######  Theorem 1.1.2.1.

Consider a set $J=\\{j_{1},\dots,j_{M}\\}$ of $M$ integers satisfying
$0<j_{1}<\dots<j_{M}<n$, a set $I=\\{i_{1},\dots,i_{m}\\}$ of $m$ integers
satisfying $0<i_{1}<\dots<i_{m}<n$ and a degree
$\mathbf{d}=(d_{1},\dots,d_{M})\in E(Fl_{J})\simeq\mathbb{N}^{M}$ such that
the collection $\\{I,J,{\mathbf{d}}\\}$ is stabilized in the sense of
Definition 1.1.1.1.

  1. i)

For $x$ general in $(Fl_{J})^{r}$, the fiber
$ev^{-1}(x)\subset\overline{\mathcal{M}_{0,r}}(Fl_{J},{\mathbf{d}})$ is a
tower of unirational fibrations over the fiber
$ev^{-1}(\pi_{I/J}(x))\subset\overline{{\mathcal{M}}_{0,r}}(Fl_{I},(\pi_{I/J})_{*}\mathbf{d})$.

  2. ii)

Let $u_{1}$, $\dots$, $u_{r}$ be elements in $W$. For $g$ general in ${\rm
GL}_{n}^{r}$ the Gromov-Witten variety
${\mathcal{W}}_{Fl_{J};u_{1},\dots,u_{r}}^{g,\mathbf{d}}$ is a rationally
connected fibration over the Gromov-Witten variety
${\mathcal{W}}_{Fl_{I};u_{1},\dots,u_{r}}^{g,(\pi_{I/J})_{*}\mathbf{d}}$.

Let $J$ be a subsrt of $I$. Denote by $W_{J}:=N_{L_{J}}(T)/T$ the Weyl group
associated with $P_{J}$. Denote by

$p:W/W_{J}\rightarrow W/W_{I}$

the natural projection. For $u$ in $W/W_{I}$, the inverse image $p^{-1}(u)$ is
the set of elements $u\cdot s$, where $s$ is an element in $W_{I}/W_{J}$. Note
that the Schubert variety $FL_{I}(u):=\overline{BuP_{I}/P_{I}}$ is independent
of the choice of a representative in $W$ of the element $u$.

######  Corollary 1.1.2.1.

Let ${\mathbf{d}}=(d_{1},\dots,d_{m})$ be a degree in
$E(Fl_{I})\simeq\mathbb{N}^{m}$ and $u_{1}$, $\dots$, $u_{r}$ be elements in
$W$ such that the following Gromov-Witten invariant is non-zero:

$n_{u_{1},\dots,u_{r}}^{{I},{\mathbf{d}}}:=\int_{\overline{{\mathcal{M}}_{0,r}}(Fl_{I},\mathbf{d})}ev_{1}^{*}[Fl_{I}(u_{1})]\cup\dots\cup
ev_{r}^{*}[Fl_{I}(u_{r})]\neq 0.$

Let $J$ and $I$ be sets of increasing positive integers and ${\mathbf{d}}$ be
a degree in $E(Fl_{J})$ such that the collection $\\{I,J,{\mathbf{d}}\\}$ is
stabilized in the sense of Definition 1.1.1.1. Then for any elements $v_{i}\in
p^{-1}(u_{i})$, for $g$ general in $GL_{n}^{r}$ the Gromov-Witten variety
$\mathcal{W}_{Fl_{J};v_{1},\dots,v_{r}}^{g,{\mathbf{d}}}$ has
$n_{u_{1},\dots,u_{r}}^{{I},{\mathbf{d}}}$ connected components, which are
irreducible rationally connected varieties.

####  1.1.3 Previous research.

Gromov-Witten varieties of ${\mathbb{P}}^{1}$ and ${\mathbb{P}}^{2}$ that are
not rationally connected varieties have been found in [IMT14]. For
${\mathbb{P}}^{2}$, the locus of
$\overline{\mathcal{M}_{0,3d-2}}(\mathbb{P}^{2},d)$ corresponding to curves of
degree $d$ going through $3d-2$ points is a curve whose genus depends on the
degree $d$, of positive genus as soon as $d>2$ [Pan97]. A recursive procedure
computing the geometric genus of $1$-dimensional Gromov-Witten varieties of
$\mathbb{P}^{r}$ was also given in [Ran01]. High degree Gromov-Witten
varieties have been shown to be rational varieties for Grassmannians by Buch
and Milhacea [BM11], and more generally for cominuscule varieties by Chaput
and Perrin [CP11a].

This chapter is organized as follows. We first prove in Section 1.2 that, when
the general fiber of the morphism $\Pi_{I/J}\times ev$ defined by (1.2) is
rationally connected, Gromov-Witten varieties of $Fl_{I}$ are rationally
connected fibrations over Gromov-Witten varieties of $Fl_{J}$ obtained by
projection. We then recall in Section 1.3 results on flags of vector bundles
over ${\mathbb{P}}^{1}$. Section 1.4 is dedicated to the study of the general
fiber of the map $\Pi_{I/J}\times ev$. We observe that when the flag variety
$Fl_{I}$ is obtained from $Fl_{J}$ by forgetting one vector space, the general
fiber of this map is a unirational variety. This result is central in the
derivation of Theorem 1.1.2.1 presented in Section 1.5. Finally, we study the
surjectivity of the morphism $\Pi_{I/J}\times ev$ in Section 1.6.

### Conventions

Schemes considered in the subsections 1.2.2 and 1.2.3 will be defined over an
algebraically closed field. Schemes considered in all other Sections will be
defined over $\mathbb{C}$.

We say a property (P) holds for a general point in a variety $X$ if (P) is
true for points belonging to a dense open subset of $X$. We say the general
fiber of a morphism $f:X\rightarrow Y$ satisfies $(P)$ if $f^{-1}(x)$
satisfies $(P)$ for points $x$ in a dense open subset of $Y$. Let $G$ be an
integral algebraic group scheme, let $Y$ be an integral scheme endowed with a
transitive $G$-action, let $Z$ be a subscheme of $Y$, and $M\rightarrow Y$ be
a morphism of integral schemes. We say a property (P) holds for a subscheme
$M\times_{Y}g\cdot Z$ of $M$ in general position in $M$ if (P) holds for
element $g$ of $G$ belonging to a dense open subset of $G$.

We consider here that a variety is not necessarily irreducible; i.e. a variety
is a reduced, finite type scheme defined over an algebraically closed field.

###  1.2 Restriction to a general subvariety

Let $X\simeq G/P$ and $X^{\prime}\simeq G/P^{\prime}$ be (generalized) flag
varieties associated with parabolic subgroups $P\subset P^{\prime}$ of a semi-
simple linear algebraic group $G$. We observe here that some properties of
Gromov-Witten varieties of $X$ and $X^{\prime}$ can be deduced from properties
of the general fiber of the morphism $\Pi\times
ev:\overline{{\mathcal{M}}_{0,r}}(X,{\mathbf{d}})\rightarrow\overline{{\mathcal{M}}_{0,r}}(X^{\prime},\pi_{*}{\mathbf{d}})\times_{(X^{\prime})^{r}}X^{r}$-cf.
Proposition 1.2.4.1. This is one of the key ingredients in the derivation of
Theorem 1.2.2.1 we provide in Section 1.5.

More precisely, the property we prove here is the following. Let $G$ be a
semi-simple algebraic group, let $Z^{\prime}$ be a projective variety with a
transitive $G$-action, let $Z$ be a projective variety over $Z^{\prime}$ such
that the morphism $Z\rightarrow Z^{\prime}$ is $G$-equivariant. Let
$\Pi:M\rightarrow M^{\prime}$ be a morphism of projective varieties over
$Z^{\prime}$. Let $M\to Z$ be a morphism of projective varieties, let $X$ be a
subvariety of $Z$. Then for $g$ general in $G$ the irreducibility
(respectively rational connectedness) of the general fiber of the restriction
map ${\Pi}_{|gX}:M_{g\cdot X}\to M^{\prime}_{g\cdot X}$induced by $\Pi$ is
implied by the irreducibility (respectively rational connectedness) of the
general fiber of the morphism $M\rightarrow M^{\prime}\times_{Z^{\prime}}Z$\-
cf. Proposition 1.2.3.1.

####  1.2.1 Rationally connected, unirational and rational schemes.

We call a scheme $X$ rationally connected if there is an irreducible rational
curve joining two general points on $X$. Note that a rationally connected
scheme is irreducible. By contradiction suppose a scheme $Y$ has several
irreducible components $Y_{i}$ and is rationally connected. We can then find
two points $y_{1}$ and $y_{2}$ that each belongs to an irreducible component
$Y_{i}$ and is not contained in any other irreducible component of $Y$ such
that there is an irreducible curve $C$ joining the points $y_{1}$ and $y_{2}$.
Then the irreducible curve $C$ belongs to both $Y_{1}$ and $Y_{2}$ and is not
contained in the intersection of the two irreducible components, which is a
contradiction.

We call a variety $Y$ rational if there exists $N>0$ such that $Y$ is
birational to ${\mathbb{P}}^{N}$. A variety $Y$ is called unirational if there
exists a dominant rational map ${\mathbb{P}}^{N}\dashrightarrow Y$. Note that
since ${\mathbb{P}}^{N}$ is rationally connected, rational and unirational
varieties are rationally connected.

####  1.2.2 Composition of rationally connected fibrations.

We call a a morphism of schemes $f:Y\to S$ a rationally connected fibration if

  * •

Each irreducible component of $Y$ dominates a different irreducible component
of $S$;

  * •

$Y$ dominates $S$;

  * •

The general fiber of $f$ is rationally connected.

We call a scheme $Y$ a rationally connected fibration over a scheme $S$ if
there exists a rationally connected fibration $Y\to S$.

######  Definition 1.2.2.1.

Let $\mathcal{C}$ be a subcategory of the category of schemes $\mathrm{Sch}$.
We call a property $(P)$ stable under composition in $\mathcal{C}$ if for
every morphisms $Y\xrightarrow{f}S$ and $S^{\prime}\xrightarrow{f^{\prime}}S$
in $\mathcal{C}$ satisfying $(P)$, the composition $f^{\prime}\circ
f:Y\rightarrow S$ also satisfies $(P)$.

Types of morphisms of schemes stable under composition are well known; for
example flat, smooth or etale morphisms are stable under composition-cf. for
example Stacks Project $62.4$. We focus here on composing rationally connected
fibrations. Grabber-Harris-Starr’s Theorem on rational connectedness implies
that the composition of rationally connected fibrations is itself a rationally
connected fibration-cf. Theorem 1.2.2.1. This property will be a key
ingredient in the derivation of Theorem 1.1.2.1 proposed in Section 1.5.

######  Theorem 1.2.2.1.

Let $\mathcal{C}$ be the category of irreducible projective schemes.

  1. (1)

The property "the general fiber is Irreducible" is stable under composition in
$\mathcal{C}$.

  2. (2)

Let $(P)$ be a property such that, for all morphisms $Y\rightarrow S$ in
$\mathcal{C}$, the following statement is verified:

if $S$ and the general fiber of the morphism $Y\rightarrow S$ satisfy $(P)$,
then $Y$ satisfies $(P)$.

Then the property "the general fiber satisfies $(P)$ and is irreducible" is
stable under composition in $\mathcal{C}$.

  3. (3)

The property "the general fiber is rationally connected" is stable under
composition in the category of irreducible complex projective schemes.

  4. (4)

Let $Y$, $Z$ and $S$ be projective complex varieties such that $Y$ is a
rationally connected fibration over $Z$ and $Z$ is a rationally connected
fibration over $S$. Then $Y$ is a rationally connected fibration over $S$.

######  Lemma 1.2.2.1.

Let $f:Y\rightarrow Z$ be a dominant morphism of schemes, of finite
presentation, where $Y$ is an equidimensional noetherian scheme. Let $V$ be a
dense open subset of $Y$. Then the intersection of the general fiber of $f$
with $V$ is dense in the fiber.

###### Proof.

First, since the question is local on $Z$, we can suppose $Z$ irreducible.
Since $Y$ is a noetherian scheme, $Y\setminus V$ can be written as the union
of a finite number of schemes, i.e. $Y\setminus V=\cup_{i=1}^{N}F_{i}$, where
the $F_{i}$ are closed subschemes of $Y$, of dimension smaller than the pure
dimension of $Y$. Denote by $f_{|F_{i}}$ the restriction of $f$ to one of the
boundary subschemes $F_{i}$. If $f_{|F_{i}}$ does not dominate $Z$, then the
preimage under $f_{|F_{i}}$ of a general point in $Z$ is empty. On the other
side, if $f_{|F_{i}}$ dominates $Z$, then the preimage under $f_{|F_{i}}$ of a
general point in $Z$ is a scheme of pure dimension smaller than
$\mathrm{dim}Y-\mathrm{dim}Z$. Indeed, since $f$ is a morphism of finite
presentation, Chevalley’s constructibility theorem implies $f(F_{i})$ contains
an open subset of $Z$; hence the subset of points $x$ in $f(F_{i})$ such that
the fiber $f_{|F_{i}}^{-1}(x)$ is of expected dimension is an open subset $V$
of $Z$, verifying ${\rm dim}f_{|F_{i}}^{-1}(x)={\rm dim}F_{i}-{\rm dim}Z$ for
all points $x$ in $V$. Thus, since $Y$ is equidimensional, the general fiber
$Y_{0}$ of $f$ is a scheme of pure dimension $\mathrm{dim}Y-\mathrm{dim}Z$,
whose intersection with $F_{i}$ is a scheme of dimension smaller than
$\mathrm{dim}Y-\mathrm{dim}Z$; this implies the intersection $Y_{0}\cap V$ is
dense in $Y_{0}$. ∎

Let $h:M\to M^{\prime}$ and $f:M\to Z$ be morphisms of schemes. We denote by
$h\times f:M\rightarrow M^{\prime}\times Z$ the morphism induced by $h$ and
$f$.

######  Proposition 1.2.2.1.

Let $M\xrightarrow{h}M^{\prime}$ and $M\xrightarrow{f}Z$ be surjective
morphisms of finite presentation of integral noetherian schemes.

Then, for $z$ general in $Z$:

  1. i)

The image by $h$ of any irreducible component of $f^{-1}(z)$ has dimension
equal to ${\rm dim}(h\times f)(M)-{\rm dim}Z$;

  2. ii)

Each irreducible component of $f^{-1}(z)$ dominates one of the irreducible
components of $h(f^{-1}(z))$.

###### Proof.

  1. i)

Consider a dense open subset $U$ of $(h\times f)(M)$ such that the fiber
$(h\times f)^{-1}(u)$ is a scheme of pure dimension
$\mathrm{dim}M-\mathrm{dim}(h\times f)(M)$ for all $u$ in $U$. Consider a
point $z$ general in $Z$, and an irreducible component $Z_{0}$ of $f^{-1}(z)$.
Denote by $Z_{0}^{\circ}$ the variety $Z_{0}$ minus the intersection of
$Z_{0}$ with other irreducible components of $f^{-1}(z)$. We first compute the
dimension of the fiber $h^{-1}(h(\mu))\cap Z_{0}^{\circ}$, where $\mu$ is a
point in $(h\times f)^{-1}(U)\cap Z_{0}^{\circ}$, before computing ${\rm
dim}h(Z_{0})$.

Computation of ${\rm dim}{h}^{-1}(h(\mu))\cap Z_{0}^{\circ}$. First, we have:

$\displaystyle{h}^{-1}(h(\mu))\cap f^{-1}(z)$ $\displaystyle=(h\times
f)^{-1}\left(h(\mu),z\right)$ $\displaystyle=(h\times f)^{-1}(h\times
f)(\mu).$

Since $(h\times f)(\mu)$ lies in $U$, this implies ${h}^{-1}(h(\mu))\cap
f^{-1}(z)$ is a scheme of pure dimension $\mathrm{dim}M-\mathrm{dim}(h\times
f)(M)$. Hence the intersection of the irreducible variety $Z_{0}^{\circ}$ with
${h}^{-1}(h(\mu))$ is a scheme of pure dimension
$\mathrm{dim}M-\mathrm{dim}(h\times f)(M)$. Indeed, the intersection of the
open subvariety $Z_{0}^{\circ}$ of $f^{-1}(z)$ with $h^{-1}(h(\mu))$ is a non
empty open subvariety of $h^{-1}(h(\mu))\cap Z_{0}$ disconnected from the
other pure dimensional irreducible components of $h^{-1}(h(\mu))\cap
f^{-1}(z)$. Hence $h^{-1}(h(\mu))\cap Z_{0}^{\circ}$ is a scheme of pure
dimension $\mathrm{dim}M-\mathrm{dim}(h\times f)(M)$.

Computation of ${\rm dim}h(Z_{0})$. Note that, since $z$ is general in $Z$,
according to Lemma 1.2.2.1, $(h\times f)^{-1}(U)\cap f^{-1}(z)$ is dense in
$f^{-1}(z)$; hence $(h\times f)^{-1}(U)\cap Z_{0}^{\circ}$ is a dense open
subset of $Z_{0}$. Consider a point $\nu$ general in $h(Z_{0})$; since $\nu$
is general and $h\left((h\times f)^{-1}(U)\cap Z_{0}^{\circ}\right)$ is dense
in the irreducible variety $h(Z_{0})$, $\nu$ belongs to $h\left((h\times
f)^{-1}(U)\cap Z_{0}^{\circ}\right)$, and we have:

$\displaystyle{\rm dim}Z_{0}-{\rm dim}h(Z_{0})$ $\displaystyle={\rm
dim}{h}^{-1}(\nu)\cap Z_{0}^{\circ}$
$\displaystyle=\mathrm{dim}M-\mathrm{dim}(h\times f)(M)$

where the first equality is deduced from the general position of $\nu$ in
$h(Z_{0})$, and the second equality holds since $\nu$ is in $h\left((h\times
f)^{-1}(U)\cap Z_{0}^{\circ}\right)$. This implies:

$\displaystyle\mathrm{dim}h(Z_{0})$
$\displaystyle=\mathrm{dim}Z_{0}-\mathrm{dim}M+\mathrm{dim}(h\times f)(M)$
$\displaystyle=\mathrm{dim}f^{-1}(z)-\mathrm{dim}M+\mathrm{dim}(h\times
f)(M),$

where the second equality holds since $z$ is general in $Z$ and thus
$f^{-1}(z)$ is equidimensional.

  2. ii)

By contradiction, if an irreducible component $Z_{0}$ of $f^{-1}(z)$ has an
image strictly contained in $h\left(f^{-1}(z)\right)$, then:

${\rm dim}h(Z_{0})<{\rm dim}h\left(f^{-1}(z)\right)={\rm dim}(h\times
f)(M)-{\rm dim}Z,$

which would contradict $i)$.

∎

###### Proof of Theorem 1.2.2.1.

  1. (1)

Let $h:Y\rightarrow S$ and $S\rightarrow Z$ be morphisms in $\mathcal{C}$
whose general fiber is irreducible. Note that since we consider properties of
general fibers, we can assume that both morphisms are dominant. Since both
morphisms are morphisms of projective schemes, they are then surjective.

Let $U$ be a dense open subset of $Z$ such that the fiber $S_{z}$ over a point
$z$ in $U$ is irreducible. For $z$ general in $Z$, according to Proposition
1.2.2.1 the image by $h:Y\rightarrow S$ of each irreducible component of
$Y_{z}$ dominates one of the irreducible components of $S_{z}$, which is
irreducible since $z$ is general in $Z$. Since, according to Lemma 1.2.2.1,
for $z$ general in $Z$ the fiber $Y_{z}$ has a dense intersection with the
dense open subset $Y_{U}$, the general fiber $Y_{z}$ is irreducible. Indeed,
if we denote by $Y_{z}^{1}$ and $Y_{z}^{2}$ two different irreducible
components of $Y$, and by $V$ the dense open subset of $S_{z}$ such that the
fiber $Y_{s}$ over a point $s$ in $V$ is irreducible and of exepected
dimension, for all $v$ in $V$ the intersection $Y_{z}^{1}\cap Y_{v}$ is equal
to $Y_{z}^{2}\cap Y_{v}$, since both are closed subschemes of maximal
dimension of an irreducible scheme. Hence the intersection of $Y_{z}$ with the
dense open subset $Y_{z}\cap Y_{V}$ is irreducible, hence $Y_{z}$ is
irreducible.

  2. (2)

Let $Y\xrightarrow{f_{1}}S$ and $S\xrightarrow{f_{2}}S^{\prime}$ be morphisms
in $\mathcal{C}$ satisfying $(P)$ and whose general fiber is irreducible.
Since we consider properties of the general fiber of $Y\rightarrow
S^{\prime}$, we can assume that $Y$ dominates $S$. Let $U$ be a dense open
subset of $S$ such that the fiber of $f_{1}$ over a point in $U$ satisfies
$(P)$. For $x$ general in $S^{\prime}$, the following properties are
satisfied:

    1. (a)

The fiber $S_{x}$ satisfies $(P)$;

    2. (b)

The fiber $S_{x}$ has a dense open intersection with $U$;

    3. (c)

The morphism $Y_{x}\rightarrow S_{x}$ is in $\mathcal{C}$;

    4. (d)

The general fiber of the morphism $Y_{x}\rightarrow S_{x}$ satisfies $(P)$;

    5. (e)

$Y_{x}$ satisfies $(P)$.

Note that proving property $(e)$ for $x$ general in $S^{\prime}$ is our goal
here. Property $(a)$ holds since $x$ is general in $S^{\prime}$. According to
Lemma 1.2.2.1 and since $x$ is general in $S^{\prime}$, property $(b)$ holds.
Property $(c)$ holds since $x$ is general in $S^{\prime}$ and according to
$(1)$ the general fiber of $Y\to S^{\prime}$ is irreducible. $(d)$ is a direct
consequence of $(b)$, the fact that the fiber of $Y_{x}\rightarrow S_{x}$ over
a point $y$ in $S_{x}$ is isomorphic to the fiber
$Y_{y}=Y\times_{S^{\prime}}\\{y\\}$, and the fact that $Y_{y}$ satisfies $(P)$
for $y$ in $U$. Finally $(e)$ is implied by $(a)$, $(c)$ and $(d)$.

  3. (3)

Note that, according to Grabber-Harris-Starr’s theorem on rational
connectedness [GHS03], if the general fiber of a morphism $Y\rightarrow S$ of
irreducible complex projective varieties is rationally connected, and the
basis $S$ is rationally connected, then $Y$ is rationally connected. Hence,
according to $2.$, the property "the general fiber is irreducible and
rationally connected" is stable under composition in $\mathcal{C}$.

  4. (4)

Let $Y$, $Z$ and $S$ be projective complex schemes, where $Y$ is a rationally
connected fibration over $Z$, and $Z$ is a rationally connected fibration over
$S$. Denote by $f:Y\rightarrow Z$ and $g:Z\rightarrow S$ the associated
morphisms. First, note that since $Y$ dominates $Z$ and $Z$ dominates $S$, $Y$
dominates $S$. Furthermore, each irreducible component $S_{i}$ of $S$ is
dominated by exactly one irreducible component $Z_{i}$ of $Z$, which itself is
dominated by exactly one irreducible component $Y_{i}$ of $Y$; hence there is
a one-to-one correspondence between irreducible components of $Y$ and $S$.
Finally, since by hypothesis the general fibers of $Y_{i}\to Z_{i}$ and
$Z_{i}\to S_{i}$ are rationally connected, according to $(3)$ the general
fiber of $Y_{i}\to S_{i}$ is a rationally connected variety. Hence $Y$ is a
rationally connected fibration over $S$.

∎

####  1.2.3 Restriction to a general subvariety.

We study here the restriction of a morphism to a general subvariety. A typical
application of the proposition below will be the study of the image of a
Gromov-Witten variety. More precisely, consider $\pi:X\rightarrow X^{\prime}$
the projection of a flag variety onto another flag variety, and denote by
$\Pi:\overline{\mathcal{M}_{0,r}}(X,\mathbf{d})\rightarrow\overline{\mathcal{M}_{0,r}}(X^{\prime},(\pi)_{*}\mathbf{d})$
the morphism induced by $\pi$ between moduli spaces of genus zero stable maps,
where $\mathbf{d}$ is the class of a curve in $X$. We thus obtain the
commutative diagram:

${\overline{\mathcal{M}_{0,r}}(X,\mathbf{d})}$${X^{r}}$${\overline{\mathcal{M}_{0,r}}(X^{\prime},(\pi)_{*}\mathbf{d})}$${(X^{\prime})^{r}}$$\scriptstyle{\Pi}$$\scriptstyle{\pi^{r}}$

to which we will apply the following proposition.

######  Proposition 1.2.3.1.

Let $G$ be an integral group scheme, let $f:M\rightarrow Z$,
$f^{\prime}:M^{\prime}\rightarrow Z^{\prime}$, $\pi:Z\rightarrow Z^{\prime}$
be projective morphisms, of integral noetherian schemes, where $Z$ and
$Z^{\prime}$ admit a transitive $G$-action, and the morphism $\pi$ is
$G$-equivariant. Let $Y$ be an integral subscheme of $Z$. Suppose there exists
a proper morphism $h:M\rightarrow M^{\prime}$ such that the following diagram
commutes :

${M}$${Z}$${M^{\prime}}$${Z^{\prime}}$$\scriptstyle{h}$$\scriptstyle{f}$$\scriptstyle{\pi}$$\scriptstyle{f^{\prime}}$

Furthermore, suppose the morphism $h\times f:M\rightarrow
M^{\prime}\times_{Z^{\prime}}Z$ induced by the morphisms
$M\xrightarrow{h}M^{\prime}$ and $M\xrightarrow{{f}}Z$ is surjective.

Consider a point $g$ general in $G$. We call $\mathcal{W}_{g}:=f^{-1}(gY)$,
$\mathcal{W}_{g}^{\prime}:=(h\times f)(\mathcal{W}_{g})$, and
$\mathcal{W}_{g}^{\prime\prime}:=h(\mathcal{W}_{g})$. We denote by
$p_{1}:M^{\prime}\times_{Z^{\prime}}Z\rightarrow M^{\prime}$ the first
projection. Then:

  1. i)

$\mathcal{W}_{g}^{\prime}=M^{\prime}\times_{Z^{\prime}}gY$ and
$\mathcal{W}_{g}^{\prime\prime}=(f^{\prime})^{-1}(g\pi(Y))$;

  2. ii)

$\mathcal{W}_{g}$, $\mathcal{W}_{g}^{\prime}$ and
$\mathcal{W}_{g}^{\prime\prime}$ are pure dimensional schemes;

  3. iii)

The restriction of $h\times f$ to any irreducible component of
$\mathcal{W}_{g}$ is dominant onto an irreducible component of
$\mathcal{W}_{g}^{\prime}$;

  4. iv)

The restriction of $p_{1}$ to any irreducible component of
$\mathcal{W}_{g}^{\prime}$ is dominant onto an irreducible component of
$\mathcal{W}_{g}^{\prime\prime}$.

Let us introduce the following notation. Let $f:Y\rightarrow Z$ and
$\varphi:Y^{\prime}\rightarrow Z$ be morphisms of schemes, where an integral
group scheme $G$ acts on $Z$. We denote by $Y\times_{Z}(gY^{\prime})$ the
fiber product induced by the morphism $\varphi_{g}:Y^{\prime}\rightarrow Z$
given by $y\rightarrow g\cdot\varphi(y)$, i.e. $Y\times_{Z}(gY^{\prime})$ is
the fiber product associated with the following commutative diagram:

${Y\times_{Z}(gY^{\prime})}$${Y^{\prime}}$${Y}$${Z}$$\scriptstyle{\varphi_{g}}$$\scriptstyle{f}$

Notice that, if $G$ acts on $Y^{\prime}$ and $\varphi$ is $G$-equivariant, the
fiber product $Y\times_{Z}(gY^{\prime})$ will simply be the usual fiber
product associated with the following commutative diagram:

${Y\times_{Z}(gY^{\prime})}$${gY^{\prime}}$${Y}$${Z}$$\scriptstyle{\varphi}$$\scriptstyle{f}$

######  Lemma 1.2.3.1.

Let $Y\rightarrow Z$ be a morphism of schemes, of finite presentation, where
$Y$ is an equidimensional noetherian scheme; let
$\varphi:Y^{\prime}\rightarrow Z$ be a morphism of integral schemes. Let $V$
be a dense open subset of $Y$. Suppose there exists an integral group scheme
$G$ such that $G$ acts transitively on $Z$.

Then, for a point $g$ general in $G$:

  1. i)

The scheme $V\times_{Z}(gY^{\prime})$ is a dense subset of
$Y\times_{Z}(gY^{\prime})$;

  2. ii)

The scheme $(gV)\times_{Z}Y^{\prime}$ is a dense subset of
$(gY)\times_{Z}Y^{\prime}$.

###### Proof.

  1. i)

Since $Y$ is a noetherian scheme, $Y\setminus V$ can be written as the union
of a finite number of irreducible schemes, i.e. $Y\setminus
V=\cup_{i=1}^{N}F_{i}$, where the $F_{i}$ are irreducible closed subschemes of
$Y$, of dimension smaller than the pure dimension of $Y$. Consider an element
$g$ general in $G$. Suppose the scheme $Y\times_{Z}(gY^{\prime})$ is not
empty. Then, according to Kleiman’s transversality theorem [Kle74], it has
pure dimension given by
$\mathrm{dim}Y+\mathrm{dim}Y^{\prime}-\mathrm{dim}Z^{\prime}$. Furthermore,
for $1\leq i\leq N$, the scheme $F_{i}\times_{Z^{\prime}}(gY^{\prime})$ is
either empty or of pure dimension smaller than
$\mathrm{dim}Y+\mathrm{dim}Y^{\prime}-\mathrm{dim}Z^{\prime}$. Hence
$V\times_{Z^{\prime}}(gY^{\prime})=(Y\setminus\cup_{i=1}^{N}F_{i})\times_{Z^{\prime}}(gY^{\prime})$
is dense in $Y\times_{Z}(gY^{\prime})$.

  2. ii)

Consider a point $g$ general in $G$. Then, according to part $i).$, the scheme
$V\times_{Z}(g^{-1}Y^{\prime})$ is a dense subset of
$Y\times_{Z}(g^{-1}Y^{\prime})$. Furthermore, notice that we have an
isomorphism $(gY)\times_{Z}Y^{\prime}\rightarrow
Y\times_{Z}(g^{-1}Y^{\prime})$. Indeed, if we denote by $g^{-1}:Z\rightarrow
Z$ the isomorphism of $Z$ induced by the action of $g^{-1}$, and by
$g:Z\rightarrow Z$ the isomorphism of $Z$ induced by the action of $g$, the
following commutative diagram

${Y^{\prime}}$${Y}$${Z}$${Z}$$\scriptstyle{\varphi}$$\scriptstyle{\varphi_{g^{-1}}}$$\scriptstyle{f_{g}}$$\scriptstyle{f}$$\scriptstyle{g^{-1}}$$\scriptstyle{g}$

implies that any morphism $T\rightarrow(gY)\times_{Z}Y^{\prime}$ factors
through $Y\times_{Z}(g^{-1}Y^{\prime})$, and reciprocally any morphism
$T\rightarrow Y\times_{Z}(g^{-1}Y^{\prime})$ factors through
$(gY)\times_{Z}Y^{\prime}$.
Finally, in the same tautological way, notice we have an isomorphism
$(gV)\times_{Z}Y^{\prime}\simeq V\times_{Z}(g^{-1}Y^{\prime})$.

∎

###### Proof of Proposition 1.2.3.1.

  1. i)

First, since, by assumption, $h\times f:M\rightarrow
M^{\prime}\times_{Z^{\prime}}Z$ is surjective,
$\mathcal{W}_{g}^{\prime}=(h\times f)\left(f^{-1}(gY)\right)=(h\times
f)(M)\cap(M^{\prime}\times_{Z^{\prime}}(gY))$ is equal to
$M^{\prime}\times_{Z^{\prime}}(gY)$. Furthermore, $h(f^{-1}(gY))$ is equal to
the image of $(h\times f)(f^{-1}(gY))$ by the first projection $p_{1}$, where
$p_{1}$ is defined by the following commutative diagram:

${M}$${(h\times
f)(M)=M^{\prime}\times_{Z^{\prime}}Z}$${Z}$${M^{\prime}}$${Z^{\prime}}$$\scriptstyle{h}$$\scriptstyle{h\times
f}$$\scriptstyle{p_{1}}$$\scriptstyle{\pi}$$\scriptstyle{f^{\prime}}$

Meanwhile, $(f^{\prime})^{-1}(\pi(gY))$ is equal to the image of
$M^{\prime}\times_{Z^{\prime}}(gY)$ by the first projection $p_{1}$, hence the
equality between $h(f^{-1}(gY))$ and $(f^{\prime})^{-1}(\pi(gY))$.
Finally, since $\pi$ is $G$-equivariant, this yields:
$\mathcal{W}_{g}^{\prime\prime}=(f^{\prime})^{-1}(g\pi(Y))$.

  2. ii)

According to Kleiman’s transversality theorem, since $g$ is general in $G$,
the schemes $\mathcal{W}_{g}=f^{-1}(gY)$,
$\mathcal{W}_{g}^{\prime}=M^{\prime}\times_{Z^{\prime}}gY$ and
$\mathcal{W}_{g}^{\prime\prime}=(f^{\prime})^{-1}(g\pi(Y))$ are either empty
or pure dimensional schemes.

  3. iii)

We consider the following commutative diagram:

${M}$${Z}$${M^{\prime}\times_{Z^{\prime}}Z}$${Z}$$\scriptstyle{h\times
f}$$\scriptstyle{f}$

Denote by $U$ a dense open subset of $(h\times f)(M)$ such that, for all $z$
in $U$, $(h\times f)^{-1}(z)$ has pure dimension equal to
$\mathrm{dim}M-\mathrm{dim}(h\times f)(M)$. Fix an irreducible component
$\mathcal{W}_{0}$ of $\mathcal{W}_{g}$. Denote by $\mathcal{W}_{0}^{\circ}$
the variety ${\mathcal{W}}_{0}$ minus the intersection of ${\mathcal{W}}_{0}$
with other irreducible components o $\mathcal{W}_{g}$. We first compute the
dimension of $(h\times f)^{-1}(x)\cap\mathcal{W}_{0}^{\circ}$, where $x$ is a
point in $(h\times f)(\mathcal{W}_{0}^{\circ})\cap U$, before computing the
dimension of $(h\times f)(\mathcal{W}_{0})$.

Computation of ${\rm dim}(h\times f))^{-1}(x)\cap\mathcal{W}_{0}^{\circ}$.
First, the fiber of the restriction $(h\times f)_{|\mathcal{W}_{g}}$ of the
morphism $h\times f$ to $\mathcal{W}_{g}$ is isomorphic to the fiber of
$h\times f$. Indeed, for $(m,z)$ in $(h\times
f)(\mathcal{W}_{g})=M^{\prime}\times_{Z^{\prime}}gY$, the fiber ${(h\times
f)_{|\mathcal{W}_{g}}}^{-1}(m,z)$ is equal to $(h\times
f)^{-1}(m,z)\cap\mathcal{W}_{g}=h^{-1}(m)\cap f^{-1}(z)\cap\mathcal{W}_{g}$,
where $z$ lies in $gY$. Meanwhile, we observe
$f^{-1}(z)\cap\mathcal{W}_{g}=f^{-1}(z)\cap f^{-1}(gY)=f^{-1}(z)$, hence
${(h\times f)_{|\mathcal{W}_{g}}}^{-1}(m)$ is equal to $(h\times f)^{-1}(m)$.

Since $x$ lies in $U$, this implies ${(h\times f)}^{-1}(x)\cap\mathcal{W}_{g}$
is a scheme of pure dimension $\mathrm{dim}M-\mathrm{dim}(h\times f)(M)$.
Hence the intersection of the irreducible variety $\mathcal{W}_{0}^{\circ}$
with ${(h\times f)}^{-1}(x)$ is a scheme of pure dimension
$\mathrm{dim}M-\mathrm{dim}(h\times f)(M)$. Indeed, the intersection of the
open subvariety $\mathcal{W}_{0}^{\circ}$ of $\mathcal{W}_{g}$ with $(h\times
f)^{-1}(x)$ is a non empty open subvariety of $(h\times
f)^{-1}(x)\cap\mathcal{W}_{0}$ disconnected from the other pure dimensional
irreducible components of $(h\times f)^{-1}(x)\cap\mathcal{W}_{g}$. Hence
$(h\times f)^{-1}(x)\cap\mathcal{W}_{0}^{\circ}$ is a scheme of pure dimension
$\mathrm{dim}M-\mathrm{dim}(h\times f)(M)$.

Computation of ${\rm dim}(h\times f)(\mathcal{W}_{0})$. Note that, since $g$
is general in $G$, according to Lemma 1.2.3.1, $(h\times
f)^{-1}(U)\cap\mathcal{W}_{g}$ is dense in $\mathcal{W}_{g}$; hence $(h\times
f)^{-1}(U)\cap\mathcal{W}_{0}^{\circ}$ is a dense open subset of
$\mathcal{W}_{0}$. Consider a general point $\nu$ in $(h\times
f)(\mathcal{W}_{0})$; since $\nu$ is general and $(h\times f)\left((h\times
f)^{-1}(U)\cap\mathcal{W}_{0}^{\circ}\right)$ is dense in the irreducible
variety $(h\times f)(\mathcal{W}_{0})$, $\nu$ belongs to $(h\times
f)\left((h\times f)^{-1}(U)\cap\mathcal{W}_{0}^{\circ}\right)$, and we have:

$\displaystyle{\rm dim}\mathcal{W}_{0}-{\rm dim}(h\times f)(\mathcal{W}_{0})$
$\displaystyle={\rm dim}{(h\times f)}^{-1}(\nu)\cap\mathcal{W}_{0}^{\circ}$
$\displaystyle=\mathrm{dim}M-\mathrm{dim}(h\times f)(M)$

where the first equality is deduced from the general position of $\nu$ in
$(h\times f)(\mathcal{W}_{0})$, and the second equality holds since $\nu$ is
in $(h\times f)\left((h\times f)^{-1}(U)\cap\mathcal{W}_{0}^{\circ}\right)$.
This implies:

$\displaystyle\mathrm{dim}(h\times f)(\mathcal{W}_{0})$
$\displaystyle=\mathrm{dim}\mathcal{W}_{0}-\mathrm{dim}M+\mathrm{dim}(h\times
f)(M)$
$\displaystyle=\mathrm{dim}\mathcal{W}_{g}-\mathrm{dim}M+\mathrm{dim}(h\times
f)(M),$

where the second equality holds since $\mathcal{W}_{g}$ is equidimensional.
Since $(h\times f)(\mathcal{W}_{0})$ is obviously contained in
$\mathcal{W}_{g}^{\prime}=(h\times f)(\mathcal{W}_{g})$, and this dimension
does not depend on the irreducible component of $\mathcal{W}_{g}$ considered,
this implies the irreducible variety $(h\times f)(\mathcal{W}_{0})$ is an
irreducible component of $\mathcal{W}_{g}^{\prime}$, which has pure dimension
equal to $\mathrm{dim}\mathcal{W}_{g}-\mathrm{dim}M+\mathrm{dim}(h\times
f)(M)$.

  4. iv)

First, denote by $\pi_{|Y}:Y\rightarrow\pi(Y)$ the restriction of the morphism
$\pi$ to $Y$. Fix a dense open subset $U$ of $\pi(Y)$ such that, for all $x$
in $U$, the fiber $\pi_{|Y}^{-1}(x)$ is a scheme of pure dimension equal to
$\mathrm{dim}Y-\mathrm{dim}\pi(Y)$.

We consider the following commutative diagram:

${M^{\prime}\times_{Z^{\prime}}Z}$${Z}$${M^{\prime}}$${Z^{\prime}}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{2}}$$\scriptstyle{\pi}$$\scriptstyle{f^{\prime}}$

where $p_{1}$ and $p_{2}$ are respectively the first and second projection
morphisms. Consider a general element $g$ in $G$. We denote by
$p_{1}^{\prime}$ the restriction of the morphism $p_{1}$ to
$\mathcal{W}_{g}^{\prime}=M^{\prime}\times_{Z^{\prime}}gY$. We want to prove
here that the image of any irreducible component of $\mathcal{W}_{g}^{\prime}$
by $p_{1}^{\prime}$ is an irreducible component of
$\mathcal{W}_{g}^{\prime\prime}=(f^{\prime})^{-1}(g\pi(Y))$. In order to do
this, we study the dimension of the fiber $(p_{1}^{\prime})^{-1}(m)$ of
$p_{1}^{\prime}$ at a point $m$ in $\mathcal{W}_{g}^{\prime\prime}$.

We will use the following observations.

######  Lemma 1.2.3.2.

Let $f^{\prime}:M^{\prime}\rightarrow Z^{\prime}$ and $\pi:Z\rightarrow
Z^{\prime}$ be projective morphisms of integral noetherian schemes. Let $Y$ be
an integral subscheme of $Z$. Suppose there exists an integral group scheme
$G$ such that $G$ acts transitively on $Z$, and $\pi$ is $G$-equivariant.
Consider the following commutative diagram:

${M^{\prime}\times_{Z^{\prime}}Z}$${Z}$${M^{\prime}}$${Z^{\prime}}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{2}}$$\scriptstyle{\pi}$$\scriptstyle{f^{\prime}}$

where $p_{1}$ and $p_{2}$ are respectively the first and second projection
morphisms.

    1. (a)

Consider an element $g$ in $G$. The fiber $p_{1}^{-1}(m)\cap{p_{2}}^{-1}(gY)$
of the restriction of the morphism $p_{1}$ to $(p_{2})^{-1}(gY)$ is isomorphic
to ${\pi}_{|Y}^{-1}(g^{-1}f^{\prime}(m))$, where we denote by
$\pi_{|Y}:Y\rightarrow\pi(Y)$ the restriction of the morphism $\pi$ to $Y$;

    2. (b)

Consider a dense open subset $U$ of $Y$. Then, for an element $g$ general in
$G$, ${p_{2}}^{-1}(gU)$ is dense in ${p_{2}}^{-1}(gY)$.

###### Proof of Lemma 1.2.3.2.

    1. (a)

Denote by $h^{\prime}$ the restriction of the morphism $p_{1}$ to
${p_{2}}^{-1}(gY)$. The fiber $(h^{\prime})^{-1}(m)$ is equal to
$\\{m\\}\times_{Z^{\prime}}(gY)$, which is isomorphic to
$\pi^{-1}(f^{\prime}(m))\cap(gY)$. Furthermore, since $\pi$ is
$G$-equivariant, the morphism $y\rightarrow g^{-1}y$ induces an isomorphism
between $\pi^{-1}(f^{\prime}(m))\cap(gY)$ and
$\pi^{-1}(g^{-1}f^{\prime}(m))\cap(Y)$. Finally, notice
$\pi^{-1}(g^{-1}f^{\prime}(m))\cap(Y)$ is equal to
${\pi}_{|Y}^{-1}(g^{-1}f^{\prime}(m))$.

    2. (b)

According to Lemma 1.2.3.1 $ii)$, since $g$ is general in $G$, the scheme
$M^{\prime}\times_{Z^{\prime}}(gU)$ is dense in
$M^{\prime}\times_{Z^{\prime}}(gY)$. Since the morphism $\pi$ is
$G$-equivariant, we have here the following equalities:
${p_{2}}^{-1}(gY)=M^{\prime}\times_{Z^{\prime}}(gY)$ and
${p_{2}}^{-1}(gU)=M^{\prime}\times_{Z^{\prime}}(gU)$. Hence the scheme
${p_{2}}^{-1}(gU)$ is dense in ${p_{2}}^{-1}(gY)$.

∎

Fix an irreducible component $\mathcal{W}_{0}^{\prime}$ of
$p_{2}^{-1}(gY)=M^{\prime}\times_{Z^{\prime}}gY$. For $\mu$ general in
$\mathcal{W}_{0}^{\prime}$, we have:

$\displaystyle{\rm dim}\mathcal{W}_{0}^{\prime}-{\rm
dim}p_{1}(\mathcal{W}_{0}^{\prime})$ $\displaystyle={\rm
dim}p_{1}^{-1}(p_{1}(\mu))\cap\mathcal{W}_{0}^{\prime}$ $\displaystyle={\rm
dim}\pi_{|Y}^{-1}\left(g^{-1}\cdot(f^{\prime}\circ p_{1})(\mu)\right)$
$\displaystyle={\rm dim}Y-{\rm dim}\pi(Y),$

where the first equality holds since the image of a dense open subset of
$\mathcal{W}_{0}^{\prime}$ by $p_{1}$ is a dense open subset of
$p_{1}(\mathcal{W}_{0}^{\prime})$, the second equality is given by Lemma
1.2.3.2 $(a)$, and the last equality holds since $(f^{\prime}\circ
p_{1})(\mu)$ is in $g\cdot\pi(U)$ for $\mu$ general in
$\mathcal{W}_{0}^{\prime}$. Indeed, according to Lemma 1.2.3.2 $(b)$,
$p_{2}^{-1}(g\pi^{-1}(U))$ is dense in $p_{2}^{-1}(gY)$, hence
$p_{2}^{-1}(g\pi^{-1}(U))\cap\mathcal{W}_{0}^{\prime}$ is dense in
$\mathcal{W}_{0}^{\prime}$, hence a point in
$(p_{2}\circ\pi)^{-1}(gU)=(f^{\prime}\circ p_{1})^{-1}(gU)$ is general in
$\mathcal{W}_{0}^{\prime}$.

Finally, notice that, since $g$ is general in $G$, any irreducible component
$\mathcal{W}_{0}^{\prime}$ of $\mathcal{W}_{g}^{\prime}$ will be of dimension
equal to $\mathrm{dim}M^{\prime}+\mathrm{dim}Y-\mathrm{dim}Z^{\prime}$, hence
:

$\mathrm{dim}p_{1}(\mathcal{W}_{0}^{\prime})=\mathrm{dim}M^{\prime}+\mathrm{dim}\pi(Y)-\mathrm{dim}Z^{\prime},$

which is equal to the pure dimension of
$\mathcal{W}_{g}^{\prime\prime}=(f^{\prime})^{-1}(g\pi(Y))$ since $g$ is
general in $G$. Since $p_{1}(\mathcal{W}_{0}^{\prime})$ is obviously contained
in $\mathcal{W}_{g}^{\prime\prime}$, this implies that
$\mathcal{W}_{0}^{\prime}$ dominates an irreducible component of
$\mathcal{W}_{g}^{\prime\prime}$.

∎

####  1.2.4 Application to Gromov-Witten varieties.

Consider a semi-simple algebraic group $G$, a maximal torus $T$ of $G$, a
Borel subgroup $B$ and parabolic subgroups $P$ and $P^{\prime}$ satisfying
$T\subset B\subset P\subset P^{\prime}$. Denote by $X:=G/P$ and
$X^{\prime}:=G/P^{\prime}$ the corresponding homogeneous varieties, and by

$\pi:X=G/P\rightarrow X^{\prime}=G/P^{\prime}$

the natural projection. For an element $u$ in the Weyl group $W:=N_{G}(T)/T$,
denote by $X(u):=\overline{BuP}/P$ the associated Schubert variety. Note that
$\pi(X(u))=X^{\prime}(u)$. Let $r\geq 0$, $u_{1}$, $\dots$, $u_{r}$ be
elements in $W$, and $\mathbf{d}$ be a degree in the semi-group $E(G/P)$ of
effective classes of curves in $H_{2}(G/P,\mathbb{Z})$. Note that the morphism
$\pi:G/P\rightarrow G/P^{\prime}$ induces a morphism:

$\Pi\times
ev:\overline{{\mathcal{M}}_{0,r}}(G/P,\mathbf{d})\rightarrow\overline{{\mathcal{M}}_{0,r}}(G/P^{\prime},\pi_{*}\mathbf{d})\times_{(G/P^{\prime})^{r}}(G/P)^{r},$

where we denote by
$\Pi:\overline{{\mathcal{M}}_{0,r}}(G/P,\mathbf{d})\rightarrow\overline{{\mathcal{M}}_{0,r}}(G/P^{\prime},\pi_{*}\mathbf{d})$
the morphism induced by $\pi$. For an element $g=(g_{1},\dots,g_{r})$ in
$G^{r}$, we will consider the following commutative diagram:

${\overline{\mathcal{M}_{0,r}}(G/P,\mathbf{d})}$${(G/P)^{r}}$${\prod_{i}g_{i}X(u_{i})}$${\overline{\mathcal{M}_{0,r}}(G/P^{\prime},\pi_{*}\mathbf{d})}$${(G/P^{\prime})^{r}}$${\prod_{i}g_{i}X^{\prime}(u_{i})}$$\scriptstyle{\Pi}$$\scriptstyle{{\rm
ev}}$$\scriptstyle{\pi}$

This commutative diagram allows us to define the following fiber products:

$\displaystyle\mathcal{W}_{g}$
$\displaystyle:=\overline{\mathcal{M}_{0,r}}(X,\mathbf{d})\times_{X^{r}}\prod_{i}g_{i}X(u_{i})$
$\displaystyle\hookrightarrow\overline{{\mathcal{M}}_{0,r}}(X,\mathbf{d})$
$\displaystyle\mathcal{W}_{g}^{\prime}$
$\displaystyle:=\overline{\mathcal{M}_{0,r}}(X^{\prime},\pi_{*}\mathbf{d})\times_{(X^{\prime})^{r}}\prod_{i}g_{i}X(u_{i})$
$\displaystyle\hookrightarrow\overline{{\mathcal{M}}_{0,r}}(X^{\prime},\pi_{*}\mathbf{d})\times
X^{r}$ $\displaystyle\mathcal{W}_{g}^{\prime\prime}$
$\displaystyle:=\overline{\mathcal{M}_{0,r}}(X^{\prime},\pi_{*}\mathbf{d})\times_{(X^{\prime})^{r}}\prod_{i}g_{i}X^{\prime}(u_{i})$
$\displaystyle\hookrightarrow\overline{{\mathcal{M}}_{0,r}}(X^{\prime},\pi_{*}\mathbf{d})$

Note that for $g=(g_{1},\dots,g_{r})$ general in $G^{r}$, the scheme
$\mathcal{W}_{g}={\mathcal{W}}_{u_{1},\dots,u_{r}}^{g,\mathbf{d}}$ is the
variety parametrizing genus zero, degree $\mathbf{d}$, stable maps to $X$
sending their $i$-th marked point within $g_{i}\cdot X(u_{i})$, and
$\mathcal{W}_{g}^{\prime\prime}={\mathcal{W}}_{u_{1},\dots,u_{r}}^{g,\pi_{*}\mathbf{d}}$
is the variety parametrizing genus zero, degree $(\pi_{*})\mathbf{d}$ stable
maps sending their $i$-th marked point within $g_{i}\cdot X^{\prime}(u_{i})$.
Indeed, according to the proof of Kleiman’s transversality theorem [Kle74],
for a point $g=(g_{1},\dots,g_{r})$ general in $G^{r}$, the natural maps
${\mathcal{W}}_{g}\to\overline{{\mathcal{M}}_{0,r}}(G/P,{\mathbf{d}})$ and
${\mathcal{W}}_{g}^{\prime\prime}\to\overline{{\mathcal{M}}_{0,r}}(G/P^{\prime},{\mathbf{d}})$
are both closed immersions. This allows us to identify the schemes
${\mathcal{W}}_{g}$ and ${\mathcal{W}}_{g}^{\prime\prime}$ with their images
in the space of stable maps.

######  Proposition 1.2.4.1.

Suppose the morphism $\Pi\times{\rm ev}$ is surjective, and suppose its
general fiber is connected.

  1. 1.

For $g$ general in $G^{r}$, $(\Pi\times
ev)(\mathcal{W}_{g})=\mathcal{W}_{g}^{\prime}$ and
$\Pi(\mathcal{W}_{g})=\mathcal{W}_{g}^{\prime\prime}$;

  2. 2.

For $g$ general in $G^{r}$, each irreducible component of the Gromov-Witten
variety $\mathcal{W}_{g}$ surjects into a different irreducible component of
$\Pi(\mathcal{W}_{g})$, i.e. there is a $1$-to-$1$ correspondence between
irreducible components of $\mathcal{W}_{g}$ and irreducible components of
$\mathcal{W}_{g}^{\prime\prime}$;

  3. 3.

Suppose the general fiber of $\Pi\times ev$ is rationally connected. Then for
$g$ general in $G^{r}$, the Gromov-Witten variety $\mathcal{W}_{g}$ is a
rationally connected fibration over both ${\mathcal{W}}_{g}^{\prime}$ and
$\mathcal{W}_{g}^{\prime\prime}$, in the sense of Definition 1.1.2.1.

######  Lemma 1.2.4.1.

Let $S$ be a normal scheme, and $S\xrightarrow{f}Y$ be a surjective morphism.
If the general fiber of $f$ is connected, then each irreducible component of
$f(S)$ is dominated by exactly one irreducible component of $S$.

###### Proof.

First, observe that, since the general fiber of $f$ is connected, two
connected components of $S$ cannot dominate the same irreducible component of
$f(S)$. Indeed, if we consider a dense open subset $U$ of $f(S)$ such that the
fiber $f^{-1}(s)$ is connected for all $s$ in $U$, two connected components of
$S$ dominating the same irreducible component of $f(S)$ would have a point in
common in their image and in $U$; these two connected components hence would
be connected within $S$. Moreover, since $S$ is normal, its connected
components are irreducible. This implies all irreducible components of $f(S)$
are dominated by exactly one irreducible component of $S$. ∎

For $u$ in $W$, we denote by $x(u):=uP/P$ the $T$-fixed point in
$X(u)=\overline{BuP/P}$ associated with $u$.

######  Lemma 1.2.4.2.

Let $u\in W$.

  1. i)

Let $f:Z\rightarrow X(u)$ be a $B$-equivariant morphism. Then there exists a
surjective map $f^{-1}(BuP/P)\rightarrow f^{-1}(x(u))$.

  2. ii)

Denote by $\pi_{|u}$ the restriction of the projection $\pi$ to $X(u)$. The
general fiber of the surjective morphism $\pi_{|u}:X(u)\rightarrow
X^{\prime}(u)$ is an irreducible unirational variety.

###### Proof.

This is implied by [CMBP13], Proposition 2.3. We include here a proof for
completeness. We denote by $U=BuP/P$ the open $B$-orbit of the $T$-fixed point
$x(u)$.

  1. i)

Denote by $s:U\rightarrow B$ a section of $B\rightarrow BuP/P$, i.e. a
morphism such that for all elements $x$ in $U$, $s(x)\cdot x(u)=x$. Such a
section exists according to [CMBP13], Proposition 2.2. Note that for $y$ in
$f^{-1}(U)$, the element $\left(s(f(y))\right)^{-1}\cdot y$ lies in
$f^{-1}(x(u))$. Indeed, since $f$ is $B$-equivariant, we have
$f(\left(s(f(y))\right)^{-1}\cdot y)=\left(s(f(y))\right)^{-1}\cdot
f(y)=x(u)$. We obtain a morphism:

$\displaystyle f^{-1}(U)$ $\displaystyle\rightarrow f^{-1}(x(u))\times U$
$\displaystyle y$ $\displaystyle\rightarrow(\left(s(f(y))\right)^{-1}\cdot
y;f(y))$

which is an isomorphism; indeed, the morphism $(p,u)\rightarrow s(u)\cdot p$
defines its inverse. Composition with the first projection induces a
surjective morphism $f^{-1}(U)\rightarrow f^{-1}(x(u))$.

  2. ii)

According to $i)$, there exists a surjective map
$\pi_{|u}^{-1}(U)\rightarrow\pi_{|u}^{-1}(x(u))$. Hence, since
$\pi_{|u}^{-1}(U)$ is a dense open subvariety of the irreducible rational
variety $X(u)$, the fiber $\pi_{|u}^{-1}(x(u))$ is an irreducible unirational
variety. Furthermore, since $\pi_{|u}$ is $B$-equivariant, the fiber
$\pi_{|u}^{-1}(x)$ at a point $x$ in $U$ is isomorphic to
$\pi_{|u}^{-1}(x(u))$. Indeed, if $x=b\cdot x(u)$, where $b$ lies in $B$, the
morphism $y\rightarrow b^{-1}\cdot y$ defines an isomorphism between
$\pi^{-1}(x)\cap X(u)$ and $\pi^{-1}(x(u))\cap X(u)$, since $X(u)$ is
$B$-stable. Hence the general fiber of $\pi_{|u}$ is isomorphic to
$\pi_{|u}^{-1}(x(u))$, which is an irreducible unirational variety.

∎

We name ${(\Pi\times
ev)}_{|\mathcal{W}_{g}}:\mathcal{W}_{g}\rightarrow\mathcal{W}_{g}^{\prime}$
the restriction of the morphism $\Pi\times ev$ to the Gromov-Witten variety
$\mathcal{W}_{g}$.

######  Lemma 1.2.4.3.

If the general fiber of $\Pi\times ev$ satisfies a property $(P)$, then for
$g$ general in $G^{r}$ the general fiber of $(\Pi\times
ev)_{|\mathcal{W}_{g}}$ also satisfies $(P)$.

###### Proof.

Name $M:=\overline{{\mathcal{M}}_{0,r}}(X,{\mathbf{d}})$. Let $U$ be a dense
open subset of $(\Pi\times ev)(M)$ such that the fiber of $\Pi\times ev$ over
a point in $U$ satisfies $(P)$. Let $(m,x)$ be a point in $U$, where $m$ lies
in
$M^{\prime}:=\overline{{\mathcal{M}}_{0,r}}(X^{\prime},\pi_{*}{\mathbf{d}})$
and $x$ lies in $X^{r}$. Note that, if $(m,x)$ is in the image $(\Pi\times
ev)({\mathcal{W}}_{g})$, then the fiber over $(m,x)$ of $(\Pi\times
ev)_{|\mathcal{W}_{g}}$ satisfies $(P)$. Indeed, we have:

$\displaystyle(\Pi\times ev)_{|\mathcal{W}_{g}}^{-1}(m,x)$
$\displaystyle=\left(M_{m}\times_{X^{r}}\\{x\\}\right)\times_{X^{r}}(g\prod_{i}X(u_{i}))$
$\displaystyle=M_{m}\times_{X^{r}}\left(\\{x\\}\times_{X^{r}}(g\prod_{i}X(u_{i}))\right)$
$\displaystyle=M_{m}\times_{X^{r}}\\{x\\}$ $\displaystyle=(\Pi\times
ev)^{-1}(m,x),$

where the third equality holds since $(m,x)$ is in $(\Pi\times
ev)({\mathcal{W}}_{g})$, and hence $x$ lies in $ev({\mathcal{W}}_{g})\subset
g\prod_{i}X(u_{i})$. Since $(m,x)$ lies in $U$, the fiber $(\Pi\times
ev)^{-1}(m,x)$ satisfies $(P)$; hence the fiber $(\Pi\times
ev)_{|\mathcal{W}_{g}}^{-1}(m,x)$ satisfies $(P)$ for all $(m,x)$ in
$(\Pi\times ev)({\mathcal{W}}_{g})\cap U$

Finally, note that for $g$ general in $G^{r}$, according to Lemma 1.2.3.1
${\mathcal{W}}_{g}\cap(\Pi\times ev)^{-1}(U)$ is dense in ${\mathcal{W}}_{g}$.
Hence $(\Pi\times ev)({\mathcal{W}}_{g})\cap U$ is dense in $(\Pi\times
ev)({\mathcal{W}}_{g})$. ∎

###### Proof of Proposition 1.2.4.1.

  1. 1.

Since the morphism $\Pi\times ev$ is surjective, this follows directly from
Proposition 1.2.3.1 $i)$, applied to
$h=\Pi:M=\overline{\mathcal{M}_{0,r}}(X,\mathbf{d})\rightarrow
M^{\prime}=\overline{\mathcal{M}_{0,r}}(X^{\prime},\pi_{*}\mathbf{d})$,
$f={\rm ev}:\overline{\mathcal{M}_{0,r}}(X,\mathbf{d})\rightarrow Z=X^{r}$,
$f^{\prime}={\rm
ev}:\overline{\mathcal{M}_{0,r}}(X^{\prime},\pi_{*}\mathbf{d})\rightarrow
Z^{\prime}=(X^{\prime})^{r}$, and from considering the integral group $G^{r}$.

  2. 2.

We proceed in two steps; we first prove that each irreducible component of
$\mathcal{W}_{g}$ surjects into a different irreducible component of
$\mathcal{W}_{g}^{\prime}$, before proving the correspondence with irreducible
components of $\mathcal{W}_{g}^{\prime\prime}$.

Correspondence between irreducible components of $\mathcal{W}_{g}$ and
$\mathcal{W}_{g}^{\prime}$. First, note that, since the space
$\overline{{\mathcal{M}}_{0,r}}(X,\mathbf{d})$ and Schubert varieties are
normal [FP96, Ram85], according to Kleiman’s transversality theorem
$\mathcal{W}_{g}=ev^{-1}(g_{1}X(u_{1})\times\dots g_{r}X(u_{r}))$ is normal
for $(g_{1},\dots,g_{r})$ general in $G^{r}$. Furthermore, since the general
fiber of $\Pi\times ev$ si connected, according to Lemma 1.2.4.3 for $g$
general in $G^{r}$ the general fiber of the restriction ${(\Pi\times
ev)}_{|\mathcal{W}_{g}}$ is connected. Hence according to Lemma 1.2.4.1 all
irreducible components of $(\Pi\times ev)(\mathcal{W}_{g})$ are dominated by
exactly one irreducible component of $\mathcal{W}_{g}$.

We then only have to ensure that there is no irreducible component of
$\mathcal{W}_{g}$ whose image is a strict closed subset of an irreducible
component of $(\Pi\times ev)(\mathcal{W}_{g})=\mathcal{W}_{g}^{\prime}$.
Proposition 1.2.3.1 $iii)$, tells us that, since $\mathcal{W}_{g}$ is in
general position, the image by $\Pi$ of any of its irreducible components is
an irreducible component of $\mathcal{W}_{g}^{\prime}$, hence the $1$-to-$1$
correspondence.

Correspondence between irreducible components of $\mathcal{W}_{g}^{\prime}$
and $\mathcal{W}_{g}^{\prime\prime}$. According to Lemma 1.2.4.2 the general
fiber of the restriction
$\pi_{u}:\prod_{i}X(u_{i})\rightarrow\prod_{i}X^{\prime}(u_{i})$ of $\pi^{r}$
to $X(u_{1})\times\dots\times X(u_{r})$ is irreducible, hence there exists an
open subvariety $U$ of $X^{\prime}(u_{1})\times\dots\times X^{\prime}(u_{r})$
such that the fiber of $\pi$ at $x$ in $U$ is irreducible. Consider the
projections $p_{1}$ and $p_{2}$ defined by the following Cartesian diagram:

${\mathcal{W}_{g}^{\prime}}$${\prod_{i}g_{i}X(u_{i})}$${\mathcal{W}_{g}^{\prime\prime}}$${\prod_{i}g_{i}X^{\prime}(u_{i})}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{2}}$$\scriptstyle{\pi_{u}}$$\scriptstyle{ev}$

Note that the fiber $p_{1}^{-1}(m)$ at a point $m$ in
$p_{2}^{-1}(g\pi_{u}^{-1}(U))$ is irreducible. Indeed, we have:

$\displaystyle p_{1}^{-1}(m)$
$\displaystyle=\\{m\\}\times_{(X^{\prime})^{r}}\left(g_{1}X(u_{1})\times\dots
g_{r}X(u_{r})\right)$ $\displaystyle\simeq\pi_{u}^{-1}(ev(m))$
$\displaystyle\simeq g\cdot\pi_{u}^{-1}(g^{-1}ev(m)),$

which is irreducible since $ev(m)$ lies in $g\pi_{u}^{-1}(U)$. Furthermore,
according to Lemma 1.2.3.2 $b)$, for $g$ general in $G^{r}$,
$\mathcal{W}_{g}^{\prime}\cap p_{2}^{-1}(g\pi_{u}^{-1}(U))$ is a dense open
subvariety of $\mathcal{W}_{g}^{\prime}=p_{2}^{-1}(\prod_{i}g_{i}X(u_{i}))$.
Hence the general fiber of $p_{1}$ is irreducible. Furthermore, note that,
since $\overline{\mathcal{M}_{0,r}}(X^{\prime},\pi_{*}\mathbf{d})$ and
Schubert varieties are normal [FP96, Ram85], according to Kleiman’s
transversality theorem

$\mathcal{W}_{g}^{\prime}=\overline{\mathcal{M}_{0,r}}(X^{\prime},\pi_{*}\mathbf{d})\times_{(X^{\prime})^{r}}\left(g_{1}X(u_{1})\times\dots
g_{r}X(u_{r})\right)$

is normal for $g=(g_{1},\dots,g_{r})$ general in $G^{r}$. According to Lemma
1.2.4.1, all irreducible components of
$p_{1}(\mathcal{W}_{g}^{\prime})=\mathcal{W}_{g}^{\prime\prime}$ are thus
dominated by exactly one irreducible component of $\mathcal{W}_{g}^{\prime}$.

Finally, Proposition 1.2.3.1 $iv)$, tells us that, since
$\mathcal{W}_{g}^{\prime}$ is in general position, the image by $p_{1}$ of any
of its irreducible components is an irreducible component of
$\mathcal{W}_{g}^{\prime\prime}$, hence the $1$-to-$1$ correspondence.

To sum up, each irreducible component of $\mathcal{W}_{g}$ surjects into a
different irreducible component of $\mathcal{W}_{g}^{\prime}$, and each
irreducible component of $\mathcal{W}_{g}^{\prime}$ surjects into a different
irreducible component of $\mathcal{W}_{g}^{\prime\prime}$. Furthermore,
$\mathcal{W}_{g}$ surjects onto $\mathcal{W}_{g}^{\prime\prime}$. There thus
is a $1$-to-$1$ correspondence between irreducible components of
$\mathcal{W}_{g}$ and irreducible components of
$\mathcal{W}_{g}^{\prime\prime}$.

  3. 3.

The only point left to prove is that the general fiber of
${\mathcal{W}}_{g}\rightarrow{\mathcal{W}}_{g}^{\prime\prime}$ is an
irreducible rationally connected variety. According to Lemma 1.2.4.2 there
exists a dense open subset $U$ of $\prod_{i}X^{\prime}(u_{i})$ such that the
fiber of $\pi_{|u}:\prod_{i}X(u_{i})\rightarrow\prod_{i}X^{\prime}(u_{i})$
over a point in $U$ is an irreducible rationally connected variety. For $g$
general in $G^{r}$, the following properties are satisfied:

    1. (a)

$\left(\overline{{\mathcal{M}}_{0,r}}(X^{\prime},\pi_{*}{\mathbf{d}})\right)_{g\cdot
U}$ is dense in
${\mathcal{W}}_{g}^{\prime\prime}=\left(\overline{{\mathcal{M}}_{0,r}}(X^{\prime},\pi_{*}{\mathbf{d}})\right)_{g\cdot\prod_{i}X^{\prime}(u_{i})}$;

    2. (b)

The general fiber of
${\mathcal{W}}_{g}^{\prime}\rightarrow{\mathcal{W}}_{g}^{\prime\prime}$ is
rationally connected;

    3. (c)

The general fiber of ${\mathcal{W}}_{g}\rightarrow{\mathcal{W}}_{g}^{\prime}$
is rationally connected;

    4. (d)

The general fiber of
${\mathcal{W}}_{g}\rightarrow{\mathcal{W}}_{g}^{\prime\prime}$ is rationally
connected.

Note that proving $(d)$ is our goal here. $(a)$ holds according to Lemma
1.2.3.1. Note that the fiber of the projection
$p_{1}:{\mathcal{W}}_{g}^{\prime}\rightarrow{\mathcal{W}}_{g}^{\prime\prime}$
over a point $m$ in
$\left(\overline{{\mathcal{M}}_{0,r}}(X^{\prime},\pi_{*}{\mathbf{d}})\right)_{g\cdot
U}$ is rationally connected; indeed, it is isomorphic to
$g\cdot(\pi_{|u})^{-1}(g^{-1}\cdot ev(m))$, where $g^{-1}\cdot ev(m)$ lies in
$U$. Hence, since according to $(a)$
$\left(\overline{{\mathcal{M}}_{0,r}}(X^{\prime},\pi_{*}{\mathbf{d}})\right)_{g\cdot
U}$ is dense in
${\mathcal{W}}_{g}^{\prime\prime}=\left(\overline{{\mathcal{M}}_{0,r}}(X^{\prime},\pi_{*}{\mathbf{d}})\right)_{g\cdot\prod_{i}X^{\prime}(u_{i})}$,
$(b)$ holds. $(c)$ holds according to Lemma 1.2.4.3. Finally, $(d)$ is deduced
from $(b)$, $(c)$ and Theorem 1.2.2.1 $(3)$.

∎

###  1.3 Balanced flags of vector bundles

Let $X$ be a flag variety. We describe here the flag of vector bundles
associated with a general point in
$\overline{{\mathcal{M}}_{0,r}}(X,\mathbf{d})$ for a degree ${\mathbf{d}}$
high enough. All definitions and results presented in this part are contained
in I. Coskun’s description of rational curves on a flag variety [Cos10], or
easily deduced from it.

####  1.3.1 Notations.

Fix two positive integers $m<n$ . Let $I=\\{i_{1},\dots,i_{m}\\}$ be a set of
integers satisfying $0<i_{1}<\dots<i_{m}<n$. Recall $Fl_{I}$ is the $m$-step
flag variety parametrizing $m$-tuples $(V_{i_{1}},\dots,V_{i_{m}})$ of vector
subspaces of $\mathbb{C}^{n}$, where the subspaces are ordered by inclusion
$0\subset V_{i_{1}}\subset\dots\subset V_{i_{m}}\subset\mathbb{C}^{n}$, and
$V_{i_{k}}$ is of dimension $i_{k}$. The class
$\mathbf{d}=(d_{1},\dots,d_{m})$ of a curve $C$ in $X$ is determined by non-
negative integers $d_{1}$, $d_{2}$,$\dots$, $d_{m}$, where $d_{i}$ is the
PlÃ¼cker degree of the projection of $C$ to the Grassmannian ${\rm
Gr}(n_{i},n)$. A point $p$ in $\mathcal{M}_{0,r}(X,\mathbf{d})$ is associated
with a morphism $f_{p}:\mathbb{P}^{1}\rightarrow X$. By the functorial
definition of flag varieties, the morphism $f_{p}:{\mathbb{P}}^{1}\rightarrow
X$ is uniquely determined by the data of a flag of vector bundles over
$\mathbb{P}^{1}$:

$\mathcal{F}_{p}=E_{1}\subset\dots\subset
E_{m}\subset{\mathcal{O}}_{{\mathbb{P}}^{1}}^{\oplus n},$

where $\mathrm{rank}E_{k}=i_{k}$ and $\mathrm{deg}E_{k}=-d_{k}$. Note that we
have $f_{p}^{*}S_{i}\simeq E_{i}$, where $S_{i}$ is the $i$-th tautological
vector bundle over $X$.

####  1.3.2 Admissible sets of sequences.

Recall $I=\\{i_{1},\dots,i_{m}\\}$ is a set of $m$ integers
$0<i_{1}<\dots<i_{m}<n$.

######  Definition 1.3.2.1.

Consider a set $\mathbf{d}=(d_{1},\dots,d_{m})$ of $m$ non negative integers.
A $(I,\mathbf{d})$-admissible set of sequences is a set

$A_{\bullet}=\\{(a_{1,j})_{j=1}^{i_{1}},(a_{2,j})_{j=1}^{i_{2}},\dots,(a_{m,j})_{j=1}^{i_{m}}\\}$

of $m$ sequences of non-negative integers of lengths $i_{1}$, $\dots$,$i_{m}$,
respectively, such that:

  1. 1.

$0\leq a_{k,j}\leq a_{k,j+1}$ for every $k$ and every $1\leq j\leq i_{k}-1$;

  2. 2.

$a_{k+1,j}\leq a_{k,j}$ for every $1\leq k\leq m-1$ and $j\leq i_{k}$;

  3. 3.

$\sum_{j=1}^{i_{k}}a_{k,j}=d_{k}$ for every $k$.

######  Definition 1.3.2.2.

A $(I,\mathbf{d})$-admissible set of sequences $A_{\bullet}=\\{(a_{k,j})\\}$
is balanced if $A_{\bullet}$ minimizes the function

$\sum_{1\leq k\leq m}\sum_{1\leq l<p\leq n_{i}}(a_{k,p}-a_{k,l})$

among the $(I,\mathbf{d})$-admissible sets of sequences.

Note that, according to [Cos10] (see the proof of Lemma $2.1$), for any set
${\mathbf{d}}=(d_{1},\dots,d_{m})$ in $\mathbb{N}^{m}$ there exists a unique
balanced $(I,{\mathbf{d}})$-admissible set of sequences $(a_{k,j})$, which can
be constructed by iteration on $k$ in the following way.

  1. $\bullet$

First, set $a_{1,j}=\lceil d_{1}/n_{1}\rceil-\epsilon_{1,j}$ where
$\epsilon_{1,j}$ is in $\\{0,1\\}$ and is chosen in such a way that
$\sum_{j=1}^{n_{1}}a_{1,j}=d_{1}$ and $a_{1,j}\leq a_{1,j+1}$ and $\lceil
q\rceil$ denote the ceiling of $q\in\mathbb{Q}$;

  2. $\bullet$

Let $1<k\leq m$. Suppose all $a_{k-1,j}$ have been constructed. Denote by
$r_{k}$ the largest index such that the element $a_{k-1,r_{k}}$ satisfies
$a_{k-1,r_{k}}\leq d_{k}/n_{k}$. Set:

$a_{k,j}=a_{k-1,j}\>\mathrm{for}\>1\leq j\leq r_{k}.$ (1.5)

Denote by $R_{k}$ the largest index $>r_{k}$ such that the element
$a_{k-1,R_{k}}$ satisfies
$a_{k-1,R_{k}}\leq(d_{k}-\sum_{j=1}^{r_{k}}a_{k-1,j})/(i_{k}-r_{k})$. Repeat
the process replacing $r_{k}$ with $R_{k}$ until all elements $a_{k-1,j}$ of
index larger than $r_{k}$ are larger than
$(d_{k}-\sum_{j=1}^{r_{k}}a_{k-1,j})/(i_{k}-r_{k})$. Then set:

$\forall r_{k}<j\leq
i_{k},\>a_{k,j}=\left\lceil\frac{(d_{k}-\sum_{j=1}^{r_{k}}a_{k-1,j})}{(a_{k}-r_{k})}\right\rceil-\epsilon_{k,j},$
(1.6)

where $\epsilon_{k,j}$ is in $\\{0,1\\}$ and is chosen in such a way that
$\sum_{j=1}^{i_{k}}a_{k,j}=d_{k}$ and $a_{k,j}\leq a_{k,j+1}$.

For simplicity, we set $d_{0}:=0$ and $n_{0}:=0$.

######  Proposition 1.3.2.1.

Let ${\mathbf{d}}=(d_{1},\dots,d_{m})$ be a set of nonnegative integers. Let
$1<k\leq m$ such that:

$\forall\>p<k,\>d_{k}\geq
i_{k}\left\lceil\frac{d_{p}-d_{p-1}}{i_{p}-i_{p-1}}\right\rceil.$ (1.7)

Then the balanced $(I,{\mathbf{d}})$-admissible set of sequences
$\\{(a_{1,j})_{j=1}^{i_{1}},\dots,(a_{m,j})_{j=1}^{i_{m}}\\}$ satisfies
$a_{k,j}=a_{k-1,j}$ for $1\leq j\leq i_{k-1}$.

###### Proof.

By iteration on $s<k$, we prove that for $1\leq s<k$ all elements $a_{s,j}$
satisfy $a_{s,j}\leq d_{k}/n_{k}$. According to (1.5) this will yield the
expected result. First, for $s=1$, since $d_{k}\geq n_{k}\lceil
d_{1}/i_{1}\rceil$, $a_{1,j}\leq d_{k}/i_{k}$. Now let $1\leq s<k$, and
suppose $a_{s-1,j}\leq d_{k}/i_{k}$ for all $1\leq j\leq n_{s-1}$. Note that,
if $r_{s}<i_{s-1}$, then there exists elements $a_{s-1,j}$ larger than
$(d_{s}-\sum_{j=1}^{r_{s}}a_{s-1,j})/(i_{s}-r_{s})$, where we use the
notations of (1.6); hence all elements $a_{s,j}$ are smaller than an element
$a_{s-1,s^{\prime}}$ for some $s^{\prime}$. Hence all elements $a_{s,j}$ then
satisfy $a_{s,j}\leq d_{k}/n_{k}$. On the other hand, if $r_{s}=i_{s-1}$, then
according to (1.6):

$\displaystyle\forall i_{s-1}<j\leq i_{s},\>\>\>\>a_{s,j}$
$\displaystyle\leq\left\lceil\frac{\left(d_{s}-\sum_{j=1}^{i_{s-1}}a_{s-1,j}\right)}{(i_{s}-i_{s-1})}\right\rceil$
$\displaystyle\leq\left\lceil\frac{(d_{s}-d_{s-1})}{(i_{s}-i_{s-1})}\right\rceil\>\>\>\>\leq\frac{d_{k}}{n_{k}}.$

∎

####  1.3.3 Balanced flag bundles.

The partial flag variety $Fl_{I}$ comes equipped with $m$ tautological bundles
$S_{1}$, $\dots$, $S_{m}$. We describe here the pullback $f_{p}^{*}S_{i}$ by a
morphism $f_{p}:\mathbb{P}^{1}\rightarrow X$ associated with a point $p$
general in $\overline{\mathcal{M}_{0,r}(X,\mathbf{d})}$.

Let $\mathbf{d}=(d_{1},\dots,d_{m})$ be an effective class of curve in
$H_{2}(Fl_{I},\mathbb{Z})$, $p$ be a point in
${{\mathcal{M}}_{0,r}}(X,\mathbf{d})$, and $f_{p}:\mathbb{P}^{1}\rightarrow X$
be its associated morphism. The pullback $f_{p}^{*}S_{s}$ of the $s$-th
tautological bundle $S_{s}$ by $f_{p}$ is a vector bundle of degree $d_{s}$
and rank $i_{s}$. According to Grothendieck’s decomposition of vector bundles,
$f_{p}^{*}S_{s}$ is isomorphic to a direct sum of line bundles

$f_{p}^{*}S_{s}\simeq\mathcal{O}_{\mathbb{P}^{1}}(-a_{s,1})\oplus\dots\oplus\mathcal{O}_{\mathbb{P}^{1}}(-a_{s,i_{s}}),$

where $\sum_{j=1}^{i_{s}-k_{s}}a_{s,j}=d_{s}$. Suppose, for a given $s$, the
$a_{s,j}$ are ordered in increasing order. Then, since, for all $1<s\leq m$,
$f_{p}^{*}S_{s-1}$ is a subbundle of $f_{p}^{*}S_{s}$, we have: $a_{s-1,j}\geq
a_{s,j}$. Hence the sequence $(a_{s,j})$ forms a $(I,\mathbf{d})$-admissible
set of sequence in the sense of Definition 1.3.2.1. We call $a_{s,j}$ the
admissible set of sequences associated with $p$.

######  Proposition 1.3.3.1.

  1. i)

The admissible set of sequences associated with a general point in
$\mathcal{M}_{0,r}(Fl_{I},\mathbf{d})$ is balanced in the sense of Definition
1.3.2.2;

  2. ii)

Suppose $\forall\>p<k,\>d_{k}\geq
n_{k}\left\lceil\frac{d_{p}-d_{p-1}}{i_{p}-i_{p-1}}\right\rceil$. Denote by
$\mathcal{F}_{\mathfrak{p}}=E_{1}\hookrightarrow\dots\hookrightarrow
E_{m}\hookrightarrow\mathcal{O}_{\mathbb{P}^{1}}^{\oplus n}$ the flag of
vector bundles associated with a general element ${\mathfrak{p}}$ in
$\overline{{\mathcal{M}}_{0,r}}(Fl_{I},\mathbf{d})$. Then $E_{k}$ splits into
$E_{k}\simeq
E_{k-1}\oplus_{\sum_{s=n_{k-1}+1}^{i_{k}}}{\mathcal{O}}_{{\mathbb{P}}^{1}}(-a_{k,s})$.

###### Proof.

  1. i)

Cf. proof of [Cos10], Proposition $2.3.$

  2. ii)

Let $\mathcal{F}_{\mathfrak{p}}=E_{1}\subset\dots\subset
E_{m}\subset\mathcal{O}_{\mathbb{P}^{1}}^{\oplus n}$ be the flag of vector
bundles associated with a general element ${\mathfrak{p}}$ in
$\overline{{\mathcal{M}}_{0,r}}(X,\mathbf{d})$. Then, according to $i)$, the
admissible set of sequences associated with $\mathcal{F}_{\mathfrak{p}}$ is
balanced. Since $d_{k}$ satisfies $\forall\>p<k,\>d_{k}\geq
i_{k}\left\lceil\frac{d_{p}-d_{p-1}}{i_{p}-i_{p-1}}\right\rceil$, we have
according to Proposition 1.3.2.1 the following sequence:

$\displaystyle 0\rightarrow$ $\displaystyle
E_{k-1}\simeq\mathcal{O}_{\mathbb{P}^{1}}(-a_{k-1,1})\oplus\dots\oplus\mathcal{O}_{\mathbb{P}^{1}}(-a_{k-1,n_{k-1}})\rightarrow$
$\displaystyle\>\>\>\>\>\>\>\>\>\>\>\>E_{k}\simeq\mathcal{O}_{\mathbb{P}^{1}}(-a_{k-1,1})\oplus\dots\oplus\mathcal{O}_{\mathbb{P}^{1}}(-a_{k-1,i_{k-1}})\oplus\mathcal{O}_{\mathbb{P}^{1}}(-a_{k,i_{k-1}+1})\oplus\dots\oplus\mathcal{O}_{\mathbb{P}^{1}}(-a_{k,n_{k}})$

which splits. Indeed, for all $j$, $l$ verifying $j\leq l$ and $j\leq
n_{k-1}<l$, we have: $a_{k,j}=a_{k-1,j}$ and $a_{k,j}\leq a_{k,l}$, hence
$a_{k-1,j}\leq a_{k,l}$ for any $l>i_{k-1}$. Hence according to Lemma 1.3.3.1
the morphisms $\mathcal{O}_{\mathbb{P}^{1}}(-a_{k-1,j})\rightarrow E_{k}$
factor through
$\oplus_{b=1}^{n_{k-1}}\mathcal{O}_{\mathbb{P}^{1}}(-a_{k,b})\hookrightarrow
E_{k}$. Finally, since the morphism $E_{k-1}\rightarrow E_{k}$ is injective,
$E_{k-1}$ is a rank $i_{k-1}$ subvector bundle of
$\oplus_{b=1}^{i_{k-1}}\mathcal{O}_{\mathbb{P}^{1}}(-a_{k,b})$; hence
$E_{k-1}\simeq\oplus_{b=1}^{n_{k-1}}\mathcal{O}_{\mathbb{P}^{1}}(-a_{k,b})$.

∎

######  Lemma 1.3.3.1.

Let $d_{1}\leq\dots\leq d_{r}$ and $d_{1}^{\prime}=d_{1}\leq\dots\leq
d_{r}^{\prime}=d_{r}\leq d_{r+1}^{\prime}\leq\dots\leq d_{r+p}^{\prime}$ be
sequences of non negative integers. Then the following sequence splits.

$\displaystyle 0\rightarrow$ $\displaystyle
E=\mathcal{O}_{\mathbb{P}^{1}}(-d_{1})\oplus\dots\oplus\mathcal{O}_{\mathbb{P}^{1}}(-d_{r})\rightarrow$
$\displaystyle\>\>\>\>\>\>\>\>\>\>\>\>F=\mathcal{O}_{\mathbb{P}^{1}}(-d_{1})\oplus\dots\oplus\mathcal{O}_{\mathbb{P}^{1}}(-d_{r})\oplus\mathcal{O}_{\mathbb{P}^{1}}(-d_{r+1}^{\prime})\oplus\dots\oplus\mathcal{O}_{\mathbb{P}^{1}}(-d_{r+p}^{\prime})$

###### Proof.

Let $k$ be the largest index such that $d_{k}<d_{r}$. Note that for $i>k$ all
the integers $d_{i}^{\prime}$ are greater than $d_{k+1}$ or equal to $d_{k+1}$
and $d_{k+1}=d_{r}>d_{k}$. Hence all integers $d_{i}^{\prime}$ are greater
than $d_{k}$ for $i>k$. Hence the induced morphism
$E_{1}:=\mathcal{O}_{\mathbb{P}^{1}}(-d_{1})\oplus\dots\oplus\mathcal{O}_{\mathbb{P}^{1}}(-d_{k})\rightarrow
F$ factors through a closed immersion
$\mathcal{O}_{\mathbb{P}^{1}}(-d_{1})\oplus\dots\oplus\mathcal{O}_{\mathbb{P}^{1}}(-d_{k})\hookrightarrow
F$. Denote by $r^{\prime}$ the largest index such that
$d_{r^{\prime}}^{\prime}=d_{r}$. Note that $r^{\prime}\geq r$. Let us now
consider the induced exact sequence

$\displaystyle 0\rightarrow$ $\displaystyle
E/E_{1}\simeq\mathcal{O}_{\mathbb{P}^{1}}(-d_{k+1})\oplus\dots\oplus\mathcal{O}_{\mathbb{P}^{1}}(-d_{r})\simeq{\mathcal{O}}_{{\mathbb{P}}^{1}}(-d_{k})^{\oplus
r-k}$ $\displaystyle\>\>\>\>\rightarrow
F/E_{1}\simeq\mathcal{O}_{\mathbb{P}^{1}}(-d_{k+1}^{\prime})\oplus\dots\oplus\mathcal{O}_{\mathbb{P}^{1}}(-d_{r+p}^{\prime})\simeq{\mathcal{O}}_{{\mathbb{P}}^{1}}(-d_{r})^{\oplus(r^{\prime}-k)}\oplus{\mathcal{O}}_{{\mathbb{P}}^{1}}(-d_{r^{\prime}+1})\oplus\dots\oplus{\mathcal{O}}_{{\mathbb{P}}^{1}}(-d_{r+p}^{\prime}).$

Since for any $i>r^{\prime}$ the integer $d_{i}^{\prime}$ is greater than
$d_{k}$, the injective morphism $E/E_{1}\rightarrow F/E_{1}$ factors through

$E/E_{1}\simeq{\mathcal{O}}_{{\mathbb{P}}^{1}}(-d_{k})^{\oplus
r-k}\hookrightarrow{\mathcal{O}}_{{\mathbb{P}}^{1}}(-d_{r})^{\oplus(r^{\prime}-k)}\hookrightarrow
F/E_{1}$

Finally observe that the morphism
$E/E_{1}\simeq{\mathcal{O}}_{{\mathbb{P}}^{1}}(-d_{k})^{\oplus
r-k}\hookrightarrow{\mathcal{O}}_{{\mathbb{P}}^{1}}(-d_{r})^{\oplus(r^{\prime}-k)}$
factors through $E/E_{1}\simeq{\mathcal{O}}_{{\mathbb{P}}^{1}}(-d_{k})^{\oplus
r-k}\hookrightarrow{\mathcal{O}}_{{\mathbb{P}}^{1}}(-d_{k})^{\oplus
r-k}\oplus{\mathcal{O}}_{{\mathbb{P}}^{1}}(-d_{k})^{\oplus
r^{\prime}-r}\hookrightarrow F/E_{1}$. ∎

###  1.4 Forgetting one vector space

We denote by $\pi_{\widehat{k}}:X\rightarrow X_{\widehat{k}}$ the morphism
forgetting the $k$-th vector space. Let ${\mathbf{d}}=(d_{1},\dots,d_{m})$ be
a degree in $E(X)$. In this section, our goal is to prove the following
results.

######  Proposition 1.4.0.1.

Suppose $\forall\>p<k,\>d_{k}\geq
n_{k}\left\lceil\frac{d_{p}-d_{p-1}}{n_{p}-n_{p-1}}\right\rceil$. Then the
general fiber of

$\Pi_{\widehat{k}}\times
ev:\overline{{\mathcal{M}}_{0,r}}(X,{\mathbf{d}})\rightarrow\left(\Pi_{\widehat{k}}\times
ev\right)(\overline{{\mathcal{M}}_{0,r}}(X,{\mathbf{d}}))\subset\overline{{\mathcal{M}}_{0,r}}(X_{\widehat{k}},(\pi_{\widehat{k}})_{*}{\mathbf{d}})\times_{(X_{\widehat{k}})^{r}}X^{r}$

is a unirational variety.

######  Corollary 1.4.0.1.

Suppose $\forall\>p<k,\>d_{k}\geq
n_{k}\left\lceil\frac{d_{p}-d_{p-1}}{n_{p}-n_{p-1}}\right\rceil$ and the map
$\Pi_{\widehat{k}}\times ev$ is surjective. Then for $x$ general in $X^{r}$,
the general fiber of the map
$\overline{{\mathcal{M}}_{0,r}}(X,{\mathbf{d}})\ni ev^{-1}(x)\to
ev^{-1}(\pi_{\widehat{k}}(x))\in\overline{{\mathcal{M}}_{0,r}}(X_{\widehat{k}},(\pi_{\widehat{k}})_{*}{\mathbf{d}})$
is a unirational variety.

We will assume in this section $\forall\>p<k,\>d_{k}\geq
n_{k}\left\lceil\frac{d_{p}-d_{p-1}}{n_{p}-n_{p-1}}\right\rceil$. Let $(p,x)$
be a point in $(\Pi_{\widehat{k}}\times{\rm
ev})({{\mathcal{M}}_{0,r}}(X,\mathbf{d}))$, where $p$ is in
${{\mathcal{M}}_{0,r}}(X_{\widehat{k}},(\pi_{\widehat{k}})*\mathbf{d})$, and
$x=(x_{1},\dots,x_{r})$ is in $X^{r}$. Denote by $\Pi_{p,x}$ the fiber of
$\Pi_{\widehat{k}}\times{\rm ev}$ at $(p,x)$, i.e. :

$\Pi_{p,x}:=\Pi_{\widehat{k}}^{-1}(p)\times_{X^{r}}x.$

In order to prove the irreducibility and unirationality of $\Pi_{p,x}$, we
define an auxilliary variety $\mathcal{Z}_{p,x}$, which dominates $\Pi_{p,x}$
when $(p,x)$ is general in the image $(\Pi_{\widehat{k}}\times
ev)({{\mathcal{M}}_{0,r}}(X,\mathbf{d}))$.

####  1.4.1 Notations.

The class $\mathbf{d}=(d_{1},\dots,d_{m})$ of a curve $C$ in $X$ is determined
by non-negative integers $d_{1}$, $d_{2}$,$\dots$, $d_{m}$, where $d_{i}$ is
the PlÃ¼cker degree of the projection of $C$ to the Grassmannian ${\rm
Gr}(n_{i},n)$. The morphism $\pi_{\widehat{k}}:X\rightarrow X_{\widehat{k}}$
induces a morphism
$\Pi_{\widehat{k}}:\overline{{\mathcal{M}}_{0,r}}(X,\mathbf{d})\rightarrow\overline{{\mathcal{M}}_{0,r}}(X_{\widehat{k}},(\pi_{\widehat{k}})_{*}\mathbf{d})$.
Denote by $\pi_{k}:X\rightarrow{\rm Gr}(n_{k},n)$ the morphism sending a
$m$-step flag $\\{0\\}\subset V_{1}\subset\dots\subset
V_{k}\subset\dots\subset V_{m}$ to the $k$-th linear subspace $V_{k}$. This
morphism induces a morphism
$\overline{{\mathcal{M}}_{0,r}}(X,\mathbf{d})\rightarrow\overline{{\mathcal{M}}_{0,r}}({\rm
Gr}(n_{k},n),d_{k})\rightarrow\overline{{\mathcal{M}}_{0,3}}({\rm
Gr}(n_{k},n),d_{k})$ into the space of genus zero stable maps with three
marked points to the Grassmannian ${\rm Gr}(n_{k},n)$. We obtain the following
commutative diagram:

${\overline{{\mathcal{M}}_{0,r}}(X,\mathbf{d})}$${\overline{{\mathcal{M}}_{0,3}}(X,\mathbf{d})}$${\overline{{\mathcal{M}}_{0,3}}({\rm
Gr}(n_{k},n),d_{k})}$${X^{r}}$${{\rm
Gr}(n_{k},n)^{r}}$${\overline{{\mathcal{M}}_{0,r}}(X_{\widehat{k}},(\pi_{\widehat{k}})_{*}\mathbf{d})}$${{X_{\widehat{k}}}^{r}}$$\scriptstyle{{\rm
ev}^{r}}$$\scriptstyle{\Pi_{\widehat{k}}}$$\scriptstyle{\pi_{\widehat{k}}\times\dots\times\pi_{\widehat{k}}}$$\scriptstyle{\pi_{k}\times\dots\times\pi_{k}}$

Furthermore, for $p$ in $\mathcal{M}_{0,r}(X,\mathbf{d})$, we will denote by:

$f_{p}:\mathbb{P}^{1}\rightarrow X$

the associated morphism. By the functorial definition of flag varieties,
$f_{p}$ is uniquely determined by the data of a flag of vector bundles over
$\mathbb{P}^{1}$:

$\mathcal{F}_{p}=E_{1}\subset\dots\subset
E_{m}\subset{\mathcal{O}}_{{\mathbb{P}}^{1}}^{\oplus n},$

where $\mathrm{rank}E_{i}=n_{i}$ and $\mathrm{deg}E_{i}=-d_{i}$. Note that we
have $f_{p}^{*}S_{i}\simeq E_{i}$, where $S_{i}$ is the $i$-th tautological
vector bundle over $X$. Finally, for a point ${\mathfrak{p}}$ in
${\mathbb{P}}^{1}$ we will denote by $(E_{i})_{\mathfrak{p}}$ the fiber at
${\mathfrak{p}}$ of the vector bundle $E_{i}$, which has a natural inclusion
$(E_{i})_{\mathfrak{p}}\subset{\mathbb{C}}^{n}$ induced by the morphism
$E_{i}\hookrightarrow{\mathcal{O}}_{{\mathbb{P}}^{1}}^{\oplus n}$.

####  1.4.2 A preliminary result.

Denote by $\mathcal{F}_{p}=E_{1}\subset\dots\subset E_{k-1}\subset
E_{k+1}\subset\dots\subset E_{m}\subset\mathcal{O}_{\mathbb{P}^{1}}^{\oplus
n}$ the flag of vector bundles over $\mathbb{P}^{1}$ associated with the point
$p$ in $\mathcal{M}_{0,r}(X_{\widehat{k}},(\pi_{\widehat{k}})_{*}\mathbf{d})$.

######  Lemma 1.4.2.1.

  1. i)

Consider a dense open subset $U$ of
$\overline{{\mathcal{M}}_{0,r}}(X,\mathbf{d})$. Then, for $(p,x)$ general in
$(\Pi_{\widehat{k}}\times ev)(\overline{{\mathcal{M}}_{0,r}}(X,\mathbf{d}))$,
$\Pi_{p,x}\cap U$ is a dense open subset of $\Pi_{p,x}$;

  2. ii)

For $(p,x)$ general in $(\Pi_{\widehat{k}}\times
ev)(\overline{{\mathcal{M}}_{0,r}}(X,\mathbf{d}))$, there exists a dense open
subset $U$ of $\Pi_{p,x}$ such that, for all $q$ in $U$, the flag of vector
bundles $\mathcal{F}_{q}$ associated with $q$ is balanced. Hence there exists
an increasing set of integers $a_{n_{k-1}+1}\leq\dots\leq a_{n_{k}}$ such that
for all $q$ in $U$, the flag $\mathcal{F}_{q}$ is of the form:

$\mathcal{F}_{q}=E_{1}\subset\dots\subset E_{k-1}\subset E_{k}\simeq
E_{k-1}\oplus_{i=n_{k-1}+1}^{n_{k}}\mathcal{O}_{\mathbb{P}^{1}}(-a_{i})\subset
E_{k+1}\subset\dots\subset E_{m}\subset\mathcal{O}_{\mathbb{P}^{1}}^{\oplus
n}.$

###### Proof.

  1. i)

Since $(p,x)$ is general in $(\Pi_{\widehat{k}}\times
ev)(\overline{{\mathcal{M}}_{0,r}}(X,\mathbf{d}))$, according to Lemma 1.2.2.1
the intersection of $\Pi_{p,x}$ with the dense open subset $U$ is dense in
$\Pi_{p,x}$.

  2. ii)

According to Proposition 1.3.3.1 $i)$, there exists a dense open subset $U$ of
$\overline{{\mathcal{M}}_{0,r}}(X,\mathbf{d})$ such that a point in $U$
corresponds to a balanced flag of vector bundles over $\mathbb{P}^{1}$.
According to $i)$, $U\cap\Pi_{p,x}$ is dense open subset of $\Pi_{p,x}$.
Finally, notice that according to Proposition 1.3.3.1 $ii)$, since
$\forall\>p<k,\>d_{k}\geq
n_{k}\left\lceil\frac{d_{p}-d_{p-1}}{n_{p}-n_{p-1}}\right\rceil$, the flag of
vector bundles associated with a point in $\Pi_{p,x}\cap U$ splits at
$E_{k-1}$.

∎

####  1.4.3 Definition of $\mathcal{Z}_{p,x}$.

Our goal here is to construct a rational variety $\mathcal{Z}_{p,x}$
dominating $\Pi_{p,x}$. First, consider the integers $a_{n_{k-1}+1}$, $\dots$,
$a_{n_{k}}$ defined in Lemma 1.4.2.1 $ii)$ and the following vector space

$V:=Hom(\oplus_{i=n_{k-1}+1}^{n_{k}}\mathcal{O}_{\mathbb{P}^{1}}(-a_{i}),\mathcal{O}_{\mathbb{P}^{1}}^{\oplus
n}).$

The flag variety $X$ is naturally embedded in $X_{\widehat{k}}\times{\rm
Gr}(n_{k},n)$ by PlÃ¼cker-embedding $X_{\widehat{k}}\times{\rm Gr}(n_{k},n)$
in a product of projective spaces, and cutting out $X$ from this product by
considering the relations given by the inclusions of flags. Note that, if we
denote by $[f_{p}:\mathbb{P}^{1}\rightarrow
X_{\widehat{k}},\\{p_{1},\dots,p_{r}\\}]$ the stable map associated with $p$,
points $q$ in $\Pi_{p,x}\cap\mathcal{M}_{0,r}(X,\mathbf{d})$ correspond to
stable maps $[f_{q}:\mathbb{P}^{1}\rightarrow X,\\{p_{1},\dots,p_{r}\\}]$
satisfying $f_{q}(p_{i})=x_{i}$ and such that the following diagram is
commutative:

${\mathbb{P}^{1}}$${X}$${X_{\widehat{k}}\times{\rm
Gr}(n_{k},n)}$${X_{\widehat{k}}}$$\scriptstyle{f_{q}}$$\scriptstyle{f_{p}}$

Furthermore, by the functorial definition of the Grassmannian, elements $u$ in
$V$ corresponding to an injective morphism

$\Phi_{u}:E_{k-1}\oplus_{i}\mathcal{O}_{\mathbb{P}^{1}}(-a_{i})\rightarrow\mathcal{O}_{\mathbb{P}^{1}}^{\oplus
n}$

define a morphism $f_{u}:\mathbb{P}^{1}\rightarrow{\rm Gr}(n_{k},n)$, where we
denote by $\Phi_{u}$ the morphism induced by the inclusion
$E_{k-1}\subset{\mathcal{O}}_{{\mathbb{P}}^{1}}^{\oplus n}$ and the morphism
$\mathcal{O}_{\mathbb{P}^{1}}(-a_{i})\rightarrow\mathcal{O}_{\mathbb{P}^{1}}^{\oplus
n}$ associated with a point $u$ in $V$. Conversally, any injective morphism
$E_{k-1}\oplus_{i}\mathcal{O}_{\mathbb{P}^{1}}(-a_{i})\rightarrow\mathcal{O}_{\mathbb{P}^{1}}^{\oplus
n}$ can, up to automorphisms of $E_{k-1}$, be obtained this way. Points in
$\Pi_{p,x}$ are thus associated with elements $u$ in $V$ such that:

  1. 1.

$\Phi_{u}:E_{k-1}\oplus_{i=n_{k-1}+1}^{n_{k}}\mathcal{O}_{\mathbb{P}^{1}}(-a_{i})\rightarrow\mathcal{O}_{\mathbb{P}^{1}}^{\oplus
n}$ is an injective morphism of vector bundles;

  2. 2.

For all $1\leq i\leq r$, $({F_{u}})_{p_{i}}=\pi_{k}(x_{i})$;

  3. 3.

$f_{p}\times f_{u}:\mathbb{P}^{1}\rightarrow X_{\widehat{k}}\times{\rm
Gr}(n_{k},n)$ factors through $X\rightarrow X_{\widehat{k}}\times{\rm
Gr}(n_{k},n)$,

where we denote by $({F_{u}})_{p_{i}}$ the fiber of the vector bundle
$F_{u}:=\Phi_{u}(E_{k-1}\oplus_{i}\mathcal{O}_{\mathbb{P}^{1}}(-a_{i}))\subset\mathcal{O}_{\mathbb{P}^{1}}^{\oplus
n}$ at the point $p_{i}\in{\mathbb{P}}^{1}$, and by $f_{u}$ the morphism
associated with the flag of vector bundles
$F_{u}\subset\mathcal{O}_{\mathbb{P}^{1}}^{\oplus n}$ by functorial definition
of the Grassmannian. Note that for $u$ in $V$ the morphism
$\Phi_{u}:E_{k-1}\oplus_{i}\mathcal{O}_{\mathbb{P}^{1}}(-a_{i})\rightarrow\mathcal{O}_{\mathbb{P}^{1}}^{\oplus
n}$ might not be an injective morphism, and the fiber $(F_{u})_{p_{i}}$ might
be of dimension less than $n_{k}$. In order to define a closed subset of $V$,
we define $\mathcal{Z}_{p,x}$ as the subset of elements $u$ in $V$ such that:

$\displaystyle\forall\>i\in\>\\{1,\dots,r\\},\>(F_{u})_{p_{i}}\subset\pi_{{k}}(x_{i}),\>\mathrm{where}\>\pi_{{k}}(x_{i})\in{\rm
Gr}(n_{k},n);$ (1.8)
$\displaystyle\forall\mathfrak{p}\in\mathbb{P}^{1},\>(F_{u})_{\mathfrak{p}}\subset({E_{k+1}})_{\mathfrak{p}},$
(1.9)

where we denote by $\pi_{k}(x_{i})$ the vector subspace of $\mathbb{C}^{n}$ of
dimension $n_{k}$ associated with the point $\pi_{k}(x_{i})$ in ${\rm
Gr}(n_{k},n)$, and by $({E_{k+1}})_{\mathfrak{p}}$ the fiber of the vector
bundle $E_{k+1}$ at $\mathfrak{p}$. Note that, when $\Phi_{u}$ defines a
morphism $\mathbb{P}^{1}\rightarrow{\rm Gr}(n_{k},n)$, Equations (1.8) and
(1.9) are equivalent to Equations $2.$ and $3.$ here above. Indeed, by
construction of $\Phi_{u}$, $(F_{u})_{\mathfrak{p}}$ satisfies
$(E_{k-1})_{\mathfrak{p}}\subset(F_{u})_{\mathfrak{p}}$ for all
${\mathfrak{p}}$ in $\mathbb{P}^{1}$, hence if $u$ satisfies Equation (1.9),
$u$ also verifies:

$\forall{\mathfrak{p}}\in\mathbb{P}^{1},\>(E_{1})_{\mathfrak{p}}\subset\dots\subset(E_{k-1})_{\mathfrak{p}}\subset(F_{u})_{\mathfrak{p}}\subset(E_{k+1})_{\mathfrak{p}}\subset\dots\subset(E_{m})_{\mathfrak{p}}\subset\mathbb{C}^{n},$

hence $\Phi_{u}$ defines a morphism to the flag variety $X$.

####  1.4.4 Irreducibility and rationality of $\mathcal{Z}_{p,x}$.

We denote by $F_{\mu}$ the subsheaf of
${\mathcal{O}}_{{\mathbb{P}}^{1}}^{\oplus r}$ associated with an element $\mu$
in $V\simeq
Hom(E_{k-1}\oplus_{i}{\mathcal{O}}_{{\mathbb{P}}^{1}}(-a_{i}),{\mathcal{O}}_{{\mathbb{P}}^{1}}^{\oplus
r})$, and by $(F_{\mu})_{\mathfrak{p}}$ the fiber over a point
${\mathfrak{p}}\in{\mathbb{P}}^{1}$ of $F_{\mu}$. We consider the usual
construction of $\mathbb{P}^{1}$ by gluing two affine schemes together. Let
$U_{1}:={\rm Spec}(\mathbb{C}[s])$, $U_{2}:={\rm Spec}(\mathbb{C}[t])$,
$U_{12}:={\rm Spec}({\mathbb{C}}[s,s^{-1}])=U_{1}\setminus\\{0\\}$, and
$U_{21}:={\rm Spec}({\mathbb{C}}[t,t^{-1}])=U_{2}\setminus\\{0\\}$. Then
$\mathbb{P}^{1}$ can be obtained by gluing $U_{1}$ and $U_{2}$ together along
$U_{12}\simeq U_{21}$, where we consider the following isomorphism:

$\displaystyle{\mathbb{C}}[s,s^{-1}]\simeq{\mathbb{C}}[t,t^{-1}],\>\>\>s\rightarrow
t^{-1}.$

For $d\in\mathbb{N}$, we view ${\mathcal{O}}_{{\mathbb{P}}^{1}}(-d)$ as the
line bundle satisfying ${\mathcal{O}}_{{\mathbb{P}}^{1}}(-d)_{|U_{i}}\simeq
U_{i}\times\mathbb{A}^{1}$ for $i=1,2$, obtained by identifying
$U_{12}\times{\mathbb{A}}^{1}={\rm Spec}[s,s^{-1}]\times{\mathbb{A}}^{1}$ and
$U_{21}={\rm Spec}[t,t^{-1}]\times{\mathbb{A}}^{1}$ via the following
isomorphism:

$(s,v)\rightarrow(s^{-1},s^{d}v).$

Equivalently we endow ${\mathcal{O}}_{{\mathbb{P}}^{1}}(-d)$ with two
trivializations on $U_{1}$ and $U_{2}$ respectively. We consider here the
natural bijection $V\simeq\oplus_{j}({\mathbb{C}}_{a_{j}}[x])^{n}$,
associating to a section in $({\mathbb{C}}_{a_{j}}[x])^{n}$ the morphism
${\mathcal{O}}_{{\mathbb{P}}^{1}}(-a_{j})\rightarrow{\mathcal{O}}_{{\mathbb{P}}^{1}}^{\oplus
n}$ described by:

$\displaystyle{\rm
Spec}{\mathbb{C}}[s]\times{\mathbb{A}}^{1}\rightarrow{\mathcal{O}}_{{\mathbb{P}}^{1}}^{\oplus
n}$ $\displaystyle(s,v)\rightarrow(s,P_{j}(s)v)$ $\displaystyle{\rm
Spec}{\mathbb{C}}[s^{-1}]\times{\mathbb{A}}^{1}\rightarrow{\mathcal{O}}_{{\mathbb{P}}^{1}}^{\oplus
n}$ $\displaystyle(s^{-1},v)\rightarrow(s^{-1},s^{-a_{j}}P_{j}(s)v)$

We may hence view a point $\mu$ in $\mathcal{Z}_{p,x}$ as a set of
$n_{k}-n_{k-1}$ vectors $P_{j}$ with coefficients in
${\mathbb{C}}_{a_{j}}[x]$. In order to simplify notations, we keep the
dependence of the polynomials $P_{j}$ on the point $\mu$ in $V$ associated
with them implicit.

######  Lemma 1.4.4.1.

  1. 1.

There exist $N\in\mathbb{N}$, $p_{1}$, $\dots$, $p_{N}\in\mathbb{P}^{1}$, such
that $\mathcal{Z}_{p,x}$ is the subspace of elements $\mu$ in $V$ satisfying
the following equations:

$\displaystyle\forall
i\in\\{1,\dots,r\\},\>(F_{\mu})_{p_{i}}\subset\pi_{{k}}(x_{i}),\>\mathrm{where}\>\pi_{{k}}(x_{i})\in{\rm
Gr}(n_{k},n);$
$\displaystyle\forall\iota\in\\{1,\dots,N\\},\>(F_{\mu})_{p_{\iota}}\subset(E_{k+1})_{p_{\iota}};$

  2. 2.

Consider a point $\mathfrak{p}$ in $\mathbb{P}^{1}$. We denote by
$V((E_{k+1})_{\mathfrak{p}})$ the subset of elements $\mu$ in $V$ satisfying
$(E_{k+1})_{\mathfrak{p}}\supset(F_{\mu})_{{\mathfrak{p}}}$. Then
$V((E_{k+1})_{\mathfrak{p}})$ is a linear subspace of $V$;

  3. 3.

$\mathcal{Z}_{p,x}$ is a linear subspace of $V$.

###### Proof.

  1. 1.

By definition, the vectors $P_{j}$ with coefficients in
${\mathbb{C}}_{a_{j}}[x]$ associated with $\mu\in V$ satisfy:

$\forall{\mathfrak{p}}\in{\mathbb{P}}^{1},\>(F_{\mu})_{\mathfrak{p}}\simeq\langle(E_{k-1})_{\mathfrak{p}},(P_{n_{k-1}+1}({\mathfrak{p}}),\dots,P_{n_{k}}({\mathfrak{p}}))\rangle,$

where we denote by $F_{\mu}$ the subvector bundle of
${\mathcal{O}}_{{\mathbb{P}}^{1}}^{\oplus r}$ associated with $\mu$. Denote by
$Q_{j}$ the $n_{k+1}$ homogeneous polynomials associated with the inclusion of
vector bundles $E_{k+1}\subset{\mathcal{O}}_{{\mathbb{P}}^{1}}^{\oplus r}$.
Then there exist $N>0$, and points $p_{1}$, $\dots$, $p_{N}$ in
${\mathbb{P}}^{1}$ such that the following statements are equivalent:

$\displaystyle\forall\>{\mathfrak{p}}\in\mathbb{P}^{1},\>(F_{\mu})_{\mathfrak{p}}\subset({E_{k+1}})_{{\mathfrak{p}}}$
$\displaystyle\Leftrightarrow$
$\displaystyle\forall\>{\mathfrak{p}}\in\mathbb{P}^{1},\>\langle
P_{n_{k-1}+1}({\mathfrak{p}}),\dots,P_{n_{k}}({\mathfrak{p}})\rangle\subset\langle
Q_{1}({\mathfrak{p}}),\dots,Q_{n_{k+1}}({\mathfrak{p}})\rangle$ (1.10)
$\displaystyle\Leftrightarrow$
$\displaystyle\forall\iota\in\\{1,\dots,N\\},\>\langle
P_{n_{k-1}+1}(p_{\iota}),\dots,P_{n_{k}}(p_{\iota})\rangle\subset\langle
Q_{1}(p_{\iota}),\dots,Q_{n_{k+1}}(p_{\iota})\rangle$

Indeed, $\mu$ satisfies the homogeneous equations (1.10) iff the polynomials
$P_{j}$ associated with $\mu$ satisfy (1.10) at $N$ different values of
$p_{\iota}=[x_{\iota},y_{\iota}]$ for $N$ large enough. Notice $N$ only
depends on the degrees $d_{k}$ and $d_{k+1}$, which are fixed here.

From now on, we fix such an $N$, and fix points $(p_{r+1},\dots,p_{N})$
general in $\mathbb{P}^{1}$. Equation (1.9) is thus equivalent to :

$\forall\iota\in\\{1,\dots,N\\},\>(F_{\mu})_{p_{\iota}}\subset({E_{k+1}})_{p_{i}}$
(1.11)

  2. 2.

We set ${\mathfrak{p}}:=[x:y]$. We will denote by $\langle E,F\rangle$ the
vector subspace of ${\mathbb{C}}^{n}$ generated by any two vector subspaces
$E$ and $F$ of ${\mathbb{C}}^{n}$. The vector subspace
$(E_{k+1})_{{\mathfrak{p}}}$ is described by linear equations. Hence:

$\displaystyle V((E_{k+1})_{{\mathfrak{p}}})$
$\displaystyle=\left\\{\mu=(P_{i,j})\in
V\mid\langle(E_{k-1})_{{\mathfrak{p}}},(F_{\mu})_{p}\rangle\subset(E_{k+1})_{{\mathfrak{p}}}\right\\}$
$\displaystyle=\left\\{\mu=(P_{i,j})\in
V\mid\langle(E_{k-1})_{{\mathfrak{p}}},P_{n_{k-1}+1}({\mathfrak{p}}),\dots,P_{n_{k}}({\mathfrak{p}})\rangle\subset(E_{k+1})_{{\mathfrak{p}}}\right\\}$

is the intersection of linear equations on the coefficients of $P_{j}$,
$n_{k-1}+1\leq j\leq n_{k}$.

  3. 3.

Fix $i$ in $\\{1\dots,N\\}$. Denote by $\pi_{\widehat{k}}(x_{i})$ the image of
$x_{i}$ in the projected flag variety $X_{\widehat{k}}$. The subset
$V_{p_{i}}$ of elements $\mu$ in $V$ satisfying
$(F_{\mu})_{p_{i}}\subset\pi_{{k}}(x_{i})$ can be described by:

$\displaystyle V_{p_{i}}:$ $\displaystyle=\left\\{\mu=(P_{i,j})\in
V\mid\langle(F_{\mu})_{p_{i}}\rangle\subset\pi_{k}(x_{i})\right\\}$
$\displaystyle=\left\\{\mu=(P_{i,j})\in V\mid\langle
P_{n_{k-1}+1}(p_{i}),\dots,P_{n_{k}}(p_{i})\rangle\subset\pi_{k}(x_{i})\right\\}.$

Since the vector space $\pi_{k}(x_{i})$ is defined by linear equations,
$V_{p_{i}}$ is defined by linear equations on the coefficients of the
polynomials $(P_{i,j})$; $V_{p_{i}}$ is a linear subspace of $V$.

Finally, according to $1.$ the variety $\mathcal{Z}_{p,x}$ is the intersection
of the linear spaces $V_{p_{i}}$, for $1\leq i\leq r$, with the linear spaces
$V((E_{k+1})_{p_{\iota}})$, for $r<\iota\leq N$. Hence $\mathcal{Z}_{p,x}$ is
a linear subspace of $V$.

∎

According to Lemma 1.4.4.1, $\mathcal{Z}_{p,x}$ is a linear subspace of $V$,
and hence is an irreducible rational variety.

####  1.4.5 Image of
$\mathcal{Z}_{p,x}\dashrightarrow\mathcal{M}_{0,r}(X,\mathbf{d})$.

First, denote by $V^{inj}$ the subset of elements of $V$ corresponding to
injective morphisms
$E_{k-1}\oplus_{i}{\mathcal{O}}_{{\mathbb{P}}^{1}}(-a_{i})\hookrightarrow{\mathcal{O}}_{{\mathbb{P}}^{1}}^{\oplus
r}$. The complementary of $V^{inj}$ is described by a set of polynomial
equations. Indeed, for $u=(P_{j})_{n_{k-1}+1\leq j\leq n_{k}}$ in $V$ the
associated morphism is injective iff
$\langle(E_{k-1})_{\mathfrak{p}},P_{n_{k-1}+1}({\mathfrak{p}}),\dots,P_{n_{k}}({\mathfrak{p}})\rangle$
is a vector space of dimension $n_{k}$ for all
${\mathfrak{p}}\in{\mathbb{P}}^{1}$. Hence $V^{inj}$ is an open subset of $V$.
Call

$\mathcal{Z}_{p,x}^{inj}:=\mathcal{Z}_{p,x}\cap V^{inj}.$

By the functorial definition of flag varieties, each element $u$ in
$\mathcal{Z}_{p,x}^{inj}$ defines a degree ${\mathbf{d}}$ morphism
$f_{u}:{\mathbb{P}}^{1}\rightarrow X$, which by construction of
$\mathcal{Z}_{p,x}$ is associated to a point in $\Pi_{p,x}$. Furthermore, by
construction of $\mathcal{Z}_{p,x}$, any point $q$ in $\Pi_{p,x}$ whose
associated flag of vector bundles is balanced is the image $(F_{u})$ of an
element $u$ in $\mathcal{Z}_{p,x}^{inj}$. Hence, according to Lemma 1.4.2.1
$ii)$, for $(p,x)$ general in $(\Pi_{\widehat{k}}\times
ev)(\overline{{\mathcal{M}}_{0,r}}(X,\mathbf{d}))$, the variety
$\mathcal{Z}_{p,x}^{inj}$ dominates $\Pi_{p,x}$.

####  1.4.6 Algebraicity of $\mathcal{Z}_{p,x}^{inj}\rightarrow\Pi_{p,x}$.

Denote by $\psi:\mathcal{Z}_{p,x}^{inj}\times\mathbb{P}^{1}\rightarrow X$ the
function sending an element $(z,{\mathfrak{p}})$ in
$\mathcal{Z}_{p,x}^{inj}\times\mathbb{P}^{1}$ to the point of $X$ associated
with $(F_{z})_{\mathfrak{p}}$. According to Lemma 1.6.1.1 (Appendix A), $\psi$
defines a morphism
$\Psi:\mathcal{Z}_{p,x}^{inj}\rightarrow\overline{{\mathcal{M}}_{0,r}}(X,\mathbf{d})$,
which dominates $\Pi_{p,x}$ according to here above.

####  1.4.7 Proof of Proposition 1.4.0.1.