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\title[The Shapiro--Shapiro conjecture for ordered positive flags]
{The Shapiro--Shapiro conjecture\\for ordered positive flags}
\author{Qiyue Tang}
\address{Zhili College, Tsinghua University, Beijing 100084, China}
\email{tangqy24@mails.tsinghua.edu.cn}
\date{November 13, 2026}
\subjclass[2020]{Primary 14M15; Secondary 14N15, 14P99, 15B48}
\keywords{Schubert calculus, total positivity, Shapiro--Shapiro conjecture, real Grassmannian}
\begin{document}
\begin{abstract}
We prove the Shapiro--Shapiro conjecture for ordered totally positive
flags in Grassmannians.
\end{abstract}
\maketitle

\section{Introduction}

A Schubert problem asks for subspaces satisfying prescribed incidence
conditions with several complete flags. Over the complex numbers,
general flags and complementary total codimension give finitely many
solutions, counted by the degree of the product of the corresponding
Schubert classes. Over the
real numbers, the solutions may occur in complex-conjugate pairs.
The question of finding real data for which all solutions are real
was emphasized by Fulton in the 1980s; Shapiro and Shapiro recount
its relation to the qualitative theory of differential equations in
\cite{ssode}. The Shapiro--Shapiro conjectures propose explicit
families of such data.

The osculating conjecture, formulated by Boris Shapiro and Michael
Shapiro in the early 1990s, uses the flags of a real rational normal
curve at distinct real parameters \cite{corrigendum,sottile}.
In the polynomial model of a Grassmannian, these incidence conditions
prescribe vanishing orders. The Wronskian is the determinant formed
from a polynomial basis and its successive derivatives; a change of
basis multiplies it by a nonzero scalar. Its roots record the Schubert
divisor conditions \cite{mtv}. Eremenko and Gabrielov proved the case corresponding to
two-dimensional polynomial spaces in 2002 \cite{eg}. Their equivalent
rational-function theorem says that a rational function whose critical
points lie on a circle maps that circle into a circle. This established
total reality for the corresponding Grassmannian problems.

Mukhin, Tarasov, and Varchenko proved the general osculating conjecture
using the Gaudin model and its Bethe algebra \cite{mtvannals}.
Their companion paper identifies the algebra of the relevant Schubert
intersection with a Bethe algebra and proves transversality for
arbitrary Grassmannian Schubert conditions at distinct real parameters
\cite{mtv}. Levinson and Purbhoo later gave a topological proof through
the Wronski map: its degree on suitable real strata is expressed by
symmetric-group characters, and its restriction over polynomials with
distinct real roots is a covering of the expected degree
\cite[Theorem~1.3 and Corollary~1.5]{lp}.

Total positivity led to a second formulation. Shapiro's letter of
26 October 1997 \cite{letter} replaces osculating flags by flags
ordered in an upper-unitriangular chart by the semigroup of strictly
totally positive matrices. Sottile states this formulation as
Conjecture 4.1 in \cite{sottile}. He proves the four-divisor problem
in $\Gr(2,\C^4)$ and gives a reduction from divisor problems to Pieri
problems \cite[Theorems 4.2--4.3]{sottile}. A Pieri condition is indexed
by a one-row partition; the reduction permits two further conditions
at the ends of the ordered tuple. The conjecture itself permits
arbitrary partitions at every flag. The precise row-frame convention
is given in Definition~\ref{def:ordered}.

The matrix formulation is connected with convex curves and
disconjugate differential equations. A fundamental system of solutions
of a linear differential equation gives the coordinate functions of
a nondegenerate projective curve. Zeros of a nonzero solution are the
intersections of this curve with the corresponding hyperplane, counted
with multiplicity. Disconjugacy bounds their number by one less than
the dimension of the solution space; the corresponding convexity
condition bounds each hyperplane intersection by the dimension of the
ambient projective space \cite[Section~2]{ssode}.
Fock and Goncharov characterized their positive flag configurations
by ordered osculating flags of smooth convex curves
\cite[Theorem~1.3]{fg}.

Shapiro and Shapiro described the
boundary of the totally positive triangular semigroup by corner minors
in 1995 \cite{ssboundary}. In their 2010 G\"okova conference article
\cite{ssode}, they formulated a local crossing conjecture for the
osculating flag of a nondegenerate curve and a fixed Schubert boundary.
Goulart and Saldanha developed triangular coordinates for locally
convex curves and proved positive triangular interpolation
\cite[Lemma~5.3 and Remark~5.4]{gs21}. They proved the crossing property, together
with persistence of singular times in continuous families
\cite[Lemmas 3.1--3.2]{gs23}. These results concern the motion of one
flag curve relative to a fixed boundary.

The convex-curve history also contains a corrected counterexample.
Sedykh and Shapiro's 2005 paper \cite{sedykh} asserted that four tangent
lines to a convex curve in real projective three-space could have no
common real transversal. Shapiro and Shapiro's 2022 corrigendum
\cite{corrigendum} identifies an incorrect multiplication order in
the example and proves that every such four-tangent configuration
has two distinct real transversals. The corrected argument uses the
positive-discriminant calculation of Arkani-Hamed, Lam, and Spradlin
\cite[Section~4]{als}.

Eremenko related the divisor secant conjecture to disconjugacy of
polynomial spaces with real-rooted Wronskians \cite{eremenko}.
Recent work also relates total positivity to the solutions themselves.
Karp characterized total nonnegativity of complete polynomial flags by
nonvanishing of their Wronskians on $(0,\infty)$, and total positivity
by nonvanishing on $[0,\infty)$ together with maximal Wronskian degrees
\cite[Theorem~1.1]{karp}. He also formulated positivity refinements of
real Schubert problems \cite{karp}.
Karp and Purbhoo constructed universal Pl\"ucker coordinates in the
symmetric-group algebra, proving these positivity statements,
the disconjugacy conjecture, and the divisor form of the secant
conjecture \cite[Theorems~1.8--1.11]{kp}. Their positive
Shapiro--Shapiro theorem concerns the Pl\"ucker coordinates of
solutions for rational-normal osculating flags
\cite[Corollary~5.30]{kp}. Their discussion
separately formulates the general secant and convex-curve
total-reality questions \cite[Sections 5.7.2--5.7.3]{kp}.

We prove the ordered-positive-flag conjecture for every Grassmannian
and arbitrary Schubert conditions of complementary total codimension.
The conclusion includes transversality, reducedness, and emptiness
when the intersection number is zero.

\subsection{Flags and Schubert conditions}

Fix an integer $N\geq2$. Vectors are rows, and ambient linear transformations act
on the right. The notation $\GL_N(\mathbb K)$ denotes the group of
invertible $N\times N$ matrices over $\mathbb K=\R$ or $\C$.
A row frame $F\in\GL_N(\R)$ represents the flag
$F_\bullet=(F_k)_{k=0}^N$, where $F_k$ is spanned by the first $k$
rows of $F$. Let $U$ be the real upper unitriangular group. Its
strictly totally positive (TP) part $\TP$ consists of the matrices for which
every minor not identically zero on $U$ is positive.
The space of strictly lower triangular matrices is denoted by $\nm$.
We write $I$ for an identity matrix, with its size indicated by a
subscript when needed.

\begin{definition}\label{def:ordered}
An $m$-tuple of real complete flags, for an integer $m\geq1$, is
\emph{ordered positive} if it has
row representatives $F_1,\ldots,F_m$ satisfying
\begin{equation}\label{eq:leftquotient}
 F_{i+1}F_i^{-1}\in\TP\qquad(1\leq i<m).
\end{equation}
\end{definition}

This is the upper-unitriangular flag-chart convention of
\cite{letter,sottile}. A common ambient change of coordinates
$v\mapsto vF_1^{-1}$ replaces the frames by $F_iF_1^{-1}$.
The identity
\[
 (F_{i+1}F_1^{-1})(F_iF_1^{-1})^{-1}=F_{i+1}F_i^{-1}
\]
shows that it preserves all left quotients and makes the first frame
$I$. All the resulting frames then belong to
$U$. We use this normalization when working with matrices.

Fix an integer $1\leq p<N$, put $q=N-p$, and write
$G=\Gr(p,\C^N)$ for the Grassmannian of complex $p$-planes,
of dimension $d=pq$. Real flags are complexified when used in $G$,
and dimensions in the Schubert conditions are complex dimensions.
A partition in the
$p\times q$ rectangle means an integer sequence
$q\geq\lambda_1\geq\cdots\geq\lambda_p\geq0$; write
$|\lambda|=\sum_j\lambda_j$. For a complete flag $F_\bullet$, set
$a_j=q+j-\lambda_j$ and define
\begin{equation}\label{eq:schubert}
 \OO_\lambda(F_\bullet)
 =\{W\in G:\dim(W\cap F_{a_j})\geq j,\ 1\leq j\leq p\}.
\end{equation}
The variety has its reduced determinantal structure. Its open
Schubert cell is
\[
 \OO_\lambda^\circ(F_\bullet)
 =\{W\in G:\dim(W\cap F_k)=\#\{j:a_j\leq k\}
                   \text{ for every }0\leq k\leq N\}.
\]
In the basis given by the rows of the flag frame, row-echelon
coordinates with rightmost pivots in columns $a_j$
give $\sum_j(a_j-j)=d-|\lambda|$ free entries. Thus this cell is
an affine space of that dimension; its closure is \eqref{eq:schubert}.
The boundary is the union of the cells indexed by partitions
$\mu\supsetneq\lambda$, where containment means
$\mu_j\geq\lambda_j$ for every $j$. In particular, each boundary cell has
larger codimension. These cells are the orbits of the stabilizer
of $F_\bullet$, and are smooth. We abbreviate
$\OO_\lambda(F_i)$ when a frame represents the flag.
Let $\sigma_\lambda=[\OO_\lambda(F_\bullet)]$ be its class in the
Chow ring $A^*(G)$; this class is independent of the flag.

\begin{theorem}\label{thm:main}
Let $N\geq2$ and $1\leq p<N$. Let $F_1,\ldots,F_m$ represent
an ordered positive tuple, and let $\lambda_1,\ldots,\lambda_m$
be partitions in the $p\times q$ rectangle. Suppose that
\[
 \sum_{i=1}^m|\lambda_i|=d,\qquad
 D=\int_G\prod_{i=1}^m\sigma_{\lambda_i}.
\]
Then the scheme-theoretic intersection
\[
 \bigcap_{i=1}^m\OO_{\lambda_i}(F_i)
\]
consists of exactly $D$ reduced real points. Each point belongs to
every prescribed open Schubert cell, and the Schubert varieties meet
transversely there. In particular, the intersection is empty if $D=0$.
\end{theorem}

Here the integral is the degree of a zero-dimensional Chow class.
The Schubert classes form an integral basis of $A^*(G)$; in particular,
$A^d(G)=\mathbb Z[\mathrm{point}]$, generated by the full-rectangle
class $\sigma_{(q^p)}$ \cite{fulton}.
A real point is a complex plane obtained by complexifying a real
$p$-plane. Transversality means that the sum map from the individual
conormal spaces to $T_W^*G$ is injective; in the displayed
top-degree problem it is an isomorphism.

\subsection{The argument}

The main step is transversality at every real common point, for
arbitrary Schubert cells at arbitrary total codimension.
A cotangent vector at a real plane $W$ is represented
by an endomorphism $\eta$ with
$\im\eta\subset W\subset\ker\eta$. Consequently any two cotangent
vectors at $W$ have zero product. A dependence among Schubert
conormals therefore gives matrices $\eta_i$ satisfying
\[
 \eta_i\eta_j=0,\qquad \sum_i\eta_i=0,\qquad
 F_i\eta_iF_i^{-1}\in\nm.
\]
We prove that all these matrices must vanish.

For this purpose we interpolate the flags by a positive simple-root
flow and deform its row frame by $I-s\sum_{t_i<t}\eta_i$.
The lower triangular jumps disappear after Gram--Schmidt
orthogonalization, leaving a continuous family of positive Frenet
curves with fixed endpoints. The local Bruhat-boundary persistence
mechanism of Goulart and Saldanha
\cite[Lemma 3.1]{gs23} prevents the family from acquiring a boundary
intersection. We prove the needed finite-dimensional version in
Section~\ref{sec:local-persistence}, using triangular factorization
and the intermediate value theorem. A differential elimination then
forces the conormal dependence to vanish.

The osculating theorem of Mukhin--Tarasov--Varchenko
\cite[Corollary 6.3]{mtv} supplies an initial configuration with the
expected number of real solutions. Proper continuation preserves
these solutions throughout the positive parameter space.
Irreducibility of the universal complex incidence excludes additional
nonreal points or positive-dimensional components. A separate
intersection-theoretic argument treats zero intersection number.

\section{Total positivity and local persistence}
\label{sec:local-persistence}

The corner-minor boundary description is due to Shapiro and Shapiro
\cite{ssboundary}. The local persistence theorem below is the
$C^1$ matrix form of the crossing mechanism in
\cite[Lemmas 3.1--3.2]{gs23}. We prove both statements using
triangular coordinates.
For a matrix $M$ and increasing index sets $I,J$, write
$M_{I,J}$ for the submatrix with those rows and columns.
Empty determinants and empty products equal one.
Write $\kk{k}=\{1,\ldots,k\}$. For a square matrix $M$ of size $n$, set
\[
 \delta_k(M)=\det M_{\{n-k+1,\ldots,n\},\kk{k}}
              \qquad(0\leq k\leq n).
\]
Thus $\delta_k$ is the southwest corner minor.
We use column frames in
$\SO_N=\{Q\in\GL_N(\R):Q^{\mathsf T}Q=I,\ \det Q=1\}$.
A
\emph{positive Frenet curve} is a $C^1$ curve satisfying
\begin{equation}\label{eq:positiveframe}
 Q^{-1}Q'=\sum_{j=1}^{N-1}\kappa_j(t)
             (E_{j+1,j}-E_{j,j+1}),\qquad \kappa_j(t)>0,
\end{equation}
where the coefficients are continuous and $E_{ij}$ is the matrix
with its only nonzero entry, equal to one, in position $(i,j)$.
The boundary under consideration is
\[
 \Sigma=\{Q\in\SO_N:\delta_k(Q)=0
                         \text{ for at least one }1\leq k<N\}.
\]

Let $U_-$ be the lower-unitriangular group and put
$\mathcal P=(\TP)^{\mathsf T}\subset U_-$. For increasing index sets
$I=\{i_1<\cdots<i_k\}$ and $J=\{j_1<\cdots<j_k\}$ in
$\{1,\ldots,N\}$, write $I\geq J$
if $i_r\geq j_r$ for every $r$, and call such a pair \emph{allowable}.
Other minors vanish identically on $U_-$. A matrix is \emph{totally
nonnegative} (TNN) if all its minors are nonnegative. Let
$\mathcal C\subset U_-$ consist of such matrices. Thus
$\mathcal P$ is defined by strict positivity of the allowable minors.

\begin{lemma}[Positive transitions]\label{lem:app-positive-transitions}
One has $\mathcal C\mathcal P\cup\mathcal P\mathcal C\subset\mathcal P$
and $\overline{\mathcal P}=\mathcal C$.
Let $L:I\to U_-$ be a $C^1$ curve on a real interval satisfying
\begin{equation}\label{eq:app-lower-flow}
 L'=LB(t),\qquad B(t)=\sum_{j=1}^{N-1}\beta_j(t)E_{j+1,j},
 \qquad \beta_j(t)>0
\end{equation}
with continuous coefficients. Then $L(u)^{-1}L(v)\in\mathcal P$ for
every $u,v\in I$ with $u<v$.
\end{lemma}
\begin{proof}
Cauchy--Binet gives nonnegative summands for every minor of a product
of TNN matrices. For $AC$ with $A\in\mathcal C$, $C\in\mathcal P$,
the summand with intermediate index $I$ in its $I,J$ minor is
$\det A_{I,I}\det C_{I,J}=\det C_{I,J}>0$. Use intermediate index $J$
for $CA$.

For the transition equation put $T(t)=L(u)^{-1}L(t)$, so
$T'=TB$ and $T(u)=I$. On the $k$th exterior power, the matrix
$T^{(k)}=(\det T_{I,J})_{I,J}$ satisfies
$(T^{(k)})'=T^{(k)}B^{[k]}$. The entry of $B^{[k]}$ from $J$ to $I$
is $\beta_j(t)$ if $I$ is obtained from $J$ by replacing $j$ by $j+1$,
with $j\in J$ and $j+1\notin J$, and is zero otherwise.
The sign is positive because the replaced index retains its position
in the increasing list. Successive integration gives the finite formula
\begin{equation}\label{eq:compound-integral}
 T^{(k)}(v)=I+
 \sum_{\ell=1}^{k(N-k)}
 \int_{u<t_1<\cdots<t_\ell<v}
 B^{[k]}(t_1)\cdots B^{[k]}(t_\ell)\,dt_1\cdots dt_\ell.
\end{equation}
Each factor raises the sum of the indices by one, so longer products
vanish. Every summand has nonnegative entries. If $I\geq J$, move
the largest index of $J$ to its target in $I$, then the next largest,
and continue downwards. This gives a permitted chain from $J$ to $I$.
In the corresponding term of \eqref{eq:compound-integral}, its steps
occur from the rightmost factor to the leftmost factor. Its integrand
is a product of strictly positive coefficients, so its integral is
positive. For $I=J$, the identity term gives the value one.
Finally, with $H=\sum_jE_{j+1,j}$, the transition result gives
$e^{\epsilon H}\in\mathcal P$ for $\epsilon>0$. For $A\in\mathcal C$,
the matrices $Ae^{\epsilon H}\in\mathcal P$ converge to $A$.
The reverse closure inclusion follows from continuity of minors.
\end{proof}

\begin{lemma}[Triangular coordinates and the boundary]
\label{lem:app-sw-boundary}
The factorization \eqref{eq:app-factorization} gives smooth
coordinates $(a_{r,j})_{1\leq j\leq r<N}\in\R_{>0}^{N(N-1)/2}$ on $\mathcal P$, with
smooth inverse. If $L\in\mathcal C\setminus\mathcal P$, then
$\delta_k(L)=0$ for some $1\leq k<N$.
\end{lemma}
\begin{proof}
Set $u_j(x)=I+xE_{j+1,j}$. Its minors are $0$, $1$, or $x$, so it is
TNN for $x\geq0$. We first prove the factorization
\begin{equation}\label{eq:app-factorization}
 \begin{aligned}
 L&=C_1\cdots C_{N-1},\qquad a_{r,j}>0,\\
 C_r&=u_r(a_{r,r})\cdots u_1(a_{r,1})
 \end{aligned}
\end{equation}
for $L\in\mathcal P$. The induction starts at size one, where
$L=(1)$ and the factor list is empty.
In fact, positivity of the allowable minors with
consecutive columns suffices. Put $b_j=L_{N,j}$, so $b_N=1$, and take
$a_{N-1,j}=b_j/b_{j+1}>0$. For $r>c$, multiplication of the elementary
factors gives
$(C_{N-1})_{r,c}=\prod_{h=c}^{r-1}a_{N-1,h}$.
The inverse factors occur in increasing index order; all products
of two or more of their off-diagonal terms vanish. Hence
\[
 C_{N-1}^{-1}=I-\sum_{j=1}^{N-1}a_{N-1,j}E_{j+1,j},
 \qquad LC_{N-1}^{-1}=\operatorname{diag}(K,1).
\]
For $I\subset\{1,\ldots,N-1\}$ of size $\ell$ and
$J=\{j,\ldots,j+\ell-1\}\subset\{1,\ldots,N-1\}$,
the exact identity is
\begin{equation}\label{eq:app-recursive-minor}
 \det K_{I,J}
 =\frac{\det L_{I\cup\{N\},\{j,\ldots,j+\ell\}}}{b_{j+\ell}}.
\end{equation}
Indeed, simultaneously subtract $(b_h/b_{h+1})$ times original column
$h+1$ from column $h$, for $j\leq h<j+\ell$, in the numerator matrix.
This determinant-one column operation leaves only $b_{j+\ell}$ in its
last row. Expansion along that row proves the identity. An allowable
minor on the left has an allowable numerator on the right, so $K$
has the required consecutive-column positivity. Induction gives
\eqref{eq:app-factorization}; the bottom-row ratios also give uniqueness. The recursion expresses
the inverse coordinates by rational functions with positive
denominators. The product map is polynomial, so both maps are smooth.

Conversely, a product in \eqref{eq:app-factorization} belongs to
$\mathcal P$. Induct on $N$ and use
$L=\operatorname{diag}(K,1)C_{N-1}$. In an allowable $I,J$ minor,
if $N\notin I$, allowability implies $N\notin J$, and the
Cauchy--Binet term with intermediate index $J$ equals
$\det K_{I,J}>0$. If $N\in I$ and $J=\{j_1<\cdots<j_k\}$, use
$H=\{j_1,\ldots,j_{k-1},N\}$. The first factor is the allowable minor
$\det K_{I\setminus\{N\},\{j_1,\ldots,j_{k-1}\}}>0$.
In the second factor, the first $k-1$ rows form a lower triangular
unit-diagonal block and vanish in the last column. Thus
$\det(C_{N-1})_{H,J}=(C_{N-1})_{N,j_k}>0$.
All other terms are nonnegative.

The corner minors in these coordinates are the monomials
\begin{equation}\label{eq:app-corner-monomials}
 \delta_k(L)=
 \prod_{r=N-k}^{N-1}\ \prod_{j=r-N+k+1}^{r}a_{r,j},
 \qquad 1\leq k<N.
\end{equation}
To see this, the first $k$ entries of the rows of $C_{N-1}$ with index
at least $k$ are proportional. In the Cauchy--Binet expansion for
$\delta_k(\operatorname{diag}(K,1)C_{N-1})$, intermediate indices
must contain $N$; the only nonzero term therefore uses
$\{1,\ldots,k-1,N\}$. Thus, with $\delta_0=1$,
\[
 \delta_k(L)=\delta_{k-1}(K)
             \prod_{j=k}^{N-1}a_{N-1,j}.
\]
Iteration proves \eqref{eq:app-corner-monomials}, which also holds for
nonnegative parameters by polynomial continuity.

Now approximate $L\in\mathcal C$ by $L_\nu\in\mathcal P$. Their
factor parameters are bounded, since
\[
 (L_\nu)_{i+1,i}=\sum_{r=i}^{N-1}a^{(\nu)}_{r,i}.
\]
A subsequence of all parameters converges to nonnegative parameters
whose product is $L$. If every $\delta_k(L)$ were nonzero, all would
be positive. Formula \eqref{eq:app-corner-monomials} would then make
every limiting parameter positive: the entire block
$a_{r,1},\ldots,a_{r,r}$ occurs in $\delta_{N-r}$. The converse
factorization assertion would give $L\in\mathcal P$, a contradiction.
\end{proof}

For a matrix $M\in\GL_N^+(\R):=\{M\in\GL_N(\R):\det M>0\}$,
column Gram--Schmidt gives the unique decomposition $M=QR$, where
$Q\in\SO_N$ and $R$ is upper triangular with positive diagonal.
Write $\operatorname{QR}_{\SO}(M)=Q$.

\begin{lemma}[Positive coordinates and normalization]
\label{lem:app-normalization}
Every $Q_*\in\SO_N$ admits a fixed upper-triangular
$A\in\GL_N^+(\R)$ such that the positive-diagonal QR action
$\Phi_A(Q)=\operatorname{QR}_{\SO}(AQ)$ preserves $\Sigma$ and positive
Frenet curves. Moreover, $\Phi_A(Q_*)$ lies in the chart with positive
leading principal minors, and its lower unitriangular LU factor
belongs to $\mathcal C$.
\end{lemma}
\begin{proof}
First write $AQ=\widetilde Q R$, where $R$ is upper triangular with
positive diagonal. If
$Q'=Q\sum_j\kappa_j(E_{j+1,j}-E_{j,j+1})=QK$, differentiation gives
\begin{equation}\label{eq:app-qr-ratios}
 \widetilde Q^{\mathsf T}\widetilde Q'
 =RKR^{-1}-R'R^{-1},\qquad
 \widetilde\kappa_j=
 \frac{R_{j+1,j+1}}{R_{jj}}\kappa_j>0.
\end{equation}
Below the diagonal only the stated first-subdiagonal entries occur;
skew symmetry supplies the upper part. For upper-triangular $A$,
\[
 \delta_k(\widetilde Q)=\delta_k(Q)
   \prod_{i=N-k+1}^{N}A_{ii}\prod_{j=1}^{k}R_{jj}^{-1}.
\]
Every diagonal factor in this identity is nonzero, so $\Phi_A$
preserves $\Sigma$. It is a smooth diffeomorphism with
inverse $\Phi_{A^{-1}}$.

On the open chart where the northwest leading minors of $Q$ are
positive, write $Q=LV$, with $L\in U_-$ and $V$ upper triangular with
positive diagonal. The nonzero leading minors give the unique LU factorization,
and their positive successive ratios make the diagonal of $V$ positive.
This chart identifies with all of $U_-$; its inverse
takes the orthogonal QR factor of $L$. Differentiation and triangularity
give
\begin{equation}\label{eq:app-lu-ratios}
 \begin{aligned}
 L^{-1}L'&=VKV^{-1}-V'V^{-1}=\sum_j\beta_jE_{j+1,j},\\
 \beta_j&=\frac{V_{j+1,j+1}}{V_{jj}}\kappa_j>0,
 \qquad \delta_k(Q)=\delta_k(L)\prod_{j=1}^{k}V_{jj}.
 \end{aligned}
\end{equation}

It remains to normalize $Q_*$. Gaussian elimination assigns to every
invertible matrix $M$ a permutation $w(M)\in S_N$, the symmetric
group on $\{1,\ldots,N\}$: its suffix-row,
prefix-column ranks are
\begin{equation}\label{eq:app-suffix-ranks}
 \operatorname{rank}M_{\{r,\ldots,N\},\{1,\ldots,c\}}
 =\#\{j\leq c:w(M)(j)\geq r\},\qquad 1\leq r,c\leq N.
\end{equation}
Let $w=w(Q_*)$ and $s_i=(i,i+1)$. Choose a reduced word
$w=s_{i_1}\cdots s_{i_\ell}$, that is, an expression with the fewest
adjacent transpositions, and set
$L_w=u_{i_1}(1)\cdots u_{i_\ell}(1)\in\mathcal C$.
We check that its permutation is $w$. Multiplication on the right by
$u_i(x)$, $x\ne0$, adds $x$ times column $i+1$ to column $i$;
all ranks in \eqref{eq:app-suffix-ranks} with $c\ne i$ are unchanged.
If the current permutation $\tau$ satisfies $\tau(i)=a<b=\tau(i+1)$,
restrict to rows $r,\ldots,N$ and let $u,v$
be the classes of columns $i,i+1$ modulo the first $i-1$ columns. The
rank formula gives
\[
 \dim\langle u\rangle=\mathbf 1_{a\geq r},\qquad
 \dim\langle u,v\rangle=\mathbf 1_{a\geq r}+\mathbf 1_{b\geq r},
\]
where $\mathbf 1$ is the indicator of the stated inequality.
For $r\leq a$ the classes are independent; for $a<r\leq b$ one has
$u=0$ and $v\ne0$; for $r>b$ both vanish. Therefore the rank with
$c=i$ increases exactly
for $a<r\leq b$. This is precisely the rank pattern of $\tau s_i$.
Every step of a reduced word increases length, so induction proves the
claim.

To obtain the required triangular factors, apply signed elimination.
Choose the lowest nonzero entry in the leftmost remaining column $c$,
and denote its row by $r_*$ and its value by $p$.
Subtract $M_{r,c}/p$ times its row from each
remaining row $r$ above it, and then subtract $M_{r_*,c'}/p$ times
the pivot column from each remaining column $c'$ to its right.
These are upper triangular row and column operations. They preserve
the suffix-row, prefix-column ranks: an upper triangular row operation
acts invertibly on every suffix of rows, and an upper triangular
column operation acts invertibly on every prefix of columns.
Delete the pivot row and column and repeat in the inherited order.
Earlier pivot rows and columns remain clear, since the active rows
and columns have zero entries in those positions.
All these elementary matrices have diagonal one and positive-diagonal
inverses.
Positive diagonal rescalings normalize the absolute values of the
pivots, leaving their signs in a signed permutation matrix.
For a determinant-one matrix this signed permutation has determinant
one. Applied to $Q_*$ and $L_w$, the procedure gives
\[
 Q_*=U_0PU_1,\qquad L_w=V_0P'V_1,
\]
with upper-triangular positive-diagonal factors and determinant-one
signed permutation matrices $P,P'$ having the same underlying
permutation. Thus $P'P^{-1}$ is diagonal and
\[
 A=V_0P'P^{-1}U_0^{-1}\in\GL_N^+(\R),\qquad
 AQ_*=L_w(V_1^{-1}U_1).
\]
The matrix $A$ is upper triangular and the last factor has positive
diagonal. Therefore $AQ_*$ and $L_w$ have the same orthogonal QR
factor, whose triangular coordinate is $L_w$, as required.
\end{proof}

\begin{theorem}[Local persistence]\label{thm:app-local-persistence}
Let $Q:I\to\SO_N$ be a $C^1$ positive Frenet curve on a real interval,
with continuous
strictly positive coefficients. If $Q(t_*)\in\Sigma$ at an interior
time, then every sufficiently uniformly close continuous
$\SO_N$-valued path on $I$ has a hit of $\Sigma$ in any prescribed
neighborhood of $t_*$.
\end{theorem}
\begin{proof}
Apply Lemma~\ref{lem:app-normalization} at $Q(t_*)$ and use a short
closed interval $[t_-,t_+]$ around $t_*$ contained in its triangular
chart. The normalized hit satisfies
$L_*=L(t_*)\in\mathcal C\setminus\mathcal P$, since both coordinate
changes preserve the vanishing of corner minors. By
\eqref{eq:app-lu-ratios} and Lemma~\ref{lem:app-positive-transitions},
\[
 L(t_+)\in\mathcal P,\qquad L(t_-)\notin\mathcal C.
\]
Indeed, the second assertion follows because otherwise
$L_*=L(t_-)\bigl(L(t_-)^{-1}L_*\bigr)$ would belong to $\mathcal P$.

Define the continuous function
\[
 g(L)=\min_{\substack{1\leq k\leq N\\ |I|=|J|=k,\ I\geq J}}
          \det L_{I,J}.
\]
Then $g(L(t_-))<0<g(L(t_+))$. A sufficiently close continuous path
stays in the same chart and retains these strict endpoint inequalities.
It has a point with $g=0$, hence a point in
$\mathcal C\setminus\mathcal P$; Lemma~\ref{lem:app-sw-boundary}
forces a zero corner minor there. The coordinate identities carry this
hit back to $\Sigma$. The times $t_-,t_+$ can be chosen arbitrarily
close to $t_*$.
\end{proof}

\begin{corollary}[Connected-family conclusion]
\label{cor:app-connected-family}
Let $S$ be a real interval, $a<b$, and let
$Q:S\times[a,b]\to\SO_N$ be continuous. Write $Q_s(t)=Q(s,t)$, with
each time curve as in Theorem~\ref{thm:app-local-persistence}. Suppose
$Q_s(a),Q_s(b)\notin\Sigma$ for every $s$. If one curve avoids $\Sigma$
on $[a,b]$, then every curve does.
\end{corollary}
\begin{proof}
The set of parameters having a hit is closed because $\Sigma$ is closed
and $[a,b]$ is compact. All hits are interior, so
Theorem~\ref{thm:app-local-persistence} makes this parameter set open
as well. Connectedness proves the assertion.
\end{proof}

\section{Positive interpolation and conormal relations}
\label{sec:stress}
The following version of positive triangular interpolation
\cite[Lemma~5.3 and Remark~5.4]{gs21} has constant controls near the
prescribed times.

\begin{lemma}[Exact positive interpolation]\label{lem:interpolate}
After the normalization in Section 1, the matrices $F_i$ can be interpolated at
distinct interior times $t_1<\cdots<t_m$ by a smooth curve
$F:[a,b]\to U$ satisfying
\begin{equation}\label{eq:frenetupper}
 F'=XF,\qquad X(t)=\sum_{j=1}^{N-1}w_j(t)E_{j,j+1},\qquad w_j(t)>0.
\end{equation}
The controls can be constant on collars of the interpolation times.
\end{lemma}

\begin{proof}
Put $J=\sum_{j=1}^{N-1}E_{j,j+1}$.
If $m=1$, the curve $F(t)=e^{(t-t_1)J}F_1$ has the required properties.
Assume $m\geq2$.
Transposing the coordinates in
Lemma~\ref{lem:app-sw-boundary} gives a factorization of each element
of $\TP$ as
\[
 T(a)=\prod_{\nu=1}^{\ell}(I+a_\nu E_{j_\nu,j_\nu+1}),
 \qquad a_\nu>0,\qquad \ell=N(N-1)/2,
\]
where the sequence of indices $j_\nu$ is fixed and the product is
written from left to right in increasing $\nu$. This is a smooth
coordinate map with smooth inverse.
On a given interval $[u,v]$, choose smooth nonnegative functions
$\rho_\nu$ of integral one, with pairwise disjoint supports in its
interior, ordered chronologically as $\rho_\ell,\ldots,\rho_1$.
Set
\[
 w_j(t)=\epsilon+\sum_{\nu:j_\nu=j}a_\nu\rho_\nu(t).
\]
Let $H'=XH$, $H(u)=I$, and denote its endpoint $H(v)$ by
$\mathcal E(a,\epsilon)$. At $\epsilon=0$, each pulse multiplies
the current matrix on the left by
$e^{a_\nu E_{j_\nu,j_\nu+1}}=I+a_\nu E_{j_\nu,j_\nu+1}$.
Thus $\mathcal E(a,0)=T(a)$. The endpoint depends smoothly on
$(a,\epsilon)$: successive integration gives finite iterated integrals,
since all coefficient matrices are strictly upper triangular.
The derivative with respect to the $\ell$ weights is invertible at
$(a,0)$ in the strictly upper triangular coordinates of $U$.
The implicit function theorem therefore supplies weights $a(\epsilon)$
with
$\mathcal E(a(\epsilon),\epsilon)=T(a)$ for all sufficiently small
$\epsilon>0$. They remain positive by continuity, so every $w_j$ is
strictly positive. Near both endpoints all controls equal $\epsilon$.

Apply this construction to each $F_{i+1}F_i^{-1}$ and write the path on
that interval as $H(t)F_i$. A single sufficiently small background works
on the finitely many intervals. On a collar of each $t_i$, both
adjacent paths equal $e^{\epsilon(t-t_i)J}F_i$, so they glue smoothly
with all derivatives.
Extend by the same positive controls before the first and after the last
specified time. The resulting matrices remain upper unitriangular.

\end{proof}

\begin{lemma}[Vanishing of a closed conormal relation]\label{lem:stress}
Let $F:[a,b]\longrightarrow U$ be a smooth curve satisfying
\eqref{eq:frenetupper}, and let
$a<t_1<\cdots<t_m<b$. Suppose real $N\times N$ matrices $\eta_i$ satisfy
\begin{equation}\label{eq:stress}
 \eta_i\eta_j=0\quad\hbox{for all }i,j,\qquad
 \sum_i\eta_i=0,\qquad
 \Delta_i:=F(t_i)\eta_iF(t_i)^{-1}\in\nm.
\end{equation}
Then $\eta_i=0$ for every $i$.
\end{lemma}

\begin{proof}
\emph{A family with fixed endpoints.}
On the intervals between the times $t_i$, define the piecewise smooth
matrices below, using their one-sided limits at each $t_i$:
\begin{equation}\label{eq:homotopy}
 S(t)=\sum_{t_i<t}\eta_i,\qquad
 Z(t)=F(t)S(t)F(t)^{-1},\qquad
 M_s(t)=F(t)(I-sS(t)),\quad s\in\R.
\end{equation}
All products in \eqref{eq:stress} vanish, so $S^2=0$ and
$(I-sS)^{-1}=I+sS$ for every real $s$. All eigenvalues of $S$ are
zero, so $\det(I-sS)=1$ and $\det M_s=1$.
Away from the times $t_i$, $M_s'=XM_s$. At such a time,
\begin{align}\label{eq:jump}
 M_s(t_i+)
 &=F(t_i)\bigl(I-s(S(t_i-)+\eta_i)\bigr)\notag\\
 &=(I-s\Delta_i)M_s(t_i-).
\end{align}
Indeed,
$(I-s\eta_i)(I-sS(t_i-))=I-s(\eta_i+S(t_i-))$
because $\eta_iS(t_i-)=0$, which proves the identity for every $s$.
The jump factor is lower unitriangular. It preserves the oriented flag
of initial row spans. Also, $S=0$ on collars of both endpoints.

\emph{Preservation of the positive Frenet equation.}
Use row Gram--Schmidt with positive diagonal:
\[
 M_s=B_sQ_s,\qquad B_s\text{ lower triangular with positive diagonal},
 \qquad Q_s\in\SO_N.
\]
At a time $t_i$, uniqueness of this factorization gives
\begin{equation}\label{eq:gsjump}
 Q_s(t_i+)=Q_s(t_i-),\qquad
 B_s(t_i+)=(I-s\Delta_i)B_s(t_i-).
\end{equation}
In particular the diagonal entries of $B_s$ agree on both sides.
Between these times,
\begin{equation}\label{eq:qrderiv}
 Q_s'Q_s^{-1}=B_s^{-1}XB_s-B_s^{-1}B_s'.
\end{equation}
The first term has no entries above the first superdiagonal: its
$(i,j)$ entry vanishes when
$j>i+1$, since a contributing index $r$ must satisfy $r\leq i$ and
$j\leq r+1$. At $(j,j+1)$ the only possible index is $r=j$, giving
\[
 (B_s^{-1}XB_s)_{j,j+1}
 =w_j\frac{(B_s)_{j+1,j+1}}{(B_s)_{jj}}=:\kappa_j(s,t)>0.
\]
The second term in \eqref{eq:qrderiv} is lower triangular. Its left side
is skew symmetric, so it is exactly the skew Jacobi matrix with these
positive superdiagonal entries. Consequently
$\Gamma_s=Q_s^{\mathsf T}$ satisfies \eqref{eq:positiveframe}.

On each of the finitely many closed time pieces, Gram--Schmidt
depends smoothly on $(s,t)$. Define $Q_s(t_i)$ by its common
one-sided limit. The values match at each $t_i$, so
$(s,t)\mapsto Q_s(t)$ is continuous on $\R\times[a,b]$.
The coefficients $\kappa_j$ match by \eqref{eq:gsjump} and continuity
of $w_j$. Therefore the skew matrices $Q_s'Q_s^{-1}$ have matching
one-sided limits. Since $Q_s$ itself matches, its one-sided
derivatives agree. Each time curve is thus $C^1$ with continuous
positive Frenet coefficients. The endpoint frames are fixed because $S$ vanishes on
endpoint collars.

\emph{Nonvanishing of leading minors.}
Let $P\in\SO_N$ be the signed reversal
\begin{equation}\label{eq:reversal}
 P_{i,N+1-i}=(-1)^{N-i},\qquad P_{ij}=0\text{ otherwise}.
\end{equation}
The row-reversal sign cancels the product of these signs, so for every
matrix $A$,
\[
 \delta_k(PA)=\det A_{\kk{k},\kk{k}}.
\]
Left multiplication by the constant $P$ preserves the logarithmic
derivative of $\Gamma_s$. Moreover,
\begin{equation}\label{eq:minors}
 \delta_k(P\Gamma_s)
 =\det (Q_s)_{\kk{k},\kk{k}}
 =\frac{\det (M_s)_{\kk{k},\kk{k}}}
 {\prod_{j=1}^k(B_s)_{jj}}.
\end{equation}
At $s=0$, $M_0=F$ is upper unitriangular, so every numerator in
\eqref{eq:minors} is one. Both endpoints have the same property for
all $s$. Corollary~\ref{cor:app-connected-family}, applied on each compact parameter
interval containing $0$, implies that every leading minor of $M_s(t)$ is nonzero for
every finite real $s$ and every $t$ between the times $t_i$,
including the one-sided limits there. At $t_i$ the minors on the two
sides agree, since its jump factor is lower unitriangular.

Since $M_s=(I-sZ)F$ and $F$ is upper unitriangular,
\[
 (M_s)_{\kk{k},\kk{k}}
 =(I_k-sZ_{\kk{k},\kk{k}})F_{\kk{k},\kk{k}}.
\]
The second factor has determinant one, so
\begin{equation}\label{eq:pencil}
 \det\bigl(I_k-s Z(t)_{\kk{k},\kk{k}}\bigr)\ne0
 \qquad(s\in\R,\ 1\leq k\leq N).
\end{equation}
The case $k=N$ also follows directly from $S^2=0$.

\emph{Triangular elimination.}
Between consecutive times $t_i$, $Z'=[X,Z]=XZ-ZX$, or entrywise,
\begin{equation}\label{eq:entryode}
 Z_{ij}'=w_i Z_{i+1,j}-w_{j-1}Z_{i,j-1},
\end{equation}
where the first term is omitted for $i=N$ and the second for $j=1$.
Equation~\eqref{eq:pencil} for $k=1$
says $1-sZ_{11}\ne0$ for every real $s$, so $Z_{11}=0$ identically on
the interval. Differentiating $Z_{11}=0$ gives $w_1Z_{21}=0$,
and hence $Z_{21}=0$. Repeating the same argument down the first
column gives $Z_{21}=\cdots=Z_{N1}=0$.

For $2\leq k\leq N$, suppose the first $k-1$ columns vanish on and below
the diagonal. The leading $k\times k$ block is upper triangular with
diagonal $0,\ldots,0,Z_{kk}$. Thus the determinant in
\eqref{eq:pencil} is $1-sZ_{kk}$, and $Z_{kk}=0$.
For $i\geq k$, the entry $Z_{i,k-1}$ is zero by induction.
Differentiating $Z_{kk}=0$ in \eqref{eq:entryode} therefore gives
$w_kZ_{k+1,k}=0$ when $k<N$. Once $Z_{ik}=0$ is known,
the same equation gives $w_iZ_{i+1,k}=0$ for $i<N$.
All $w_i$ are positive, so the entire $k$th column vanishes on and
below the diagonal. Induction makes $Z$ strictly upper triangular on
every interval. Its one-sided limits at the times $t_i$ have the
same property. Their difference is $\Delta_i$, which by assumption is
strictly lower triangular. Hence $\Delta_i=0$, and therefore $\eta_i=0$.
\end{proof}

\section{Real Schubert transversality}\label{sec:realtransversality}

\begin{proposition}\label{prop:realtransverse}
For any ordered positive tuple and any Schubert-cell labels, the real
cells meet simultaneously transversely at every real common point.
This assertion allows arbitrary codimension sums.
\end{proposition}

\begin{proof}
For a real row $p$-plane $W\subset V=\R^N$, the tangent space of
$\Gr(p,V)$ is $\operatorname{Hom}(W,V/W)$. Its dual is
$\operatorname{Hom}(V/W,W)$, paired by the trace of the composition.
Thus the cotangent space is represented by matrices
\begin{equation}\label{eq:cotangent}
 \eta\in\operatorname{End}(V),\qquad
 \im\eta\subset W\subset\ker\eta.
\end{equation}
Here image and kernel refer to the right action on row vectors.
An ambient infinitesimal right
action $A$ is paired with $\eta$ by $\Tr(\eta A)$. Any two matrices
in \eqref{eq:cotangent} have zero product in either order.

The stabilizer of the standard initial-row flag for the right action is
the invertible lower triangular group. Its orbits in the Grassmannian
are the Schubert cells, and the orbit tangent space is the image of its
Lie algebra under the infinitesimal action. A cotangent vector is
conormal precisely when it annihilates every lower triangular
infinitesimal action $Y$. Since
\[
 \Tr(\eta Y)=\sum_{i,j}\eta_{ij}Y_{ji},
\]
this annihilator consists of strictly lower triangular matrices. For
the flag represented by $F_i$, use the coordinate change $v\mapsto
vF_i^{-1}$. The cotangent representative becomes
$F_i\eta F_i^{-1}$. Thus the conormal space to its Schubert cell is
\[
 \{\eta:\im\eta\subset W\subset\ker\eta,
             \ F_i\eta F_i^{-1}\in\nm\}.
\]

If simultaneous transversality failed at $W$, the sum map from the
direct sum of the cell conormal spaces into $T_W^*\Gr(p,V)$ would have
a nonzero kernel. It would supply matrices $\eta_i$, not all zero,
with $\sum_i\eta_i=0$. They satisfy every hypothesis of
Lemma~\ref{lem:stress}, after exact interpolation by
Lemma~\ref{lem:interpolate}. This is impossible.
\end{proof}

\begin{corollary}\label{cor:boundary}
If the sum of cell codimensions exceeds $d$, there is no real common
point. In a closed Schubert intersection of total prescribed codimension
$d$, every real point belongs to every prescribed open cell. At such a
point the complex scheme-theoretic intersection is a reduced point.
\end{corollary}

\begin{proof}
Independent conormal spaces have total dimension at most $d$. At a
boundary point, at least one actual cell has codimension strictly larger
than the prescribed one. This would make the total exceed $d$.
Otherwise all varieties are smooth there. Their real conormal spaces
complexify to the corresponding complex conormal spaces, so the complex
intersection is transverse of dimension zero, with local ring $\C$.
\end{proof}

\section{The universal incidence and exhaustion of the complex fiber}

The universal complex incidence will show that $D$ transverse real
points exhaust a fiber when $D>0$.

Let $\Fl(\C^N)$ denote the variety of complete flags. Set
$B=\Fl(\C^N)^m$ and, initially with its reduced structure, let
\[
 \mathcal X=\{(W,E_1,\ldots,E_m)\in G\times B:
                  W\in\OO_{\lambda_i}(E_i)\ \forall i\}.
\]
Write $\pi:\mathcal X\to B$ for the projection.

\begin{lemma}\label{lem:universal}
The variety $\mathcal X$ is irreducible and proper over $B$, and
\[
 \dim\mathcal X=d+m\dim\Fl(\C^N)-\sum_i|\lambda_i|.
\]
For top-degree data with $D>0$, $\pi$ is dominant and generically finite
of degree $D$.
\end{lemma}

\begin{proof}
Let $H=\GL_N(\C)$ act on row subspaces on the right.
Fix a flag $E$ and a $p$-plane $W_0$, and denote their stabilizers by
$Q$ and $P$. These groups are irreducible: in adapted bases they are
triangular or block triangular groups with invertible diagonal blocks.
Put
\[
 Z_i=\{E':W_0\in\OO_{\lambda_i}(E')\}.
\]
The map $a:H\to G$, $a(g)=W_0g^{-1}$, is a Zariski locally trivial
bundle whose fiber through $g_0\in H$ is $g_0P$, isomorphic to $P$.
For local sections,
choose a basis adapted to $W_0$. On a Grassmannian graph chart the
map $g\mapsto W_0g$ has the section
\[
 \operatorname{rowspan}(I_p\ Z)\longmapsto
 \begin{pmatrix}I_p&Z\\0&I_q\end{pmatrix}.
\]
Inverting this matrix gives a section of $a$; the same construction
applies on the other graph charts.
Consequently $a^{-1}(\OO_{\lambda_i}(E))$ is irreducible of dimension
$\dim H-|\lambda_i|$. Indeed,
$W_0g^{-1}\in\OO_{\lambda_i}(E)$ is equivalent to
$W_0\in\OO_{\lambda_i}(Eg)$.
Its image under $g\mapsto Eg$ is $Z_i$, and the
fiber over $Eg_0\in Z_i$ is $Qg_0$, isomorphic to $Q$.
Hence $Z_i$ is irreducible
of dimension $\dim\Fl(\C^N)-|\lambda_i|$.

The map
\[
 H\times\prod_i Z_i\longrightarrow\mathcal X,\qquad
 (g,E'_1,\ldots,E'_m)\longmapsto(W_0g,E'_1g,\ldots,E'_mg)
\]
is surjective. Indeed, for a target $(W,E_1,\ldots,E_m)$ choose
$g_0$ with $W_0g_0=W$. Its full fiber is parametrized isomorphically by
\[
 h\longmapsto
 \bigl(hg_0,E_1g_0^{-1}h^{-1},\ldots,E_mg_0^{-1}h^{-1}\bigr),
 \qquad h\in P.
\]
The source is irreducible, and subtracting
$\dim P=\dim H-d$ gives the dimension formula.
Properness follows because $G$ is projective.
Kleiman's transversality theorem in characteristic zero
\cite{kleiman}, applied successively to the finitely many smooth
Schubert cells, makes their intersections transverse for general
flags. Boundary strata in a top-degree problem have negative expected
dimension and hence are absent for general flags. Thus there is a
nonempty Zariski open subset $B^\circ$ whose fibers contain
exactly $D$ distinct transverse points. Their intersection cycle is the
Schubert product. For $D>0$ this proves the last assertion.
\end{proof}

\begin{lemma}[Analytic exhaustion]\label{lem:exhaust}
Let $\pi:X\to B$ be a dominant morphism of irreducible complex
algebraic varieties, with exactly $D$ points in every fiber over a
nonempty Zariski open $B^\circ$. Suppose a fiber over $b$ contains $D$
distinct points where $\pi$ is a local complex analytic isomorphism.
Those points exhaust the entire underlying fiber over $b$.
\end{lemma}

\begin{proof}
Suppose there is an additional point $x$. Choose pairwise disjoint
analytic neighborhoods $U_1,\ldots,U_D,U_x$ of the specified points and
$x$. The first $D$ points give analytic inverse sections $s_i$; shrink
their common base neighborhood to $V\ni b$ so that every $s_i(V)$ is
contained in its designated $U_i$.

The set $U_x\cap\pi^{-1}(V)$ is a nonempty analytic open subset of
$X$. The nonempty Zariski open $\pi^{-1}(B^\circ)$ is analytically dense
in irreducible $X$, so the former set contains a point $x'$ over
$b'\in V\cap B^\circ$. That fiber contains the $D$ distinct points
$s_i(b')$ and also $x'$, a contradiction. Analytic density holds at
singular points as well: a proper algebraic subset has smaller dimension
and cannot contain a nonempty analytic open subset.
\end{proof}

At a real solution $W_*$ covered by Corollary~\ref{cor:boundary},
this local analytic isomorphism holds over the full complex base $B$.
To see it, fix the flag tuple $(E_1,\ldots,E_m)$ and choose local
holomorphic sections $g_i(E)$ of the flag action with
$E_i g_i(E)=E$ and $g_i(E_i)=I$. Such sections are given by the
usual triangular flag charts. Since $W_*$ lies in the open cell,
the germ of $\OO_{\lambda_i}(E_i)$ at $W_*$ is smooth of codimension
$|\lambda_i|$. Choose local holomorphic equations $f_{i,a}$,
$1\leq a\leq|\lambda_i|$, generating its ideal and having independent
differentials. Near $(W_*,E_1,\ldots,E_m)$ the moving conditions are
\[
 f_{i,a}\bigl(Wg_i(E'_i)^{-1}\bigr)=0
 \qquad(1\leq i\leq m,\ 1\leq a\leq|\lambda_i|).
\]
There are $d$ equations. Their derivative in the $d$ coordinates of
$G$ is invertible by transversality. The holomorphic implicit function
theorem makes their common zero scheme the graph of a unique
holomorphic function of all flag coordinates. Thus $\pi$ is a local
analytic isomorphism at $W_*$.

\section{Positive intersection number}
\begin{theorem}[Mukhin--Tarasov--Varchenko {\cite[Corollary 6.3]{mtv}}]
\label{thm:mtv}
Let $t_1,\ldots,t_m$ be distinct real numbers, and let
$F_\bullet(t_i)$ be the osculating complete flags of the rational
normal curve in $\R^N$. For nonzero partitions $\lambda_i$ in the
$p\times q$ rectangle, of total size $d$, suppose
$D=\int_G\prod_i\sigma_{\lambda_i}>0$. The intersection
$\bigcap_i\OO_{\lambda_i}(F_\bullet(t_i))$
contains $D$ distinct real points in the prescribed open cells,
and the conditions meet transversely at those points.
\end{theorem}
\begin{proof}[Conversion from the polynomial formulation]
Let $\R[u]_{<N}$ be the vector space of polynomials of degree less
than $N$, and let $e_1,\ldots,e_N$ be the standard row basis.
The real linear isomorphism
\[
 \Phi(e_{k+1})=\frac{(N-1)!}{(N-1-k)!}\,u^{N-1-k}
 \qquad(0\leq k<N)
\]
sends $\gamma(t)=(1,t,\ldots,t^{N-1}/(N-1)!)$ to
$(u+t)^{N-1}$. The first $j$ rows of $e^{tJ}$,
$J=\sum_{j=1}^{N-1}E_{j,j+1}$, are
$\gamma(t),\ldots,\gamma^{(j-1)}(t)$. Their images span
\[
 (u+t)^{N-j}\R[u]_{<j}.
\]
Thus the flag becomes the polynomial osculating flag at $b=-t$.
For $a_j=q+j-\lambda_j$, its $a_j$th subspace consists of polynomials
divisible by $(u-b)^{p-j+\lambda_j}$. The open Schubert condition is
therefore the condition that the increasing vanishing orders of a
basis are
\[
 \lambda_p,\ \lambda_{p-1}+1,\ \ldots,\ \lambda_1+p-1.
\]
These are the finite labels in \cite[Corollary 6.3]{mtv}, with their
plane rank equal to $p$ and polynomial ambient dimension equal to $N$.
Choose their highest-weight partition to be the full rectangle
$(q^p)$. Its complementary partition is zero, so their extra condition
at infinity is an open cell of codimension zero.
Write $\mathfrak{gl}_p$ for the Lie algebra of complex $p\times p$
matrices. The corollary gives as many distinct real transverse points as the
multiplicity of the irreducible $\mathfrak{gl}_p$-module of highest
weight $(q^p)$ in the tensor product with highest weights $\lambda_i$.
By the Littlewood--Richardson rule, this multiplicity is the
coefficient of the Schur polynomial $s_{(q^p)}$ in
$\prod_i s_{\lambda_i}$, where $s_\lambda$ denotes the Schur
polynomial in $p$ variables. The same rule computes the Schubert
product, so this coefficient is $D$ \cite{fulton}.
The real isomorphism $\Phi^{-1}$ carries these points and their
transverse intersections to the stated flags.
\end{proof}

\begin{proposition}\label{prop:positive}
Theorem~\ref{thm:main} holds when $D>0$.
\end{proposition}

\begin{proof}
After normalizing the first frame to $I$, the positive parameter space
is $\mathcal B_+=(\TP)^{m-1}$, parametrized by the independent
increments $G_i=F_{i+1}F_i^{-1}$.
The frames are recovered recursively from $F_1=I$ and
$F_{i+1}=G_iF_i$. This base is a connected smooth manifold by the
coordinates of Lemma~\ref{lem:app-sw-boundary}; for $m=1$ it is a point.
It contains an osculating tuple
\[
 F_i^0=e^{t_iJ},\qquad J=\sum_{j=1}^{N-1}E_{j,j+1},\qquad
 0=t_1<t_2<\cdots<t_m.
\]
These are row flags of the rational normal curve
$(1,t,t^2/2!,\ldots,t^{N-1}/(N-1)!)$. The increments
$e^{(t_{i+1}-t_i)J}$ are strictly totally positive.

Theorem~\ref{thm:mtv} supplies $D$ distinct real points at this tuple.
Zero labels may be omitted in that application, since
$\OO_0=G$ and $\sigma_0=1$. Corollary~\ref{cor:boundary} places the
resulting points in every prescribed open cell of the full tuple.
They give $D$ complex local inverse branches over $B$, so
Lemmas~\ref{lem:universal} and \ref{lem:exhaust} show that they exhaust
the starting complex fiber.

The real incidence over the positive parameter space is closed in its
product with the compact real Grassmannian. Its projection is proper.
By Corollary~\ref{cor:boundary} and the relative implicit function
theorem, this projection is a local diffeomorphism. Each fiber is
discrete and compact, hence finite. Choose disjoint local inverse
neighborhoods of its points. Properness allows the base neighborhood
to be shrunk so that its entire inverse image lies in their union:
otherwise additional preimages over a sequence converging to the
base point would have a limiting preimage outside that union.
Thus the projection is a finite covering of its image, which is open
by local invertibility and closed by properness. The image contains the
osculating tuple and the base is connected. Thus every positive tuple
has exactly $D$ real solutions.

At any target tuple those $D$ real points provide $D$ complex local
sections of $\pi$. Lemmas~\ref{lem:universal} and \ref{lem:exhaust}
exclude all other complex points, including every possible excess
component. Finally, Corollary~\ref{cor:boundary} identifies the original
scheme-theoretic local ring at each point with $\C$. Hence the entire
intersection scheme is exactly $D$ reduced real points.
\end{proof}

\section{Zero intersection number}

Positive-degree completions first control intersections with a nonzero
Schubert product. For an $m$-tuple $\alpha=(\alpha_1,\ldots,\alpha_m)$
of partitions in the $p\times q$ rectangle, write
\[
 P_\alpha=\prod_i\sigma_{\alpha_i},\qquad
 Y_\alpha=\bigcap_i\OO_{\alpha_i}(F_i).
\]
Let $h$ be the Pl\"ucker hyperplane class, and for an irreducible
subvariety $T\subset G$ set $\deg_h(T)=\int_T h^{\dim T}$.

\begin{samepage}
\begin{lemma}\label{lem:nonzero-product}
Suppose $P_\beta\ne0$, with total codimension
$c\leq d$, and set $r=d-c$. For ordered positive flags the support of
$Y_\beta$ is pure of dimension $r$, and every irreducible component
contains a point in all prescribed open Schubert cells.
\end{lemma}
\end{samepage}

\begin{proof}
For $r=0$, the assertion follows from
Proposition~\ref{prop:positive}. Suppose $r>0$.
For general flags, Kleiman's theorem represents $P_\beta$ by an
effective cycle of dimension $r$. Because $P_\beta\ne0$, this cycle
is nonzero. Its degree for the ample Pl\"ucker class is positive, so
\[
 D'=\int_G P_\beta h^r>0.
\]
Append $r$ new ordered positive flags after the last original flag,
for example by repeated left multiplication by $e^J$. At each impose
a simple Schubert divisor $H_j$, which is a hyperplane section of the
Pl\"ucker embedding. Proposition~\ref{prop:positive} makes
\[
 Y_\beta\cap H_1\cap\cdots\cap H_r
\]
a finite reduced set of real points.

Every component $C$ of $Y_\beta$ has dimension at least $r$ by the
intersection dimension inequality in the smooth variety $G$.
In the Pl\"ucker projective space, a projective variety of positive
dimension meets every hyperplane, and intersecting with one
hyperplane decreases dimension by at most one. Applying this statement
$r$ times shows that
$C\cap H_1\cap\cdots\cap H_r$ is nonempty and has dimension at least
$\dim C-r$. Finiteness of the completed intersection therefore implies
$\dim C=r$.

Every such $C$ contains a point of the completed intersection, and
that point is real. It lies in all the original open cells by
Corollary~\ref{cor:boundary} applied to the completed top-degree
problem. Thus $C$ contains a point in all the prescribed open cells.
\end{proof}

\begin{proposition}\label{prop:zero}
If $\prod_i\sigma_{\lambda_i}=0$ and $\sum_i|\lambda_i|\leq d$,
then the corresponding ordered-positive closed Schubert intersection
is empty.
\end{proposition}

\begin{proof}
Starting with empty diagrams, build the target diagrams by increasing
one part by one at each step, preserving the partition inequalities
and the rectangle bound. Let $\gamma$ be the first tuple whose
product is zero, and let $\beta$ be the preceding tuple. Exactly one
condition, say the $j$th, changes; its Schubert variety is replaced by
a codimension-one boundary
\[
 \OO_{\gamma_j}(F_j)\subsetneq\OO_{\beta_j}(F_j).
\]
Put $c=\sum_i|\beta_i|$ and $r=d-c\geq1$.
Lemma~\ref{lem:nonzero-product} says that the support of $Y_\beta$ is
pure $r$-dimensional, with no component contained in this boundary.
Every component of $Y_\gamma$ therefore has dimension at most $r-1$.
The ordinary intersection dimension inequality gives the reverse
inequality for every nonempty component. Thus, if $Y_\gamma$ is
nonempty, the $\gamma$ conditions meet properly in dimension $r-1$.

Work in the smooth variety $G^m$, with the small diagonal map
$\delta:G\to G^m$, $\delta(W)=(W,\ldots,W)$. Its image is a regular
embedding of codimension $(m-1)d$. The
irreducible variety
\[
 A=\prod_i\OO_{\gamma_i}(F_i)
\]
has dimension $md-(c+1)$ and meets this diagonal properly in dimension
$r-1$, with intersection identified with $Y_\gamma$.
By positivity of proper intersection multiplicities for
a regular embedding \cite[Lemma 43.16.3]{stacks}, every component $T$
has a strictly positive coefficient $e_T$ in the intersection cycle.
The regular sequence is supplied by the small diagonal: locally in
smooth coordinates on $G$, it consists of the differences of each
coordinate in the last $m-1$ factors with that in the first factor.
These $(m-1)d$ coordinate differences form a regular sequence in
the local ring of the smooth ambient variety.
The diagonal formula for intersection products gives, in the Chow group
$A_{r-1}(G^m)$ of $(r-1)$-dimensional cycles modulo rational equivalence,
\[
 [\delta(G)]\cdot[A]
 =\sum_T e_T[\delta(T)]
 =\delta_*\!\left(\Bigl(\prod_i\sigma_{\gamma_i}\Bigr)\cap[G]\right)=0.
\]
If the intersection were nonempty, pair its effective cycle with the
appropriate power of the ample class $\sum_i\operatorname{pr}_i^*h$,
where $\operatorname{pr}_i:G^m\to G$ is the $i$th projection.
Since $\delta^*(\sum_i\operatorname{pr}_i^*h)=mh$, its degree would be
\[
 m^{r-1}\sum_T e_T\deg_h(T)>0,
\]
contradicting the zero class. Therefore $Y_\gamma$ is empty. Every later
tuple imposes stronger conditions, so the target intersection is empty.
An empty underlying support also means an empty intersection scheme.
\end{proof}

In top degree,
$\prod_i\sigma_{\lambda_i}=D[\mathrm{point}]$ in $A^d(G)$.
Thus $D=0$ gives the hypothesis of Proposition~\ref{prop:zero}.
Together with Proposition~\ref{prop:positive}, this proves
Theorem~\ref{thm:main}.


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\end{document}