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\title{Bigness of the tangent bundles of projective bundles over curves}

\author{\firstname{Jeong-Seop} \lastname{Kim}}
\address{School of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro Dongdaemun-gu, Seoul 02455, Republic of Korea}
\email{jeongseop@kias.re.kr}
\thanks{This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2022R1A2B5B03002625)}
\CDRGrant{MSIT No. 2022R1A2B5B03002625}

\begin{abstract}
In this short article, we determine the bigness of the tangent bundle $T_X$ of the projective bundle $X=\mathbb{P}_C(E)$ associated to a vector bundle $E$ on a smooth projective curve $C$.
\end{abstract}

\begin{document}
\maketitle


\section{Introduction}

In this article, all varieties are defined over the field of complex numbers $\CC$. After Mori's proof of the Hartshorne conjecture on ample tangent bundles~\cite{Mor79}, it has been asked to characterize a smooth projective variety $X$ with certain positivity of its tangent bundle $T_X$. For example, a conjecture proposed by Campana and Peternell asks whether the homogeneous varieties are the only smooth Fano varieties $X$ with nef $T_X$, and the conjecture is settled for dimension three~\cite{CP91}, four~\cite{CP93, Hwa06, Mok02} (see also~\cite[Corollary~4.4]{MOSWW15}), and five~\cite{Wat14, Kan17}. Recently, a~series of works done by H\"{o}ring, Liu, Shao~\cite{HLS20}, and H\"{o}ring, Liu~\cite{HL21} investigates smooth Fano varieties $X$ with big $T_X$ as follows.
\begin{theo}[{\cite{HLS20, HL21}}]
Let $X$ be a smooth Fano variety.
\begin{enumerate}
\item \label{1.1} If $X$ has dimension~$2$, then $T_X$ is big if and only if $(-K_X)^2\geq 5$.
\item \label{1.2} If $X$ has dimension~$3$ and Picard number $1$, then $T_X$ is big if and only if $(-K_X)^3\geq 40$.
\item \label{1.3} If $X$ has Picard number $1$, and if $X$ contains a rational curve with trivial normal bundle, then $T_X$ is not big unless $X$ is isomorphic to the quintic del Pezzo threefold.
\end{enumerate}
\end{theo}

The second statement is extended to the following case.
\begin{theo}[{\cite{KKL22}}]
Let $X$ be a smooth Fano variety of dimension~$3$ and Picard number $2$. Then $T_X$ is big if and only if $(-K_X)^3\geq 34$.
\end{theo}

These results make use of a special divisor on the projective bundle $\PP_X(T_X)$, called the total dual VMRT $\breve\Cc$ (see~\cite{HR04, OSW16}). In~\cite{HLS20}, they find a formula for $\breve\Cc$, which can be written as follows in the case where $X$ attains a conic bundle structure $X\to Y$.
\[
[\breve\Cc]
\sim \zeta+\Pi^*K_{X/Y}
\]
where $\Pi: \PP_X(T_X)\to X$ is the projection and $\zeta$ is the tautological divisor on $\PP_X(T_X)$. In other words, $\breve\Cc$~arises as the divisor on $\PP_X(T_X)$ corresponding to the natural subsheaf $T_{X/Y}\to T_X$ of rank $1$.

In this article, we deal with a question on the bigness of $T_X$ in the case of the projective bundles $X=\PP_C(E)$ over a smooth projective curve $C$. When $E$ has rank $2$, $X$ becomes a ruled surface, and the classification of $X$ with big $T_X$ is a consequence of some known facts. Indeed, if $E$ is semi-stable, then $h^0(S^k T_X)$ is bounded above by a sum of dimensions of certain families of curves on $X$, whose bound can be obtained from a remark of~\cite{Ros02} (see Remark~\ref{semi-stable_rank_two}). Otherwise, if $E$ is unstable, then the bigness of $T_X$ easily follows from the formula introduced above (cf.~\cite[Remark~2.4]{KKL22}). However, when the rank of $E$ gets larger, we cannot apply the formula because $X\to C$ is not a conic bundle.

In the case of higher ranks, when $E$ is unstable, we can find a rank $1$ subsheaf of $S^k T_X$ instead of $T_X$ to conclude that $T_X$ is big. Also, when $E$ is semi-stable, by computing an upper bound of $h^0(S^k T_X)$, we~can determine the bigness of $T_X$ according to the stability of $E$ as follows.

\begin{enonce*}
{Main Theorem}
Let $C$ be a smooth projective curve and $E$ be a vector bundle on $C$. Then the projective bundle $X=\PP_C(E)$ has big tangent bundle $T_X$ if and only if $E$ is unstable or $C=\PP^1$.
\end{enonce*}

The proof is divided into two parts; the case $E$ is semi-stable (Theorem~\ref{semi-stable_case}), and the case $E$ is unstable (Theorem~\ref{unstable_case}). The exceptional case $C=\PP^1$ is explained in Remark~\ref{trivial_case}. It is worth noting that the result is no longer true for varieties other than curves; there exist stable bundles $E$ of rank $2$ on $\PP^2$ such that one of $E$ gives big $T_X$ whereas another choice of $E$ gives not big $T_X$ for $X=\PP_{\PP^2}(E)$ (see No.~24, 27, and 32 of Table~1 in~\cite{KKL22}; No. 24 is the only case with non-big $T_X$, and see also~\cite{SW90}).



\section{Preliminaries}

Let $X$ be a smooth projective variety of dimension $n>0$ and $V$ be a vector bundle of rank $r\geq 2$ on $X$. In this article, $\PP_X(V)$ denotes the projective bundle with the projection $\Pi: \PP_X(V)\to X$ in the sense of Grothendieck. That is, for the tautological line bundle $\Oo_{\PP_X(V)}(1)$ on $\PP_X(V)$, we have
\[
\Pi_*\Oo_{\PP_X(V)}(m)=
\begin{cases}
S^mV & \text{for $m\geq 0$,}\\
0 & \text{for $m< 0$}
\end{cases}
\]
where the $0$-th power is taken to be $S^0 V=\Oo_X$ for convenience.

For an integer $m> -r$ and vector bundle $W$ on $X$,
\[
R^i\Pi_*(\Pi^*W\otimes\Oo_{\PP_X(V)}(m)) =W\otimes R^i\Pi_*\Oo_{\PP_X(V)}(m) =0
\quad \text{for all $i>0$}.
\]
Thus, when $m> -r$,
\[
H^i(\PP_X(V),\Pi^*W\otimes\Oo_{\PP_X(V)}(m))
\cong H^i(X,W\otimes\Pi_*\Oo_{\PP_X(V)}(m))
\quad \text{for all $i\geq 0$}.
\]
In particular, $H^0(\Pi^*W\otimes\Oo_{\PP_X(V)}(-1))=0$.

\subsection{Bigness of Vector Bundle}

In this article, we define certain positivity of a vector bundle by the same positivity of the tautological line bundle on the projective bundle associated to the given vector bundle. The definition may differ, depending on the article; for example, there are distinct notions of bigness of vector bundles; L-big and V-big (see~\cite{BKKMSU15}).

\begin{defi*}
A vector bundle $V$ is said to be \emph{ample} (resp., \emph{nef}, \emph{big}, \emph{effective}, and \emph{pseudo-effective}) on~$X$ if the tautological line bundle $\Oo_{\PP_X(V)}(1)$ is ample (resp., nef, big, effective, and pseudo-effective) on~$\PP_X(V)$.
\end{defi*}

%\pagebreak

\begin{rema}
Recall that a line bundle $L=\Oo_X(D)$ on $X$ is big if and only if it satisfies one of the followings (see~\cite[Section~2.2]{Laz04}).
\begin{itemize}
\item $h^0(L^k)\sim k^n$ (which is the maximum possible).
\item $mD\sim_{\mathrm{lin}} A+E$ for some integer $m>0$, ample divisor $A$, and effective divisor $E$ on $X$.
\item $D$ lies in the interior of the closure $\overline{\Eff}(X)\subseteq N^1(X)$ of the cone of effective divisors (as bigness is well-defined under numerical equivalence).
\end{itemize}
If $L$ is a line bundle on $X$, then the following holds.
\begin{itemize}
\item $L$ is big if and only if $L^{\otimes k}$ is big for some integer $k>0$.
\end{itemize}
If $V$ is a vector bundle on $X$, then the following holds (see also~\cite[Section~6.1]{Laz04b}).
\begin{itemize}
\item $V$ is big if and only if $h^0(S^k V)\sim k^{n+r-1}$ (which is the maximum possible). In particular, $T_X$~is big if and only if $h^0(S^k T_X)\sim k^{2n-1}$.
\end{itemize}
\end{rema}

We will denote by $\zeta$ the tautological divisor on $\PP_X(V)$. Also, for a divisor $B$ on $X$, we will denote by $Bf$ the divisor $\Pi^*B$ on $\PP_X(V)$.

\begin{lemm}[cf. {\cite[Lemma~2.3]{HLS20}}]\label{peff_plus_big_on_base_is_big}
Let $V$ be a vector bundle on a normal projective variety $X$. Let $k>0$ and $B$ be a divisor on $X$. If $k\zeta+(B-D)f$ is pseudo-effective for some big divisor $D$ on $X$, then $k\zeta+Bf$ is big on $\PP_X(V)$. In particular, if $S^k V\otimes \Oo_X(-D)$ is effective for some big divisor $D$ on $X$, then $V$ is big on $X$.
\end{lemm}

\begin{proof}
Let $\zeta'=k\zeta+ Bf$. Assume that $\zeta'-Df$ is pseudo-effective. Note that $mD\sim A+N$ for some $m>0$, ample divisor $A$, and effective divisor $N$ on $X$. By~\cite[Proposition~1.45]{KM98}, there exists an integer $n> 0$ such that $\zeta'+nAf$ is ample on $X$ because $\zeta'$ is $\Pi$-ample. Then $(mn+1)\zeta'$ is written by a sum of ample and pseudo-effective divisors as
\[
(mn+1)\zeta' =(\zeta'+nAf)+mn(\zeta'-Df)+nNf.
\]
Thus $(mn+1)\zeta'$ is big, and it implies that $\zeta'=k\zeta+Bf$ is big on $\PP_X(V)$.
\end{proof}

As an application of the lemma, we present a proof of the following fact.

\begin{prop}\label{bigness_of_product}
Let $X$ and $Y$ be smooth projective varieties with big tangent bundles $T_X$ and $T_Y$. Then the tangent bundle $T_{X\times Y}$ of $X\times Y$ is big.
\end{prop}

\begin{proof}
Let $B$ and $D$ be big and effective divisors on $X$ and $Y$, respectively. As $T_X$ and $T_Y$ are big, there exist integers $m,\,n>0$ such that $S^m T_X(-B)$ and $S^n T_Y(-D)$ are effective by Kodaira's Lemma. Note that $T_{X\times Y}=p^*T_X\oplus q^*T_Y$ for the natural projections $p:X\times Y\to X$ and $q:X\times Y\to Y$, and $p^*B+q^*D$ is a big divisor on $X\times Y$. Since $S^{m+n}T_{X\times Y}$ contains $S^m p^*T_X\otimes S^n q^*T_Y$ as a direct summand, we~have
\[
H^0(S^{m+n}T_{X\times Y}\otimes \Oo_{X\times Y}(-p^*B-q^*D))
\supseteq H^0(S^m p^*T_X\otimes\Oo_{X\times Y}(-p^*B)\otimes S^n q^*T_Y\otimes\Oo_{X\times Y}(-q^*D))
\neq 0.
\]
Thus $S^{m+n}T_{X\times Y}\otimes\Oo_{X\times Y}(-(p^*B+q^*D))$ is effective, and hence $T_{X\times Y}$ is big by Lemma~\ref{peff_plus_big_on_base_is_big}.
\end{proof}


\subsection{Stability of Vector Bundle} In this article, stability is defined in the sense of Mumford and Takemoto. For the definitions introduced in this section, we add a mild condition (torsion-freeness) from the definitions in the reference~\cite[Chapter~1]{HL10}.

Let $Y$ be a smooth projective variety and $E$ be a torsion-free coherent sheaf on $Y$. Then there exists an open dense subset $U\subseteq Y$ such that $Y\setminus U$ has codimension at least two and $E|_U$ is locally free. The \emph{rank} of $E$ is defined by $\rk E=\rk E|_U$.

\begin{defi*}
Fix an ample divisor $H$ on $Y$. For a torsion-free coherent sheaf $E$ on $Y$, the \emph{$H$-slope} of $E$ is defined by
\[
\mu_H(E)=\frac{\deg_H E}{\rk E}
\]
where the $H$-degree of $E$ is defined by $\deg_H E=c_1(E).H^{n-1}$.

Let $E$ be a torsion-free coherent sheaf of rank $r>0$ on $Y$. Then $E$ is said to be \emph{$\mu_H$-stable} (resp., \emph{$\mu_H$-semi-stable}) if for every coherent subsheaf $F$ of $E$ with $0<\rk F< r$,
\[
\mu_H(F)<\mu_H(E)
\quad\text{(resp., $\mu(F)\leq \mu(E)$)}.
\]
Also, $E$ is said to be \emph{$\mu_H$-unstable} if it is not $\mu_H$-semi-stable. If there is no confusion in the choice of~$H$, then we denote it by $\mu$-stable (resp. $\mu$-semi-stable, $\mu$-unstable), or stable (resp. semi-stable, unstable) in the case where $Y$ is a curve.
\end{defi*}

\begin{rema}
The followings are some known facts on the $\mu$-stability and slope of vector bundles $E$ and $F$ on $Y$. For the proofs, we may refer~\cite[Chapter~3]{HL10}.
\begin{itemize}
\item If $E$ and $F$ are $\mu$-semi-stable and $\mu(E)<\mu(F)$, then $\Hom(F,E)=0$.
\item If $E$ and $F$ are $\mu$-semi-stable, then $E\otimes F$ is $\mu$-semi-stable.
\item If $E$ is $\mu$-semi-stable, then $S^m E$ is $\mu$-semi-stable for all $m>0$.
\item $\rk(S^m E)=\binom{m+r-1}{r-1}$, $c_1(S^m E)=c_1(E)^{\otimes\binom{m+r-1}{r}}$, and $\mu(S^mE)=m\cdot \mu(E)$.
\item Assume that $E$ fits into the following exact sequence of vector bundles on $Y$.
\[
0\to F\to E\to Q\to 0
\]
If $\mu(F)=\mu(E)=\mu(Q)$, then $E$ is $\mu$-semi-stable if and only if both $F$ and $Q$ are $\mu$-semi-stable.
\item $E$ is $\mu$-semi-stable if and only if its dual $E^\vee$ is $\mu$-semi-stable, and $\mu(E^\vee)=-\mu(E)$.
\end{itemize}
\end{rema}

For a torsion-free coherent sheaf $E$ on $Y$, there exists a canonical filtration
\[
0=E_0\subset E_1\subset
\cdots\subset E_k=E,
\]
which satisfies
\begin{itemize}
\item $E_i/E_{i-1}$ is $\mu$-semi-stable (also, torsion-free) for all $0<i\leq k$, and
\item $\mu(E_{i+1}/E_i)<\mu(E_i/E_{i-1})$ for all $0<i<k$.
\end{itemize}
This filtration is called the \emph{Harder-Narasimhan filtration} of $E$. We call $F=E_1$ the \emph{maximal destabilizing subsheaf} of $E$. When $E$ is $\mu$-unstable, we must have $\mu(F)>\mu(E)$. Also, it follows from the definition that $E/F$ is torsion-free. In the case of curves $Y=C$, a coherent sheaf is torsion-free if and only if it is locally free, so we can further say that $E/F$ is locally free.


\section{Semi-Stable Case}

In this section, let $Y$ be a smooth projective variety of dimension $n>0$, and fix an ample divisor $H$ on $Y$. Let $E$ be a vector bundle of rank $r>0$ on $Y$. We denote by $X=\PP_Y(E)$ the projective bundle associated to $E$ with the projection $\pi: \PP_Y(E)\to Y$, and by $\Oo_X(\xi)$ the tautological line bundle on $X$. Then, after taking symmetric powers to the relative Euler sequence
\[
0
\to \Oo_X
\to \pi^*E^\vee\otimes\Oo_X(\xi)
\to T_{X/Y}
\to 0,
\]
we obtain the following exact sequence on $X$.
\begin{equation}\label{symmetric_product_of_relative_Euler_sequence}
0
\to S^{m-1}\pi^*E^\vee\otimes\Oo_X((m-1)\xi)
\to S^m\pi^*E^\vee\otimes\Oo_X(m\xi)
\to S^mT_{X/Y}
\to 0
\end{equation}
By pushing forward the exact sequence via $\pi$, we have the following exact sequence on $Y$.
\[
0
\to S^{m-1}E^\vee\otimes S^{m-1}E
\to S^mE^\vee\otimes S^mE
\to \pi_*S^mT_{X/Y}
\to 0
\]

\begin{lemm}\label{semi-stability_of_push-forward_of_relative_tangent_bundle}
Let $X=\PP_Y(E)$ and $\pi: \PP_Y(E)\to Y$ be the projection. If $E$ is $\mu$-semi-stable, then $\pi_*S^m T_{X/Y}$ is a $\mu$-semi-stable bundle of $\deg_H\pi_*S^mT_{X/Y}=0$ on $Y$.
\end{lemm}

\begin{proof}
Note that $S^m E^\vee\otimes S^m E$ is $\mu$-semi-stable for all $m>0$ because $E$ is $\mu$-semi-stable. Moreover, we have $\deg_H\pi_*S^mT_{X/Y}=0$ due to the above exact sequence and
\[
\deg_H (S^mE^\vee\otimes S^m E) =\rk (S^m E)\cdot\deg_H (S^mE^\vee)+\rk (S^m E^\vee)\cdot\deg_H (S^m E) =0.
\]
Since $\pi_*S^mT_{X/C}$ is a quotient of a $\mu$-semi-stable bundle of the same $H$-slope, it is $\mu$-semi-stable.
\end{proof}

\begin{prop}\label{semi-stable_general_case}
Assume that $T_Y$ is $\mu$-semi-stable and $\deg_H T_Y<0$. If $E$ is $\mu$-semi-stable, then the tangent bundle $T_X$ of $X=\PP_Y(E)$ is not big.
\end{prop}

\begin{proof}
Since the projection $\pi: X\to Y$ is a smooth morphism, there is the following exact sequence of vector bundles on $X$.
\[
0
\to T_{X/Y}
\to T_X
\to \pi^*T_Y
\to 0
\]
From this exact sequence, we can find a bound of the dimension of the global sections of $S^k T_X$ as follows.
\[
h^0(S^k T_X)\leq\sum_{m=0}^k h^0(S^mT_{X/Y}\otimes S^{k-m}{\pi^*T_Y})
\]

By the assumption, $\left(S^{k-m}{T_Y}\right)^\vee$ is $\mu$-semi-stable, and $\deg_H \left(S^{k-m}{T_Y}\right)^\vee>0$. Due to Lemma~\ref{semi-stability_of_push-forward_of_relative_tangent_bundle}, $\pi_*S^mT_{X/Y}$ is $\mu$-semi-stable, and $\deg_H S^m T_{X/Y}=0$. So we have
\[
h^0(S^m T_{X/Y}\otimes S^{k-m}{\pi^*T_Y}) =h^0(\pi_* S^m T_{X/Y}\otimes S^{k-m}{T_Y}) =\dim \Hom(\left(S^{k-m}{T_Y}\right)^\vee,\pi_* S^m T_{X/Y}) =0
\]
whenever $0\leq m<k$. Thus
\[
h^0(S^k T_X)
\leq h^0(S^k T_{X/Y}) =O(k^{n+r-2}).
\]
That is, $T_X$ is not big.
\end{proof}

%\pagebreak

\begin{theo}\label{semi-stable_case}
Let $C$ be a smooth projective curve of genus $g>0$ and $E$ be a vector bundle on $C$. If~$E$ is semi-stable, then the tangent bundle $T_X$ of $X=\PP_C(E)$ is not big.
\end{theo}

\begin{proof}
If $g\geq 2$, then $\deg T_C<0$ and $T_C$ is stable as every line bundle is stable. So $T_X$ is not big by Proposition~\ref{semi-stable_general_case}. Otherwise, if $g=1$, then $T_C=\Oo_C$. By~\cite[Lemma~15]{Ati57}, $h^0(S^mT_{X/C})=h^0(\pi_*S^mT_{X/C})$ is bounded above by the number of indecomposable direct summands of $\pi_*S^mT_{X/C}$ as $\pi_*S^mT_{X/C}$ is a semi-stable bundle of degree~$0$ on $C$. Thus we have
\[
h^0(S^m T_{X/C})
\leq \rk(\pi_*S^mT_{X/C}) =\rk(S^mE^\vee\otimes S^mE)-\rk(S^{m-1}E^\vee\otimes S^{m-1}E).
\]
After telescoping, we can conclude that
\[
h^0(S^k T_X)
\leq\sum_{m=0}^k h^0(S^mT_{X/C})
\leq\rk(S^k E^\vee\otimes S^k E) =O(k^{2r-2}).
\]
That is, $T_X$ is not big as $X$ has dimension $r=\rk E$.
\end{proof}

\begin{rema}\label{semi-stable_rank_two}
Let $E$ be a semi-stable bundle of rank $2$ on a smooth projective curve $C$ of genus $g>0$. Then $T_{X/C}$ is a line bundle on $X=\PP_C(E)$, and
\[
S^mT_{X/C} ={T_{X/C}}^{\otimes m}
\cong\Oo_X(2mC_0)
\]
for some $\QQ$-divisor $C_0$ on $X$ with ${C_0}^2=0$. For a divisor $\bb$ on $C$, we denote by $\bb f$ the divisor $\pi^*\bb$ on~$X$.

Let $D\sim 2mC_0+\bb f$ for some $m>0$. If $\deg \bb<0$, then $h^0(\Oo_X(D))=0$ because there is no effective divisor $D$ on $X$ with $D^2<0$ (cf.~\cite[Section~1.5.A]{Laz04}). Assume that $\deg \bb=0$ and $D$ is effective. If $D$ is integral, then it is known from~\cite[Remark in p.~122]{Ros02} that $h^0(\Oo_X(D))=1$. Otherwise, if $D$ is not integral, then $D$ is written in a sum of effective divisors linearly equivalent to $kC_0+\aa f$ for some $k>0$ and $\deg \aa=0$. In this case, we can find an upper bound of $h^0(\Oo_X(2mC_0))$ by the same remark and the fact that $E$ splits once we have $h^0(\Oo_X(C_0+\aa f))\geq 2$ for some divisor $\aa$ on $C$ with $\deg \aa=0$~\cite[Lemma~5.4]{NR69}.
\end{rema}

\begin{rema}\label{trivial_case}
If $g=0$ and $E$ is semi-stable, then $C=\PP^1$ and $E=\Oo_{\PP^1}(a)^{\oplus r}$ for some $a\in\ZZ$. Thus $X=\PP_C(E)\cong \PP^{r-1}\times\PP^1$ and $T_X$ is big by Lemma~\ref{bigness_of_product}.
\end{rema}

\begin{rema}
Using the result on curves, we can state the non-bigness of $T_X$ under some special assumptions on $Y$ and $E$ (cf. Proposition~\ref{semi-stable_general_case}). Assume that $Y$ has a fibration $p: Y\to B$ over a smooth base~$B$ whose general fiber $f$ is a smooth curve of genus $g>0$. If $E|_f$ is semi-stable on a general fiber $f$, then the tangent bundle $T_X$ of $X=\PP_Y(E)$ is not big.

Suppose that $T_X$ is big. Let $Z=\PP_f(E|_f)$ and $\pi_f:Z\to f$ be the induced projection. Then, for general $Z=\PP_f(E|_f)$, $T_X|_Z$ is big, and it implies that $T_Z$ is big. Indeed, from the exact sequence
\[
0
\to T_Z
\to T_X|_Z
\to N_{Z|X}
\to 0,
\]
we have $N_{Z|X}\cong {\pi_f}^*N_{f|Y}\cong {\pi_f}^*\Oo_f^{\oplus n-1}\cong \Oo_Z^{\oplus n-1}$, and it gives the following bound.
\[
h^0(S^k T_X|_Z)
\leq \sum_{m=0}^k h^0(S^m T_Z\otimes S^{k-m}(\Oo_f^{\oplus n-1})) = \sum_{m=0}^k O(k^{n-2})\cdot h^0(S^m T_Z) = O(k^{n-1})\cdot h^0(S^k T_Z)
\]
However, as $E|_f$ is assumed to be semi-stable, $Z=\PP_f(E|_f)$ cannot have big $T_Z$ due to Theorem~\ref{semi-stable_case}. By the contradiction, $T_X$ is not big.
\end{rema}


\section{Unstable Case}

In this section, we concentrate on the case where $Y$ is a smooth projective curve $C$ of genus $g\geq 0$. We continue to use the notation in the previous section.

\begin{prop}\label{line_bundle_unstabilization}
If $E$ is unstable, then $S^m E$ is \emph{unstabilized} by a line subbundle for some $m>0$; there exists a line subbundle $L$ of $S^m E$ with $\mu(L)>\mu(S^m E)$.
\end{prop}

\begin{proof}
Let $F$ be the maximal destabilizing subbundle of $E$. Then $\mu(F)>\mu(E)$ as $E$ is unstable. Also, the quotient $Q$ of $E$ by $F$ is locally free, so we obtain the following exact sequence of vector bundles on $C$.
\[
0
\to F
\to E
\to Q
\to 0
\]
By taking symmetric powers to the exact sequence,
\[
0
\to S^m F
\to S^m E
\to S^{m-1}E\otimes Q
\to S^{m-2}E\otimes \wedge^2 Q
\to \cdots
\to S^{m-\rk Q}E\otimes \wedge^{\rk Q} Q
\to 0,
\]
we can observe that $S^m F$ is a subbundle of $S^m E$. Note that $\mu(S^m F)-\mu(S^m E)=m\cdot (\mu(F)-\mu(E))>0$.

According to~\cite{MS85}, for each $m>0$, there exists a line subbundle $L$ of $S^m F$ satisfying
\[
\mu(S^m F)-\mu(L)
\leq \frac{\rk(S^m F)-\rk(L)}{\rk(S^m F)\cdot\rk(L)}\cdot g < g.
\]
So we can find a line subbundle $L$ of $S^m F$ such that
\[
\mu(L)-\mu(S^m E) =\left\{\mu(S^m F)-\mu(S^m E)\right\}-\left\{\mu(S^m F)-\mu(L)\right\} >m\cdot (\mu(F)-\mu(E))-g>0
\]
by taking $m>0$ large enough. As $S^m F$ is a subbundle of $S^m E$, $L$ is also a nonzero subbundle of~$S^m E$. Hence we obtain a line subbundle $L$ of $S^m E$ satisfying $\mu(L)>\mu(S^m E)$ for some $m>0$.
\end{proof}

\begin{lemm}\label{bigness_by_degree}
Let $m>0$ and $\bb$ be a divisor on $C$ with $b=\deg \bb$. If $m\mu(E)+b>0$, then $m\xi+\bb f$ is big on $\PP_C(E)$.
\end{lemm}

\begin{proof}
Assume that $m\mu(E)+b>0$. If $E$ is semi-stable, then $m\xi+\bb f$ is ample on $\PP_C(E)$ by~\cite[Theorem~3.1]{Miy87}, and hence $m\xi+\bb f$ is big on $\PP_C(E)$.

Otherwise, if $E$ is unstable, then there exists the maximal destabilizing subbundle $F$ of $E$. Note that $F$ is semi-stable and $\mu(F)>\mu(E)$. Let $\eta=\Oo_{\PP_C(F)}(1)$ be the tautological line bundle on $\PP_C(F)$. Then $m\eta+\bb f$ is big on $\PP_C(F)$ by the previous argument, and so $km\eta+(k\bb-P)f$ is effective for some $k>0$ and $P\in C$ by Kodaira's Lemma. Thus $S^{km}F\otimes \Oo_C(k\bb-P)$ is effective, and from the inclusion
\[
S^{km}F\otimes \Oo_C(k\bb-P)
\to S^{km}E\otimes \Oo_C(k\bb-P),
\]
we can observe that $S^{km}E\otimes \Oo_C(k\bb- P)$ is effective as well. That is, $km\xi+(k\bb-P)f$ is effective for some $k>0$ and $P\in C$. Therefore, $km\xi+k\bb f$ is big by Lemma~\ref{peff_plus_big_on_base_is_big}, and it implies that $m\xi+ \bb f$ is big on $\PP_C(E)$.
\end{proof}

%\pagebreak

\begin{theo}\label{unstable_case}
If $E$ is unstable, then the tangent bundle $T_X$ of $X=\PP_C(E)$ is big.
\end{theo}

\begin{proof}
Since $E^\vee$ is also unstable, there exists an integer $m>0$ such that $S^m E^\vee$ has a line subbundle $L\to S^m E^\vee$ with $\mu(L)>\mu(S^m E^\vee)=-m\mu(E)$ by Proposition~\ref{line_bundle_unstabilization}. By twisting $\Oo_X(m\xi)$ after pulling-back the inclusion $L\to S^m E^\vee$ via $\pi$, it gives a nonzero subbundle
\begin{equation}\label{the_subbundle}
\pi^*L\otimes\Oo_X(m\xi)\to \pi^*S^mE^\vee\otimes\Oo_X(m\xi).
\end{equation}
Note that there cannot exist a nonzero morphism $\pi^*L\otimes \Oo_X(m\xi)\to S^{m-1}\pi^*E^\vee\otimes\Oo_X((m-1)\xi)$ as $\pi^*(S^{m-1}E^\vee\otimes L^{-1})\otimes\Oo_X(-\xi)$ never has a global section. Thus~\eqref{the_subbundle} induces a nonzero subsheaf
\begin{equation}\label{the_subsheaf}
\pi^*L\otimes \Oo_X(m\xi)\to S^m T_{X/C}
\end{equation}
via~\eqref{symmetric_product_of_relative_Euler_sequence} as follows.
\[
\xymatrix@C1pc@R1.7pc{ &&\pi^*L\otimes \Oo_X(m\xi) \ar[d] \ar@{-->}[dr]&&\\
0 \ar[r] & S^{m-1}\pi^*E^\vee\otimes \Oo_X((m-1)\xi) \ar[r] & S^m\pi^*E^\vee\otimes \Oo_X(m\xi) \ar[r] & S^mT_{X/C} \ar[r] & 0 }
\]
Because $S^mT_{X/C}$ is a subbundle of $S^m T_X$, \eqref{the_subsheaf} induces a nonzero subsheaf
\[
\pi^*L\otimes \Oo_X(m\xi)\to S^m T_X,
\]
and hence $S^mT_X\otimes \Oo_X(-m\xi-\bb f)$ becomes effective for $\Oo_X(\bb)=L$. Since $m\mu(E)=\mu(S^m E)+b>0$ for $b=\deg \bb=\mu(L)$, the divisor $m\xi+\bb f$ is big on $X$ by Lemma~\ref{bigness_by_degree}. Thus, by applying Lemma~\ref{peff_plus_big_on_base_is_big}, we can conclude that $T_X$ is big.
\end{proof}


\subsection*{Acknowledgement}
I would like to thank my thesis advisor Prof. Yongnam Lee for suggesting this problem and giving valuable comments. I also thank Chih-Wei Chang for pointing out an error in the previous manuscript, Insong Choe for introducing helpful references, and the anonymous referee for carefully reading the manuscript and giving suggestions on improving the clarity.

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