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Hello everybody please help me with this doubt:

Let $f:\mathbb{P}^{1} \rightarrow \mathrm{Sym}^{d}(X)$, where $X$ is a smooth projective surface, $\mathbb{P}^{1}$ is a projective line and $\mathrm{Sym}^{d}(X)$ is the $d$-symmetric product, i.e., the quotient of $\Sigma_{d}$ (permutation group) acting on $X^d$.

I know that $f^{*}T_{\mathrm{Sym}^{d}(X)}$ is a coherent sheaf in $\mathbb{P}^{1}$ then by https://arxiv.org/pdf/0911.4473.pdf (proposition 5.4.2) $$ f^{*}T_{\mathrm{Sym}^{d}(X)}=\mathcal{O}_{\mathbb{P}^1}(a_1) \oplus \ldots \oplus \mathcal{O}_{\mathbb{P}^1}(a_r) \oplus \mathcal{T}$$ where $\mathcal{T}$ is the torsion sheaf. My questions are

(a)$\; r=2d-1?$

(b) What could i said about $\dim H^{0}(\mathbb{P}^1,f^{*}T_{\mathrm{Sym}^{d}(X)})$?

I know that $$H^{0}(\mathbb{P}^1,f^{*}T_{\mathrm{Sym}^{d}(X)})=H^{0}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}}(a_1)) \oplus \ldots \oplus H^{0}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}}(a_r)) \oplus H^{0}(\mathbb{P}^{1},\mathcal{T}) ,$$ with $\dim H^{0}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}}(a_i))=a_{i}+1$ if and only if $a_{i} \geq 0$ and $H^{0}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}}(a_i))=0$ \, for \,$a_{i} < 0$.

Is $H^{0}(\mathbb{P}^{1},\mathcal{T})=0?$ and is there a way to compute those $a_i$? or maybe is there a relationship between them ?

If anybody has a good example to compute $f^{*}T_{\mathrm{Sym}^{d}(X)}$ or maybe for a specific surface it will be very useful.

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    $\begingroup$ (a) Why $2d-1$? $\quad r=\dim\operatorname{Sym}^d(X)=2d\,.\qquad\qquad \ \ $ (b): $H^0(\mathbb{P}^1,\mathscr{T})$ is always nonzero for a torsion sheaf. $\endgroup$ – abx Mar 18 at 19:56
  • $\begingroup$ Why $H^{0}(\mathbb{P}^{1},\mathcal{T}) \neq 0 $?, i know that if $ f: \mathbb{P}^{1} \rightarrow X$ where $X$ is smooth of dimension $d$ then $f^{*}T_X$ is a locally free sheaf of rank d...but in our case $Sym^{d}(X)$ is not smooth then $f^{*}T_{Sym^{d}(X)}$ is coherent sheaf ..which is his rank ? Dear @abx $\endgroup$ – alberth NUÑEZ SULLCA Mar 18 at 21:54
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    $\begingroup$ The rank depends on $f$ of course. Do you know what a skyscraper sheaf is? Maybe learning some basic algebraic geometry would not hurt. $\endgroup$ – abx Mar 19 at 5:24

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