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.
