Tangent bundle of symmetric product of surface

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.

• (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. – abx Mar 18 at 19:56
• 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 – alberth NUÑEZ SULLCA Mar 18 at 21:54
• The rank depends on $f$ of course. Do you know what a skyscraper sheaf is? Maybe learning some basic algebraic geometry would not hurt. – abx Mar 19 at 5:24