I'm reading Mumford's & Oda's [Algebraic Geometry II][1]  and I'm confused about explanations on geometric intuition of sections $H^0(\Theta_X, X)$ of the tangent sheaf on page 287:


Let $X$ a geometrically irreducible smooth  and proper curve of genus $g$ over a field $k$ and let $\Theta_X := \mathcal{Hom}(\Omega_X ^1 , \mathcal{O}_X) \cong \mathcal{O}_X(-K)$ the tangent sheaf to $X$. applying Riemann-Roch we obtain $$\dim_k H^0(\Theta_X, X)=$$

 \begin{cases} 
      3 & g=0, \text{ } i.e. X \cong \mathbb{P}^1_k \\
      1 & g=1 \\
      0 & g \ge 2
   \end{cases}

The author makes some comments on *geometrical* interpretation of these sections, which I don't understand:

>...In fact, the three sections of $\Theta$ when $X \cong \mathbb{P}^1_k$
come from the *infinitesimal section* of the
$3$-dimensional group scheme $PGL_{2,k}$ acting on $\mathbb{P}^1_k$; the one section of $\Theta$ when $g = 1$ comes from the infinitesimal action of $X$ on itself (make sense, since $g=1$ says that $X$ is elliptic curve; i.e. $X$ obtains structure of a group scheme) , and the absence of sections when $g > 1$ is reflected in the
fact that the group of automorphisms of such curves is finite. ...

could anybody explain how these sections concretly arise for $g=0,1$ or respectively why in case $g>1$ the finiteness of automorphism group of $X$ inply that there a no such sections.

the matter is that we need to construct somehow a certain bundle $V$ over $X$ with projection $p:V \to X$. such that the sections $s \in H^0(\Theta_X, X)$ give rise for 'geometric' sections $s:X \to V$ with $p \circ s=id_X$. 

my first guess is that since $\Theta_X$ is invertible, we can construct a bundle $V:= \mathbb{P}(\Theta_X)= Proj(Sym(\Theta_X))$ and each section $s \in H^0(\Theta_X, X)$ induces a 'geometric' section $s:X \to V=\mathbb{P}(\Theta_X)$. the problem that I don't see how this construction is related to the described actions by $PGL_{2,k}$ or $X$ itself as described in the book. therefore I have a lot of doubts that the construction I tried to explain coinsides with the one sketched in the book. could anybody describe the construction the authors have sketched in the text above giving geometric interpretation of sections $H^0(\Theta_X, X)$?





 [1]: https://www.google.com/search?client=firefox-b-d&q=Algebraic%20Geometry%20II%20%28a%20penultimate%20draft%29