$\DeclareMathOperator\Hom{Hom}$I have some questions on a proof of **Proposition II.3.10** & notations from Rational Curves
on Algebraic Varieties by Janos Kollar (page 117).

We work in setting exposed in **3.1 Definition** (p. 113):

Let $C$ be a proper curve without embedded points (for us a curve is an integral scheme of dimension $1$, proper over $k$, all of whose local rings are regular), $X$ a smooth variety and $f: C \to X$ a morphism. Let $B \subset C$ be a closed subscheme with ideal sheaf $I_B$ and $g = f \rvert _B$.

The Proposition uses a couple of notations (same notations as above):

Let $F: C \times \Hom(C,X,g) \to X$ be the universal morphism. For later applications we also consider the induced morphism $F^{(2)}: C \times C \times \Hom(C,X,g) \to X \times X$ (not relevant for us).

Let $p, q \in C$ be closed points and $f : C \to X$ a morphism such that $f \rvert _B = g$.

If $p, q \not \in B$, then by (1.2.16) and (1.9)

$$T_{C \times \Hom(C,X,g)} \otimes k(p, [f])=T_C \otimes k(p) + H^0(C, f^*T_X \otimes I_B).$$

Few words on notations: $T_?$ is the tangent sheaf (the dual to Kähler $\Omega_?$), $\Hom(C,X g)= \{f \in \Hom(C,X) \vert f \rvert _B=g \}$ and $[f] \in \Hom(C,X g)$.

Let $df(s): T_C \otimes k(s) \to T_X \otimes k(f(s))$ be the differential of $f$ at $s \in C$,

$$\phi(p,f): H^0(C, f^*T_X \otimes I_B) \to f^*T_X \otimes k(p)$$

the evaluation map.

The following is a reformulation of (1.2.19):

3.4 Proposition(Notation as above). Then$$dF(p,[f])=df(p)+ \phi(p,f).$$

Now we are ready for Prop. 3.10:

3.10 Proposition(Notation as in (3.3)). Assume that $C \cong \mathbb{P}^1$ and $\lvert B \rvert \le 2$ and write $f*T_X \otimes I_B = \sum \mathcal{O}(a_i)$. Then$$\#\{i \vert a_i \ge 0 \} = \operatorname{rank} dF(p, [f]) \ \forall p \in \mathbb{P}^1 - B.$$

Proof.Since $\vert B \vert \le 2$, $H^0(C, T_{\mathbb{P}^1} \otimes I_B) \to T_{\mathbb{P}^1} \otimes k(p)$ is surjective (???). Therefore$$\operatorname{rank} \phi(p,f) =\operatorname{rank} dF(p, [f]).$$

**Questions:**

**Q₁**: Why does $\lvert B \rvert \le 2$ imply $H^0(C, T_{\mathbb{P}^1} \otimes I_B) \to T_{\mathbb{P}^1} \otimes k(p)$ is surjective? Firstly what is $\lvert B \rvert$? The cardinality of $B$ as set or the dimension of dimensional linear system induced by $B$ as subscheme?

I tend to say that it's the latter one. Nevertheless, why does it imply surjectivity of $H^0(C, T_{\mathbb{P}^1} \otimes I_B) \to T_{\mathbb{P}^1} \otimes k(p)$?

**Q₂**: What is exactly the rank $\operatorname{rank} \phi(p,f)$ of $\phi(p,f): H^0(C, f^*T_X \otimes I_B) \to f^*T_X \otimes k(p)$? Rank as what? Linear maps? Over which field?

`$rank\ \phi$`

, using`\operatorname`

will both give the appropriate font and fix the spacing: $\operatorname{rank} \phi$`$\operatorname{rank} \phi$`

. If you use a named operator repeatedly, then you can declare it, e.g.,`\DeclareMathOperator\rank{rank}`

(although I seem to remember that`\rank`

is reserved in TeX). Also,`>`

quoting works fine on display math. I have edited accordingly. $\endgroup$ – LSpice Feb 5 at 3:45