$\DeclareMathOperator\Hom{Hom}$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?

$\DeclareMathOperator\Hom{Hom}$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?

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 curve 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 $g = f \vert _B$.

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 $f \vert _B = g$.

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 $\Omega_?$), $Hom(C,X g)= \{f \in Hom(C,X) \vert f \vert _B=g \}$ 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 $\vert B \vert \le 2$ and write $f*T_X \otimes I_B = \sum \mathcal{O}(a_i)$. Then

$$\#\{i \vert a_i \ge 0 \} = rank dF(p, [f]) \ \forall p \in \mathbb{P}^1 - B.$$

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

$$rank \ \phi(p,f) =rank \ dF(p, [f])$$

**Questions: **

Q_1: Why $\vert B \vert \le 2$ imply $H^0(C, T_{\mathbb{P}^1} \otimes I_B) \to T_{\mathbb{P}^1} \otimes k(p)$ is surjective? firstly what is $\vert B \vert$? 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 later one. Nevertheless, why does it imply surjectivity of $H^0(C, T_{\mathbb{P}^1} \otimes I_B) \to T_{\mathbb{P}^1} \otimes k(p)$?

Q_2: What is exactly the rank $rank \ \phi(p,f)$ 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?

$\DeclareMathOperator\Hom{Hom}$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?