I have a question on an unequality used in the proof of the Very weak Riemann-Roch on curves in Janos Kollar's Lecture on Resolution of Singularities (page 14):

1.13 (Very weak Riemann-Roch on curves). Let $C$ be an irrecucible, reduced, projective curve over an alg closed field $k$. We claim that for any ample line bundle $L$,

$$h^0(C,L^m) \ge m \cdot deg \ L +1 -\binom {\operatorname{deg} \ L-1}{2}$$

for $ m \ge \operatorname{deg} \ L$.

 

(proof) Indeed, embed $C$ into $\mathbb{P}^n$ by $L$ (since $L$ ample that works), and then project it generically to a plane curve of degree $\operatorname{deg} \ L$, $\pi: C \to C' \subset \mathbb{P}^2$

few explanations: i.e. $\pi$ is the composition of the embedding $i:C \to \mathbb{P}^n$ and and projection of $\mathbb{P}^n$ to a plane $\cong \mathbb{P}^2$; it's only a rational map but if we restrict it to $C$ and chose the projection planes in clever way (see my comments below) then the projection $C \dashrightarrow \mathbb{P}^2$ becomes a morphism. $C'$ is the image of $C$ and $C'$ is birational to $C$. we continue the proof...

Now, for $m \ge \operatorname{deg} \ L$,

$$h^0(C,L^m) \ge h^0(C', \mathcal{O}_{\mathbb{P}^2}(m) \vert _{C'}) (???) \\ \ge h^0(\mathbb{P}^2, \mathcal{O}_{\mathbb{P}^2}(m)) -h^0(\mathbb{P}^2, \mathcal{O}_{\mathbb{P}^2}(m- \operatorname{deg} \ L)) \\ \ge m \cdot \operatorname{deg} \ L +1 - \binom {\operatorname{deg} \ L-1}{2}. $$

[...]

Question: I not understand the first unequality

$$h^0(C,L^m) \ge h^0(C', \mathcal{O}_{\mathbb{P}^2}(m) \vert _{C'}).$$

more concretly, I don't see how $\pi$ induces an appropriate exact sequence which allows to relate dimensions of sheaf cohomology groups $H^0(C,L^m)$ and $ H^0(C', \mathcal{O}_{\mathbb{P}^2}(m) \vert _{C'})$

In contrast, the second equality is clearly induced by $0 \to O(m-\operatorname{deg} \ L) \to O(m) \to O(m) \vert _{C'} \to 0$. Ok, on first one I have no idea.

A couple words about the projection $\mathbb{P}^n \dashrightarrow \mathbb{P}^2$: that's nothing but the composition of the projections explaned in example 1.9 page 12: Let $p \in \mathbb{P}^n$ and $\mathbb{P}^{n-1} \cong H \subset \mathbb{P}^n$ a hyperplane not containing $p$. This induces the projection $\pi_{p,H}: \mathbb{P}^n \dashrightarrow \mathbb{P}^{n-1}$ (in the book the construction is described concretly). Moreover the book tells that sophisticated choises of points and hyperplanes allows to project down a curve $C$ birationally to $C'$ by iterating the process above & restricting to $C$.

I have a question on an unequality used in the proof of the Very weak Riemann-Roch on curves in Janos Kollar's Lecture on Resolution of Singularities (page 14):

1.13 (Very weak Riemann-Roch on curves). Let $C$ be an irrecucible, reduced, projective curve over an alg closed field $k$. We claim that for any ample line bundle $L$,

$$h^0(C,L^m) \ge m \cdot deg \ L +1 -\binom {\operatorname{deg} \ L-1}{2}$$

for $ m \ge \operatorname{deg} \ L$.

(proof) Indeed, embed $C$ into $\mathbb{P}^n$ by $L$ (since $L$ ample that works), and then project it generically to a plane curve of degree $\operatorname{deg} \ L$, $\pi: C \to C' \subset \mathbb{P}^2$

few explanations: i.e. $\pi$ is the composition of the embedding $i:C \to \mathbb{P}^n$ and and projection of $\mathbb{P}^n$ to a plane $\cong \mathbb{P}^2$; it's only a rational map but if we restrict it to $C$ and chose the projection planes in clever way (see my comments below) then the projection $C \dashrightarrow \mathbb{P}^2$ becomes a morphism. $C'$ is the image of $C$ and $C'$ is birational to $C$. we continue the proof...

Now, for $m \ge \operatorname{deg} \ L$,

$$h^0(C,L^m) \ge h^0(C', \mathcal{O}_{\mathbb{P}^2}(m) \vert _{C'}) (???) \\ \ge h^0(\mathbb{P}^2, \mathcal{O}_{\mathbb{P}^2}(m)) -h^0(\mathbb{P}^2, \mathcal{O}_{\mathbb{P}^2}(m- \operatorname{deg} \ L)) \\ \ge m \cdot \operatorname{deg} \ L +1 - \binom {\operatorname{deg} \ L-1}{2}. $$

[...]

Question: I not understand the first unequality

$$h^0(C,L^m) \ge h^0(C', \mathcal{O}_{\mathbb{P}^2}(m) \vert _{C'}).$$

more concretly, I don't see how $\pi$ induces an appropriate exact sequence which allows to relate dimensions of sheaf cohomology groups $H^0(C,L^m)$ and $ H^0(C', \mathcal{O}_{\mathbb{P}^2}(m) \vert _{C'})$

In contrast, the second equality is clearly induced by $0 \to O(m-\operatorname{deg} \ L) \to O(m) \to O(m) \vert _{C'} \to 0$. Ok, on first one I have no idea.

A couple words about the projection $\mathbb{P}^n \dashrightarrow \mathbb{P}^2$: that's nothing but the composition of the projections explaned in example 1.9 page 12: Let $p \in \mathbb{P}^n$ and $\mathbb{P}^{n-1} \cong H \subset \mathbb{P}^n$ a hyperplane not containing $p$. This induces the projection $\pi_{p,H}: \mathbb{P}^n \dashrightarrow \mathbb{P}^{n-1}$ (in the book the construction is described concretly). Moreover the book tells that sophisticated choises of points and hyperplanes allows to project down a curve $C$ birationally to $C'$ by iterating the process above & restricting to $C$.

I have a question on an unequality used in the proof of the Very weak Riemann-Roch on curves in Janos Kollar's Lecture on Resolution of Singularities (page 14):

1.13 (Very weak Riemann-Roch on curves). We claim that for any ample line bundle $L$,

$$h^0(C,L^m) \ge m \cdot deg \ L +1 -\binom {\operatorname{deg} \ L-1}{2}$$

for $ m \ge \operatorname{deg} \ L$.

(proof) Indeed, embed $C$ into $\mathbb{P}^n$ by $L$ (since $L$ ample that works), and then project it generically to a plane curve of degree $\operatorname{deg} \ L$, $\pi: C \to C' \subset \mathbb{P}^2$

few explanations: i.e. $\pi$ is the composition of the embedding $i:C \to \mathbb{P}^n$ and and projection of $\mathbb{P}^n$ to a plane $\cong \mathbb{P}^2$; it's only a rational map but if we restrict it to $C$ and chose the projection planes in clever way (see my comments below) then the projection $C \dashrightarrow \mathbb{P}^2$ becomes a morphism. $C'$ is the image of $C$ and $C'$ is birational to $C$. we continue the proof...

Now, for $m \ge \operatorname{deg} \ L$,

$$h^0(C,L^m) \ge h^0(C', \mathcal{O}_{\mathbb{P}^2}(m) \vert _{C'}) (???) \\ \ge h^0(\mathbb{P}^2, \mathcal{O}_{\mathbb{P}^2}(m)) -h^0(\mathbb{P}^2, \mathcal{O}_{\mathbb{P}^2}(m- \operatorname{deg} \ L)) \\ \ge m \cdot \operatorname{deg} \ L +1 - \binom {\operatorname{deg} \ L-1}{2}. $$

[...]

Question: I not understand the first unequality

$$h^0(C,L^m) \ge h^0(C', \mathcal{O}_{\mathbb{P}^2}(m) \vert _{C'}).$$

more concretly, I don't see how $\pi$ induces an appropriate exact sequence which allows to relate dimensions of sheaf cohomology groups $H^0(C,L^m)$ and $ H^0(C', \mathcal{O}_{\mathbb{P}^2}(m) \vert _{C'})$

In contrast, the second equality is clearly induced by $0 \to O(m-\operatorname{deg} \ L) \to O(m) \to O(m) \vert _{C'} \to 0$. Ok, on first one I have no idea.

A couple words about the projection $\mathbb{P}^n \dashrightarrow \mathbb{P}^2$: that's nothing but the composition of the projections explaned in example 1.9 page 12: Let $p \in \mathbb{P}^n$ and $\mathbb{P}^{n-1} \cong H \subset \mathbb{P}^n$ a hyperplane not containing $p$. This induces the projection $\pi_{p,H}: \mathbb{P}^n \dashrightarrow \mathbb{P}^{n-1}$ (in the book the construction is described concretly). Moreover the book tells that sophisticated choises of points and hyperplanes allows to project down a curve $C$ birationally to $C'$ by iterating the process above & restricting to $C$.

I have a question on an unequality used in the proof of the Very weak Riemann-Roch on curves in Janos Kollar's Lecture on Resolution of Singularities (page 14):

1.13 (Very weak Riemann-Roch on curves). Let $C$ be an irrecucible, reduced, projective curve over an alg closed field $k$. We claim that for any ample line bundle $L$,

$$h^0(C,L^m) \ge m \cdot deg \ L +1 -\binom {\operatorname{deg} \ L-1}{2}$$

for $ m \ge \operatorname{deg} \ L$.

(proof) Indeed, embed $C$ into $\mathbb{P}^n$ by $L$ (since $L$ ample that works), and then project it generically to a plane curve of degree $\operatorname{deg} \ L$, $\pi: C \to C' \subset \mathbb{P}^2$

few explanations: i.e. $\pi$ is the composition of the embedding $i:C \to \mathbb{P}^n$ and and projection of $\mathbb{P}^n$ to a plane $\cong \mathbb{P}^2$; it's only a rational map but if we restrict it to $C$ and chose the projection planes in clever way (see my comments below) then the projection $C \dashrightarrow \mathbb{P}^2$ becomes a morphism. $C'$ is the image of $C$ and $C'$ is birational to $C$. we continue the proof...

Now, for $m \ge \operatorname{deg} \ L$,

$$h^0(C,L^m) \ge h^0(C', \mathcal{O}_{\mathbb{P}^2}(m) \vert _{C'}) (???) \\ \ge h^0(\mathbb{P}^2, \mathcal{O}_{\mathbb{P}^2}(m)) -h^0(\mathbb{P}^2, \mathcal{O}_{\mathbb{P}^2}(m- \operatorname{deg} \ L)) \\ \ge m \cdot \operatorname{deg} \ L +1 - \binom {\operatorname{deg} \ L-1}{2}. $$

[...]

Question: I not understand the first unequality

$$h^0(C,L^m) \ge h^0(C', \mathcal{O}_{\mathbb{P}^2}(m) \vert _{C'}).$$

more concretly, I don't see how $\pi$ induces an appropriate exact sequence which allows to relate dimensions of sheaf cohomology groups $H^0(C,L^m)$ and $ H^0(C', \mathcal{O}_{\mathbb{P}^2}(m) \vert _{C'})$

In contrast, the second equality is clearly induced by $0 \to O(m-\operatorname{deg} \ L) \to O(m) \to O(m) \vert _{C'} \to 0$. Ok, on first one I have no idea.

A couple words about the projection $\mathbb{P}^n \dashrightarrow \mathbb{P}^2$: that's nothing but the composition of the projections explaned in example 1.9 page 12: Let $p \in \mathbb{P}^n$ and $\mathbb{P}^{n-1} \cong H \subset \mathbb{P}^n$ a hyperplane not containing $p$. This induces the projection $\pi_{p,H}: \mathbb{P}^n \dashrightarrow \mathbb{P}^{n-1}$ (in the book the construction is described concretly). Moreover the book tells that sophisticated choises of points and hyperplanes allows to project down a curve $C$ birationally to $C'$ by iterating the process above & restricting to $C$.