Let $X_k$ be $\mathbb{P}^2$ blown up at $k$ points (where $k$ is $0$ to $8$). Let $\beta \in H_2(X_k, \mathbb{Z}) $ be a homology class given by $$ \beta := n L + m_1 E_1 + \ldots + m_k E_k $$ where $L$ is the homology class of a line and $E_i$ are the exceptional divisors. My question is as follows:

Which homology classes are $\textit{indecomposable}$? By definition, a homology class is indecomposable if:

a) It can be represented by a non constant holomorphic map $u:\mathbb{P}^1 \longrightarrow X_k $ and

b) It cannot be written as $\beta = \beta_1 + \ldots \beta_n$ for some $n \geq 2$ such that each $\beta_i$ has a non constant holomorphic representative (as a map from $\mathbb{P}^1 $ to $X_k$).

My motivation for asking the question is as follows: I am explicitly trying to work out what is $N_{\beta}$, the number of rational curves in $X_k$ (through the right number of generic points) that represent the class $\beta$. Kontsevich and Mannin have given a recursive formula for this number in their paper (page 29)

http://www.ihes.fr/~maxim/TEXTS/WithManinCohFT.pdf

In order to actually calculate what is $N_{\beta}$, we need some initial conditions. I think the initial condition is that $N_{\beta} =1$ if $\beta$ is indecomposable.

$\textbf{Added Later:}$ Based on Mark's observation (and one further question I have about Kontsevich Mannin's paper) I have posted a separate question on mathoverflow

Are genus zero Gromov Witten Invariants on Del-Pezzo surfaces enumerative?

Let $X_k$ be $\mathbb{P}^2$ blown up at $k$ points (where $k$ is $0$ to $8$). Let $\beta \in H_2(X_k, \mathbb{Z}) $ be a homology class given by $$ \beta := n L + m_1 E_1 + \ldots + m_k E_k $$ where $L$ is the homology class of a line and $E_i$ are the exceptional divisors. My question is as follows:

Which homology classes are $\textit{indecomposable}$? By definition, a homology class is indecomposable if:

a) It can be represented by a non constant holomorphic map $u:\mathbb{P}^1 \longrightarrow X_k $ and

b) It cannot be written as $\beta = \beta_1 + \ldots \beta_n$ for some $n \geq 2$ such that each $\beta_i$ has a non constant holomorphic representative (as a map from $\mathbb{P}^1 $ to $X_k$).

My motivation for asking the question is as follows: I am explicitly trying to work out what is $N_{\beta}$, the number of rational curves in $X_k$ (through the right number of generic points) that represent the class $\beta$. Kontsevich and Mannin have given a recursive formula for this number in their paper (page 29)

http://www.ihes.fr/~maxim/TEXTS/WithManinCohFT.pdf

In order to actually calculate what is $N_{\beta}$, we need some initial conditions. I think the initial condition is that $N_{\beta} =1$ if $\beta$ is indecomposable.

$\textbf{Added Later:}$ Based on Mark's observation (and one further question I have about Kontsevich Mannin's paper) I have posted a separate question on mathoverflow

Are genus zero Gromov Witten Invariants on Del-Pezzo surfaces enumerative?

Let $X_k$ be $\mathbb{P}^2$ blown up at $k$ points (where $k$ is $0$ to $8$). Let $\beta \in H_2(X_k, \mathbb{Z}) $ be a homology class given by $$ \beta := n L + m_1 E_1 + \ldots + m_k E_k $$ where $L$ is the homology class of a line and $E_i$ are the exceptional divisors. My question is as follows:

Which homology classes are $\textit{indecomposable}$? By definition, a homology class is indecomposable if:

a) It can be represented by a non constant holomorphic map $u:\mathbb{P}^1 \longrightarrow X_k $ and

b) It cannot be written as $\beta = \beta_1 + \ldots \beta_n$ for some $n \geq 2$ such that each $\beta_i$ has a non constant holomorphic representative (as a map from $\mathbb{P}^1 $ to $X_k$).

My motivation for asking the question is as follows: I am explicitly trying to work out what is $N_{\beta}$, the number of rational curves in $X_k$ (through the right number of generic points) that represent the class $\beta$. Kontsevich and Mannin have given a recursive formula for this number in their paper (page 29)

http://www.ihes.fr/~maxim/TEXTS/WithManinCohFT.pdf

In order to actually calculate what is $N_{\beta}$, we need some initial conditions. I think the initial condition is that $N_{\beta} =1$ if $\beta$ is indecomposable.

Let $X_k$ be $\mathbb{P}^2$ blown up at $k$ points (where $k$ is $0$ to $8$). Let $\beta \in H_2(X_k, \mathbb{Z}) $ be a homology class given by $$ \beta := n L + m_1 E_1 + \ldots + m_k E_k $$ where $L$ is the homology class of a line and $E_i$ are the exceptional divisors. My question is as follows:

Which homology classes are $\textit{indecomposable}$? By definition, a homology class is indecomposable if:

a) It can be represented by a non constant holomorphic map $u:\mathbb{P}^1 \longrightarrow X_k $ and

b) It cannot be written as $\beta = \beta_1 + \ldots \beta_n$ for some $n \geq 2$ such that each $\beta_i$ has a non constant holomorphic representative (as a map from $\mathbb{P}^1 $ to $X_k$).

My motivation for asking the question is as follows: I am explicitly trying to work out what is $N_{\beta}$, the number of rational curves in $X_k$ (through the right number of generic points) that represent the class $\beta$. Kontsevich and Mannin have given a recursive formula for this number in their paper (page 29)

http://www.ihes.fr/~maxim/TEXTS/WithManinCohFT.pdf

In order to actually calculate what is $N_{\beta}$, we need some initial conditions. I think the initial condition is that $N_{\beta} =1$ if $\beta$ is indecomposable.

$\textbf{Added Later:}$ Based on Mark's observation (and one further question I have about Kontsevich Mannin's paper) I have posted a separate question on mathoverflow

Are genus zero Gromov Witten Invariants on Del-Pezzo surfaces enumerative?