\begin{document}

\title{Errata for paper\\
``The Di Francesco--Itzykson--G\"ottsche\\
Conjectures for Node Polynomials of $\mathbb{P}

\noindent (I) Lemma 2.1 of this paper contains a small error: Since $\mathscr{L}^2$ is always even for a line bundle $\mathscr{L}$ on an Abelian surface, we only get that $n_1$ is an integer \textit{or} a half-integer. Fortunately, this is easily repaired, and does not affect the rest of the paper. We propose a corrected statement and proof of this lemma: \\

\noindent \textbf{Lemma 2.1.} Suppose $n_{i}, 1 \leq i \leq 4,$ are rational numbers satisfying the following requirements:
\begin{enumerate}[(1)]
\item $n_1$ and $n_4$ are integers;
\item for all polarized, smooth, irreducible projective surfaces $(S,\mathscr{L}),$ the number $n_{1}\partial + n_{2} k + n_{3} s + n_{4} x$ is an integer.
\end{enumerate}
Then each $n_{i} \in \mathbb{Z}.$

\begin{proof}
Recall that $\mathbb{P}^{2}$ has Chern numbers $\mathscr{K}_{\mathbb{P}^{2}}^{2} = 9$ and $c_{2}(\mathbb{P}^{2}) = 3.$ Blowing up $\mathbb{P}^{2}$ in a point yields a surface $\mathbb{F}_1$ which has Chern numbers 8 and 4. Let $\mathscr{L}$ be the structure sheaf on $\mathbb{P}^{2},$ then from (2) it follows that $9n_3 + 3n_4 \in \mathbb{Z}.$ Next, consider the structure sheaf on $\mathbb{F}_1,$ so that we get $8n_3 + 4n_4 \in \mathbb{Z}.$ Hence, taking the difference, $n_3-n_4 \in \mathbb{Z}.$ But $n_4$ is an integer, so $n_3$ is as well. Now, let $S$ be the Hirzebruch surface $\mathbb{F}_{3},$ and denote by $E$ the class of the exceptional curve, and $F$ the class of a fiber. Let $\mathscr{L}$ be $\mathscr{O}(1,1) := \mathscr{O}(E+F).$ Then
\begin{eqnarray*}
k & = & \mathscr{K}_{S} \cdot \mathscr{L} \\
& = & (-2E - 5F) \cdot (F + E) = -2EF - 2E^2 - 5F^2 -5FE \\
& = & -2 - 2\cdot (-3) - 0 - 5 = -1,
\end{eqnarray*}
since the canonical divisor is $-2E-(n+2)F$ on $\mathbb{F}_{n}.$ It follows from (1) and (2) that $n_2$ is also an integer.
\end{proof}

So we must prove that $n_1$ and $n_4,$ i.e., the coefficients of respectively $\partial$ and $x$ in the $q^{n}$-term of $\frac{d}{dq}\log \phi(S,\mathscr{L})(q),$ are integers. The fact that $n_1$ is an integer follows direcly from the computation of the $D_{n},$ cf. Eqn. (3.9). Now, a similar procedure, only using a K3 surface $S$ with $\mathscr{L} = \mathscr{O}_{S}$ so that the Chern numbers $\partial, k, s$ are 0 while $x \neq 0,$ allows for an explicit proof that also $n_4,$ the coefficient of $x,$ is an integer, through the G\"ottsche--Yau--Zaslow formula. The process is exactly as in the proof of Prop. 3.2, only with sligthly different power series.\\

\noindent (II) The notion of $r$-ample is mentioned throughout the paper. The correct expression is $r$-very ample.

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