= Solution
For a <symplectic ball> centered at the origin, its <symplectic blowup> replaces a smaller concentric ball by the disk bundle in $\mathcal O_{\mathbb{CP}^{n-1}}(-1)$. The zero section is the <exceptional divisor> $E\cong\mathbb{CP}^{n-1}$; the blowdown collapses $E$ to the original point and is a <symplectomorphism> away from $E$. The new <symplectic form> agrees with $\omega_0$ outside the surgery region, while its integral over a line in $E$ specifies the blowup size. In real dimension four, $E\cong\mathbb{CP}^1$ and $E^2=-1$.
A <Lefschetz pencil> on the compact oriented four-manifold $X$ is a map
$$
f:X\setminus B\longrightarrow\mathbb{CP}^1
$$
with finite base locus $B$, such that orientation-compatible complex coordinates give the local model $f(z_1,z_2)=[z_1:z_2]$ at a base point and $f(z_1,z_2)=z_1^2+z_2^2$ at each critical point. Blowing up all $b=|B|$ base points produces a <Lefschetz fibration> $\widetilde X\to S^2$ whose regular fiber is the closed smooth fiber of the pencil.
If that fiber has <genus> $g$ and there are $m$ critical points, then
$$
\chi(\widetilde X)=\chi(S^2)\chi(\Sigma_g)+m
=2(2-2g)+m.
$$
Every blowup raises the <Euler characteristic> by one, so $\chi(\widetilde X)=\chi(X)+b$. Therefore the <Euler characteristic of a Lefschetz pencil> is
$$
\boxed{\chi(X)=4-4g+m-b}.
$$
Now let $X=\mathbb{CP}^2$ with its <Fubini-Study form>. A symplectic smooth fiber $F$ represents $dH\in H_2(\mathbb{CP}^2;\mathbb Z)$ for some positive integer $d$, because its symplectic area is positive. The <Symplectic adjunction formula>, using $K_{\mathbb{CP}^2}=-3H$, gives
$$
2g(F)-2=F^2+K_{\mathbb{CP}^2}\mathbin{\cdot}F=d^2-3d,
$$
and hence
$$
\boxed{g(F)=\frac{(d-1)(d-2)}2},\qquad d=1,2,3,\ldots.
$$
These values are $0,0,1,3,6,10,\ldots$. They are all attained: two sufficiently general homogeneous polynomials of degree $d$ generate a pencil of complex plane curves with only the singularities allowed in a <Lefschetz pencil>, and its smooth fibers are symplectic. Thus the displayed list is exactly the list described by the <Genus of a symplectic Lefschetz pencil on the complex projective plane>.
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