= Solution
A <symplectic blowup> replaces a small <symplectic ball> around the point by an <exceptional divisor> $\mathbb{CP}^{n-1}$, for $\dim_{\mathbb R}X=2n$. One can construct its form by <symplectic reduction>. In a <Darboux chart>, consider $\mathbb C^n\times\mathbb C$ with the <circle action>
$$
e^{it}\cdot(z,w)=(e^{it}z,e^{-it}w),
\qquad \mu(z,w)=\frac{|z|^2-|w|^2}{2}.
$$
On $\mu^{-1}(R^2/2)$ the action is free because $z\ne0$. Its quotient therefore carries a smooth <symplectic form>. Where $w\ne0$, choose the representative with $w$ positive real; the reduced form is the original form on $|z|>R$. Where $w=0$, the quotient is the <Hopf fibration> quotient of $|z|=R$, namely $\mathbb{CP}^{n-1}$. The local quotient is the tautological complex line bundle near its zero section, hence the local blowup. Glue this model to the exterior of the ball to obtain a <symplectic form> on the blowup.
The area of a projective line in the exceptional divisor is $\varepsilon=\pi R^2$. For a closed $X$, changing this size changes the <symplectic volume>:
$$
\operatorname{Vol}(\widetilde X,\widetilde\omega)
=\operatorname{Vol}(X,\omega)-\frac{\varepsilon^n}{n!}.
$$
Consequently \b[the blowup is not determined up to symplectomorphism without a size choice]. The underlying smooth blowup is fixed, but the symplectic construction has a parameter, as in <symplectic blowup size changes volume>.
For the blowdown map $\pi$ and exceptional divisor $E$, the <First Chern class> is
$$
\boxed{c_1(T\widetilde X)=\pi^*c_1(TX)-(n-1)\operatorname{PD}[E]}.
$$
In real dimension four this becomes $\pi^*c_1(TX)-\operatorname{PD}[E]$, the <first Chern class formula for a symplectic blowup>.
The tangent bundle of the standard four-<torus> is a trivial complex rank-two bundle, so its <symplectic canonical class> is zero. For a generic fiber $F$ of a symplectic <Lefschetz pencil>, the <symplectic adjunction formula> gives $2g-2=F^2$. Two generic fibers intersect precisely at the base points, each with local <intersection number> one in the pencil model $[z_1:z_2]$. Hence
$$
\boxed{\#\{\text{base points}\}=F^2=2g-2}.
$$
This is <base-point count for a Lefschetz pencil on a symplectic four-torus>.
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