Let $(M,\omega,T^2,\mu)$ be a compact connected symplectic toric $4$-manifold, and let $\Delta=\mu(M)\subset\mathfrak t^*$ be its moment polygon with the integral affine structure determined by the weight lattice of $T^2$. Let $p$ be a $T^2$-fixed point and set $v=\mu(p)$. Choose integral affine coordinates $(x_1,x_2)$ on a neighbourhood of $v$ in $\mathfrak t^*$ such that $v$ is sent to $(0,0)$ and the two facets of $\Delta$ adjacent to $v$ are sent locally to the positive coordinate axes.

Use the normalization in which the standard Hamiltonian $T^2$-action on $\mathbb C^2$ has moment map

\begin{align*} \mu_{\mathbb C^2}(z_1,z_2)=(\pi |z_1|^2,\pi |z_2|^2). \end{align*}

Then there exists $\varepsilon_0>0$ such that, for every $0<\varepsilon<\varepsilon_0$, the toric symplectic blow-up of $(M,\omega,T^2,\mu)$ at $p$ of capacity $\varepsilon$ has moment polygon obtained from $\Delta$ by replacing the corner at $v$ with the facet

\begin{align*} x_1+x_2=\varepsilon. \end{align*}

Equivalently, in the chosen integral affine coordinates near $v$, the blown-up moment polygon is

\begin{align*} \Delta\cap\{x_1+x_2\ge \varepsilon\}, \end{align*}

so the two adjacent facets are cut at integral-affine distance $\varepsilon$ from $v$.