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We consider the special roles of the zero loci of the Weierstrass invariants$g_2(\tau(z))$, $g_3(\tau(z))$ in F-theory on an elliptic fibration over $P^1$or a further fibration thereof. They are defined as the zero loci of thecoefficient functions $f(z)$ and $g(z)$ of a Weierstrass equation. They arethought of as complex co-dimension one objects and correspond to the two kindsof critical points of a dessin d'enfant of Grothendieck. The $P^1$ base isdivided into several cell regions bounded by some domain walls extending fromthese planes and D-branes, on which the imaginary part of the $J$-functionvanishes. This amounts to drawing a dessin with a canonical triangulation. Weshow that the dessin provides a new way of keeping track of mutualnon-localness among 7-branes without employing unphysical branch cuts or theirbase point. With the dessin we can see that weak- and strong-coupling regionscoexist and are located across an $S$-wall from each other. We also present asimple method for computing a monodromy matrix for an arbitrary path by tracingthe walls it goes through.

A dessin on the base: a description of mutually non-local 7-branes without using branch cuts