Reading on the tropical approach to enumerative geometry I have come across the claim: given a projective toric surface from a polygon P, we can consider a tautological bundle of algebraic / holomorphic functions with newton polygon contained in P (i.e. take functions that in the dense torus can be written as a combination of monomial corresponding only to points in P).

The claim I've come across (in Mikhalkin's paper [1] (section on severi varieties) or in Shustin's book [2] (proof of lemma 2.42 page 59-61) is that if we take the zeroes of such a function from our bundle, then in the general case it is a curve that is transversal to the boundary divisors of the toric surface. Moreover, "Any curve (which is not a toric divisor) crosses at least two toric divisors; take the Newton polygon of a curve equation, then its intersections with the toric divisors are as follows: for each side of that Newton polygon equipped with the outer normal, take the parallel supporting line for the polygon P defining the toric surface, if the intersection is a vertex, then the curve passes through the corresponding intersection point of toric divisors, if the intersection is a side, then the curve crosses the corresponding toric divisor; for a Newton segment (binomial curve), we take it twice - with both normals." [I quote an explanation given to me on what are 'rules' of intersection]

A) I could not find any proof or explanation why it is true. I am currently less interested in a fully rigorous proof but more in an explanation why this is true.

B) Another question come to my mind, does any other curve on the toric surface must come from zeroes of a function from the tautological bundle? if not, why do authors focus so much on curves that are zeroes of this bundle in the context of counting curves via n points? (i.e. GW invariants)?

Thanks!

And i apologize if this was a bit cumbersome way of asking my question, when things are not clear enough, even asking properly is somewhat difficult :)

References