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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)?


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 :)


[1] Mikhalkin's papaer
[2] Shustin's (et al) book

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up vote 2 down vote accepted

The answer to your question B is yes and no :) You see: replacing $P$ by $kP$ for any $k \geq 1$ gives rise to the same toric surface (the latter is precisely the $k$-uple Veronese embedding of the former). And, the defining polynomial of any curve will fit (up to translation) in $kP$ for a sufficiently large $P$.

For the phenomenon in your first question, I can give an answer in the case that the underlying field $\mathbb{k}$ is algebraically closed. Here it goes: let $f$ be the Laurent polynomial defining the curve $C$ and $S$ is an edge of the Newton Polygon $N$ of $f$. Let $\nu := (p, q)$ be an outward pointing (with respect to $N$) normal to $S$. For simplicity assume $q > 0$. Then there is a branch of $C$ with degree-wise Puisuex series of the form: $\gamma(t) = (t^p, \sum_{k = 0}^\infty a_k t^{q_k})$, where $q = q_0 > q_1 > \cdots $ are rational numbers with bounded denominators. Now let $\psi_P: X_P \to \mathbb{P}^N$ (where $N := |P \cap \mathbb{Z}^n| - 1$) be the embedding of the toric variety defined by the monomials in $P$, i.e. the restriction of $\psi_P$ to $(\mathbb{k}^*)^n$ is given by: $\psi_P(x) := [x^{\alpha_0}: \cdots : x^{\alpha_N}]$, where $P \cap \mathbb{Z}^n = \lbrace \alpha_0, \ldots \alpha_N \rbrace$. Let $Q$ be the face (i.e. edge or vertex) of $P$ such that $\nu$ is an outer normal to $Q$. W.l.o.g. assume that $Q \cap \mathbb{Z}^n = \lbrace \alpha_0, \ldots, \alpha_q \rbrace$, $q < N$. Then precisely the first $q+1$ coordinates of $ x := \lim_{t \to \infty} \psi_P(\gamma(t))$ are non-zero, i.e. $x$ belongs to the subvariety of $X_P$ determined by $Q$.

Remark: Usually in the books on toric varieties, inner normals are used instead of outer normals. That gives rise to a usual Puiseux series (the exponents being increasing , as opposed to the one in the preceding paragraph). But then the point at infinity (on the curve) is approached as $t \to 0$. I prefer that one approaches the point at infinity as $t$ approaches infinity as well.

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Wow, Thanks :) You really cleared up many things for me :) But I still have some bits of things that puzzle me: I understand that the notion of a parametrization of a branch (with Puiseux series) is a local one, around a point. how do we use it to get some data not near the point around which we developed the series? By the way, could you refer me to some book/article/survey where this is discussed? I could'nt find it in Fulton's book (maybe I missed it due to different terminology or something?) Thanks again – Blade Apr 23 '11 at 12:57

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