Monodromy of rational curves on toric surfaces

Abstract

For an ample line bundle L on a complete toric surface X, we consider the subset V L ⊂ |L| of irreducible, nodal, rational curves contained in the smooth locus of X. We study the monodromy map from the fundamental group of V L to the permutation group on the set of nodes of a reference curve C ∈ V L. We identify a certain obstruction map Ψ X defined on the set of nodes of C and show that the image of the monodromy is exactly the group of deck transformations of Ψ X , provided that L is sufficiently big (in the sense we make precise below). Along the way, we construct a handy tool to compute the image of the monodromy for any pair (X, L). Eventually, we present a family of pairs (X, L) with small L and for which the image of the monodromy is strictly smaller than expected.

A note on the Severi problem for toric surfaces

“…For planar curves, the monodromy acts as the full symmetric group by [11]. More generally, this is the case if the toric surface is smooth at (at least) one of its zero-dimensional orbits, and the polarization is ample enough; see [18] for details. Thus, the upper bound in such cases is one.…”

“…In 2013, the second author found first examples of reducible Severi varieties on toric surfaces, initially in positive characteristic [27], and then in characteristic zero [28]. A different type of examples was discovered recently by the first author in his study of the monodromy action on the set of nodes of rational curves on toric surfaces [18].…”

“…The goal of the current paper is to make a step towards a classification of toric surfaces admitting reducible Severi varieties, and towards a description of the corresponding irreducible components. We unify the examples of [18,27,28], and introduce certain invariants that distinguish between components of Severi varieties on toric surfaces. As a result, we are able to provide lower bounds on the numbers of the irreducible components.…”

Sparse polynomial equations and other enumerative problems whose Galois groups are wreath products

“…It is a widely known empirical phenomenon that Galois groups of interesting enumerative problems (including the ones mentioned in Example 2.1 below, see for instance [24,27,29] and [15]) are either imprimitive or "trivial" (i.e. symmetric or alternating).…”

“…Remark 4. 24 In other words, f is in the A-bifurcation set if Z ( f γ ) is not regular for some γ . In particular, the definition of the A-bifurcation set does not depend on the choice of the fan compatible with A.…”