Abstract
We introduce a new technique to prove connectivity of subsets of covering spaces (so called inductive connectivity), and apply it to Galois theory of problems of enumerative geometry. As a model example, consider the problem of permuting the roots of a complex polynomial f(x) = c_0 + c_1 x^{d_1} + cdots + c_k x^{d_k} by varying its coefficients. If the GCD of the exponents is d, then the polynomial admits the change of variable y=x^d, and its roots split into necklaces of length d. At best we can expect to permute these necklaces, i.e. the Galois group of f equals the wreath product of the symmetric group over d_k/d elements and {mathbb {Z}}/d{mathbb {Z}}. We study the multidimensional generalization of this equality: the Galois group of a general system of polynomial equations equals the expected wreath product for a large class of systems, but in general this expected equality fails, making the problem of describing such Galois groups unexpectedly rich.
Highlights
Proposition 1.5 For d := GCD(a1, . . . , ak), we have G A = (Z/dZ) Sa/d, i.e. the monodromy group G A includes all permutations of the roots of the Eq (1) that preserve the necklace structure. It was proved in [9, Theorem 2.1], motivated by the study of linear recurrent sequences over functional fields. We shall deduce it as a special case of a certain new general fact about the monodromy of enumerative problems: it is a consequence of Wreath product theorem 3.7 and Lemma 3.8, see Sect. 3.2
Already in dimension 2, there exist non-reduced supports Afor which the inclusion (2) is proper, see Example 1.13 below. This observation turns the determination of Galois groups of the form G Ainto an unexpectedly rich and challenging problem that can be addressed in two steps: (A) determine the irreducible non-reduced supports Afor which (2) is an isomorphism, (B) compute the G Awhenever (2) is a proper inclusion
The Galois group G of the φ-wreath enumerative problem U is contained in W := D G. This observation directly follows from the definitions of the wreath problem and the wreath group, and is widely known in the algebraic Galois theory
Summary
We will denote the wreath product by H G This group can be seen as the group of all permutations σ of the set H × S, satisfying the following properties: 1) σ can be included into the commutative diagram. Ak), we have G A = (Z/dZ) Sa/d , i.e. the monodromy group G A includes all permutations of the roots of the Eq (1) that preserve the necklace structure It was proved in [9, Theorem 2.1], motivated by the study of linear recurrent sequences over functional fields. We shall deduce it as a special case of a certain new general fact about the monodromy of enumerative problems: it is a consequence of Wreath product theorem 3.7 and Lemma 3.8, see Sect. Remark 1.6 It would be interesting to give this fact an algebraic proof and to understand to what extent it survives the positive characteristic
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