This is my report on the pdf file 071213-Vetro-v1.pdf authored by Francesca Vetro, entitled "On Hurwitz spaces of coverings with one special fiber." To an outsider to the goals of the paper, it might seem dry. Still, I recommend publication of the paper. The problem of figuring Hurwitz space components is valuable in many applications, and the author setsup situations that naturally arise thanks to a cool classification result of Kanev. Then the paper handles those situations with considerable combinatorial agility. The report has three sections: I. THE SETUP OF THE PAPER: II. WHAT THE AUTHOR PROVES: III. EDITORIAL COMMENTS AND A QUESTION THE AUTHOR MIGHT ANSWER BETTER: Sincerely, the referee I. THE SETUP OF THE PAPER: In studying algebraic curves it might seem excitingly prestigious to deal with the moduli of curves of genus g in one lump (that is, Teichm\"uller theory approaches). In practical applications, however, serious progress comes most often from dealing with curves through covering space theory. I retain much (but not all) of the author's notation in this report. The study of curve moduli through their period matrices illustrates that. Using the Poincare complete reducibility theorem, tremendous numbers of papers used degree 2 covers X \to X' of curves X' of genus g to form a component (called a Prym) of the Jacobian variety of X. The periods P'_{X'} of these Pryms can be expressed from the periods P_X of X. Then, plugging P'_{X'} into general period relations of Riemann (for an abelian variety of the dimension of the Prym) gives equations satisfied by P_{X'}. Such equations were used for over a 120 years to differentiate between the periods of a general abelian variety of dimension g and the Jacobians of curves of genus g. The author mentions Pryms -- much more briefly than I just did -- as one serious motivation for her study. If you further move the curve X' around in special families, then the nature of the Prym's is just one question you might have about the nature of the family. A covering space approach to considering families would use a branched cover of one curve Y by another: \phi': X' \to Y. Then, fix Y (a reference curve), but vary X' by varying just the branch points on Y. Example: Take Y = P^1 (the projective line), use simple branching (at most two sheets come together over a branch point), and use enough branch points, and your family will contain the general curve of genus g. To make this work well, you need all the possible \phi', but by a method that won't force you into having to write out equations. That is the author's approach: Describing connected components of these families of covers \phi': X' \to Y, in the service say of understanding the different types of Pryms that might arise from this situation. So, in each case Vetra is considering \phi' together with an unramified (connected) degree 2 cover \phi: X \to X'. She allows the branch points to be any set of r points on Y. The algebraic invariants now forced on the author are the nature of the branching B of \phi', and the monodromy group G of the cover X\to Y (Galois group of the smallest Galois cover of Y factoring through X) from forming the composite of \phi and \phi'. These will be constant as you move the branch points of \phi'. So, if you want connected components, you can phrase everything the author does using the pair (B,G): It makes sense to describe the Hurwitz space of all such covers of Y. The author's goal is to describe the components of that Hurwitz space with (Y,B,G) fixed. Indeed, Riemann's Existence Theorem shows that only the genus g of Y will matter. That is, you can drag the whole Hurwitz space construction along a deformation of Y. Since G is the monodromy group of a cover of Y, it comes automatically with a permutation representation T of degree 2d where d is the degree of \phi'. So, evaluating what the author does comes down to how impressed you are with Vetra's choices of (B,G) and whether these connectedness results seem substantial. I think they are. My comments are intended to improve the readability of the paper, so I hope they are understandable and then taken to heart. Part of my positive recommendation comes from the savvy use of groups. I use some notation below in making recommendations to the author. Note: G maps to G' (always S_d in this paper), the monodromy group of \phi'. Indeed G is a subgroup of the semidirect product, (Z/2)^d\xs G', of (Z/2)^d and G'. The classical people who did Pryms -- I don't mention names, and of course, I exclude some -- mostly knew no group theory (not a whit). Intuitively they understood that some Pryms (from covers X\to X') coming from an X' were different from others. Yet, they didn't have the tools to make those distinctions, valuable in all considerations of Hurwitz spaces. One virtue of this paper is that it does. I say how in the next section. II. WHAT THE AUTHOR PROVES: The author lists some previous connectedness results, starting with those that deal with just \phi'. Those stay close to Clebsch's 1872 simple branching result in that they use r branch points, all but one of which give simple branching. The author lists results where r-1 branch points are simple, and one is given by any fixed partition (I call this \be) of the integer d. The index i(\be) (the author calls this |\be|), as it arises in the Riemann-Hurwitz formula, is significant. The author starts with Y=P^1 by Natanzon and Kluitmann (independently; here it is automatic that G'=S_d, so long as the 2-cycles are transitive), then by Kanev and Vetro (separate papers) where Y has genus g\ge 1, and G=S_d). In Kanev-Vetro it is no longer automatic that G'=S_d even if the subgroup of S_d generated by entries of a Hurwitz system is transitive. For example, you could use any transitive subgroup G' of S_d that is a quotient of the fundamental group of Y to allow having r=0. So Vetro adds G'=S_d as part of the condition on G. Vetro showed there is just one (Hurwitz space) component when (*) r-1 + i(\be) \ge 2d. Biggers-Fried (Y=P^1) and Kanev (general Y) considered the Prym situation where B is r points of simple branching and G is a Weyl group. When Y=P^1 (and X\to P^1) is simple, the Weyl group D_d was forced (from connectedness of X). One way to write D_d is as the index 2 subgroup of (Z/2)^d\xs S_d consists of elements whose 1st slot entries in (Z/2)^d sum to 0. Vetro says everything in terms of root systems. In this case D_b contains reflections from the long roots, but not from the short roots. When you switch to Y of genus g\ge 1, you can get those long root reflections. On p. 2, the overly long 3rd paragraph summarizes nice results that should have a reader realize the author is onto something. Especially that hypothesis (*) is of great significance. Although this calls for the following digression. Here the author allows \phi (the degree 2 map) to ramify, and that allows the branch cycles (local monodromys in the author's language) to include reflections in both the long and short roots. In that case exactly two groups can arise as G (assuming (*)). When it is D_d (the same as Biggers-Fried) you get just one component, and when G is B_d, the whole semidirect product, (Z/2)^d\xs S_d, you get 2^{2g}-1 components. In the cases in this paper, \phi does not ramify, but g\ge 1. Then there are three possibilities for G, the two above, and one other, the group generated by the natural copy of S_d in (Z/2)^d\xs S_d and the element from (Z/2)^d that has 1 in every slot. Then, the B_d case and the other case have 2^{2g}-1 components (again assuming (*)). III. EDITORIAL COMMENTS AND A QUESTION THE AUTHOR MIGHT ANSWER BETTER: III.1. COMMENTS ON THE WRITING: 1. The author's introduction is more like my section II, very unlike my section I. A little mixing with section I talk might make the paper more accessible. As it stands it looks specialized. In fact it should appeal to many interested in such connectedness results. Example: That there are just three possible groups G in case \phi' does not ramify is a result of Kanev-Lange. The author quotes this on p. 6. I'd suggest saying something about this in the intro, for it shows the cleverness of the Weyl group approach. 2. Break the 3rd paragraph of p. 2 into two parts (a little like in section II above), separating the case \phi is ramified -- allowing reflections with respect to short roots as local monodromy generators --- from the case \phi is unramified, g\ge 1, and three Weyl groups can arise. 3. In quoting Kanev's results in Prop. 1, I can't see the point in naming a braid, and then it's inverse, separately. The braids form a group in their actions on "Hurwitz systems" and the whole notation is done very inefficiently. I do the easiest case to give an example: The elementary move \sigma_i fixes all \lambda s and \mu s, all the t_i s with i \ne j or j+1, and it sends (t_j, t_{j+1}) to (t_jt_{j+1}t_j^{-1},t_j). I understand the others are more complicated, but the author can simplify easily. There is an unnecessary formality that should use a statement like the paragraph at the top of p. 6 early. Likewise, Defs. 1-4 are inefficient and much too long. Example: Before Def. 3 define the product of commutations from (\lambda_1,\mu_1,..., \lambda_g,\mu_g) using a simpler notation [\blambda,\bmu] where \blambda and \bmu are bold. Also, before the definition, remind of the classical case when g=0. For Def. 3: A Hurwitz system with values in H is a 2g+n-tuple (t_1,...,t_n;\lambda_1,\mu_1,..., \lambda_g,\mu_g)=(\bt,\blambda,\bmu) \in H^{2g+n} satisfying t_1\cdots t_n=[\blambda,\bmu]. Its entries generate the monodromy group of the system. We equivalence (\bt,\blambda,\bmu) and h(\bt,\blambda,\bmu)h^{-1} for h\in H. Use [\bt,\blambda,\bmu] for this equivalence class. 4. The author's first language is not English, though her English is good. Still, it is possible that part of my problem with Def. 1 is with English. Here is how I would say what is a positive cycle in W(B_d) using the natural image h: S_d \to (Z/2)^d\xs S_d in the style I discussed the 3rd group in the last paragraph of section II. A positive e-cycle in W(B_d) is an element of order e of form (a_1,...,a_d; \sigma) where \sigma is an e-cycle of S_d, and the a_is are 0 unless i is supported in \sigma. Example: With d=3, (0,1,1; (1 2 3)) is a positive 3-cycle, but (1,1,1; (1 2 3)) is not. We say it is negative. III.2. THE FINAL CONCLUSIONS OF THEOREMS 1 AND 2: I'm certain that with simplifications in notation that I've suggested above, these results will not seem so technical. The crucial goal of the author should be to assure the reader can assimilate why the braid representatives listed in Props. 3 and 4 are braid inequivalent. It would also be nice if the statement -- end of Theorem 1 -- that "we can replace .... with a product of disjoint positive cycles, but not with a negative cycle" could be said more geometrically. Ditto with the corresponding statement at the end of Theorem 2.