Questions tagged [hurwitz-theory]

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Which invertible linear maps preserve the set of Hurwitz stable matrices?

Let $V = M_n(\mathbb{R})$ be the set of all $n\times n$ matrices with real elements and $V_{-}$ be a subset of Hurwitz stable matrices, i.e. matrices such that all their eigenvalues have strictly ...
Fiktor's user avatar
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1 answer
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Conditions for a block matrix to be Hurwitz stable

Consider the following block matrix: $$ A = \begin{bmatrix} 0 & I\\ -M & -I \end{bmatrix} $$ Suppose matrix $M$ is positive definite and satisfies $M\succeq \alpha I$, where $\alpha>0$ is a ...
Evan's user avatar
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1 vote
2 answers
259 views

Reference request for widely used theorem

I am looking for a reference to the theorem that any oriented closed surface of genus $g$ is a 2-fold cover of $S^2$ (branched over 2$g$+2 points).
John Rached's user avatar
1 vote
0 answers
150 views

Is this an explicit construction of a Hurwitz space with Galois group Z/p, p distinct branch points, and inertia group Z/(p-1)?

I am desperately confused and would like a sanity check that the following moduli space/stack is a Hurwitz space/stack. I would also appreciate any references on the topic of the explicit construction ...
Catherine Ray's user avatar
5 votes
0 answers
254 views

Connected relative Gromov Witten invariants

I am currently interested to compute relative Gromov Witten invariants(GW) over $\mathbb{P}^1$. In the paper https://arxiv.org/pdf/math/0204305.pdf eq 3.1 gives the count of relative disconnected GW ...
GGT's user avatar
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2 votes
0 answers
101 views

Cut and Join for Hurwitz number with multiple spin

Let me introduce some background of cut and join equation for spin Hurwitz number with the completed cycle as mentioned in https://arxiv.org/pdf/1103.3120.pdf We fix two partition $\mu $ and $\nu$ of ...
GGT's user avatar
  • 685
2 votes
0 answers
65 views

A version of Hurwitz' theorem in terms of Euler characteristic

Page 203 of Farb and Margalit's Primer on Mapping Class Groups contains the result: Let $g ≥ 2$. The order of any finite subgroup of $MCG(S_g)$ is at most $84(g − 1)$. I've been told by my ...
Yossarian's user avatar
4 votes
0 answers
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Generalising definition of Hurwitz number of compactified moduli space of curve

I am asking mostly for reference if such a definition exists in the literature. I am also interested in the count if it appears somewhere. Let $\mu:=(\mu_1,\ldots , \mu_n)\vdash d$ for positive ...
GGT's user avatar
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4 votes
1 answer
480 views

fundamental domains in H^2 containing large balls

I would like to construct a genus $g$ surface regularly tiled by triangles (for example by 238 triangles). Edmunds-Ewing-Kulkarni prove that the only obstruction to doing this is Euler characteristic ...
Arielle Leitner's user avatar
8 votes
0 answers
212 views

Are spin Hurwitz numbers $r$-spin Hurwitz numbers?

(I think the answer is no, but I'm not sure.) In Hurwitz theory, one counts $n$-fold branched covers $\Sigma'\to\Sigma$ of a Riemann surface $\Sigma$ with fixed ramification data around each branch ...
Arun Debray's user avatar
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1 vote
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Cycles in the Chow ring of the moduli of curves coming from Hurwitz theory

Are there interesting cycles (other then the famous ones such as: Harris and Mumford's gonality divisor, the Gieseker-Petri divisor [which can be realized as the branch locus of a forgetful map from ...
Nati's user avatar
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2 votes
1 answer
373 views

holomorphic automorphisms of universal cover of configuration spaces

Hello everyone, I have been trying (without success) to determine the following. Let $P$ denote the space of monic polynomials of degree $n$ with complex coefficients, which have distinct roots. It ...
Venkataramana's user avatar
4 votes
3 answers
372 views

Automorphism of finite groups and Hurwitz spaces

If $G$ is a finite group, embedded as a transitive subgroup of $S_n$ for some $n$, will every automorphism of $G$ extend to an inner automorphism of $S_n$? I'm trying to connect the language that's ...
Will Chen's user avatar
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5 votes
1 answer
340 views

Spaces parametrizing ramified covers of surfaces

Let $\Sigma$ be a surface (let's say oriented and of finite type). We can consider the configuration space $F(\Sigma,n)$ of $n$ ordered distinct points on $\Sigma$, i.e. $\Sigma^n\setminus \Delta$ ...
Dan Petersen's user avatar
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