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Search: MSC category 14H10 ( Families, moduli (algebraic) )

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1. CJM Online first

Xia, Eugene Z.
The algebraic de Rham cohomology of representation varieties
The $\operatorname{SL}(2,\mathbb C)$-representation varieties of punctured surfaces form natural families parameterized by monodromies at the punctures. In this paper, we compute the loci where these varieties are singular for the cases of one-holed and two-holed tori and the four-holed sphere. We then compute the de Rham cohomologies of these varieties of the one-holed torus and the four-holed sphere when the varieties are smooth via the Grothendieck theorem. Furthermore, we produce the explicit Gauss-Manin connection on the natural family of the smooth $\operatorname{SL}(2,\mathbb C)$-representation varieties of the one-holed torus.

Keywords:surface, algebraic group, representation variety, de Rham cohomology
Categories:14H10, 13D03, 14F40, 14H24, 14Q10, 14R20

2. CJM 2016 (vol 69 pp. 613)

Moon, Han-Bom
Mori's Program for $\overline{M}_{0,7}$ with Symmetric Divisors
We complete Mori's program with symmetric divisors for the moduli space of stable seven-pointed rational curves. We describe all birational models in terms of explicit blow-ups and blow-downs. We also give a moduli theoretic description of the first flip, which has not appeared in the literature.

Keywords:moduli of curves, minimal model program, Mori dream space
Categories:14H10, 14E30

3. CJM 2009 (vol 61 pp. 109)

Coskun, Izzet; Harris, Joe; Starr, Jason
The Ample Cone of the Kontsevich Moduli Space
We produce ample (resp.\ NEF, eventually free) divisors in the Kontsevich space $\Kgnb{0,n} (\mathbb P^r, d)$ of $n$-pointed, genus $0$, stable maps to $\mathbb P^r$, given such divisors in $\Kgnb{0,n+d}$. We prove that this produces all ample (resp.\ NEF, eventually free) divisors in $\Kgnb{0,n}(\mathbb P^r,d)$. As a consequence, we construct a contraction of the boundary $\bigcup_{k=1}^{\lfloor d/2 \rfloor} \Delta_{k,d-k}$ in $\Kgnb{0,0}(\mathbb P^r,d)$, analogous to a contraction of the boundary $\bigcup_{k=3}^{\lfloor n/2 \rfloor} \tilde{\Delta}_{k,n-k}$ in $\kgnb{0,n}$ first constructed by Keel and McKernan.

Categories:14D20, 14E99, 14H10

4. CJM 2008 (vol 60 pp. 297)

Bini, G.; Goulden, I. P.; Jackson, D. M.
Transitive Factorizations in the Hyperoctahedral Group
The classical Hurwitz enumeration problem has a presentation in terms of transitive factorizations in the symmetric group. This presentation suggests a generalization from type~$A$ to other finite reflection groups and, in particular, to type~$B$. We study this generalization both from a combinatorial and a geometric point of view, with the prospect of providing a means of understanding more of the structure of the moduli spaces of maps with an $\gS_2$-symmetry. The type~$A$ case has been well studied and connects Hurwitz numbers to the moduli space of curves. We conjecture an analogous setting for the type~$B$ case that is studied here.

Categories:05A15, 14H10, 58D29

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