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The celebrated Zariski Cancellation Problem asks as to when the existence of an isomorphism If the cancellation does not hold then

We derive generalizations of McShane’s identity for higher ranked surface group representations by studying a family of mapping class group invariant functions introduced by Goncharov and Shen, which generalize the notion of horocycle lengths. In particular, we obtain McShane-type identities for finite-area cusped convex real projective surfaces by generalizing the Birman–Series geodesic scarcity theorem. More generally, we establish McShane-type identities for positive surface group representations with loxodromic boundary monodromy, as well as McShane-type inequalities for general rank positive representations with unipotent boundary monodromy. Our identities are systematically expressed in terms of projective invariants, and we study these invariants: we establish boundedness and Fuchsian rigidity results for triple and cross ratios. We apply our identities to derive the simple spectral discreteness of unipotent-bordered positive representations, collar lemmas, and generalizations of the Thurston metric.

Let M¯g,A[n]\\overline {\\mathcal {M}}_{g,A[n]} be the moduli stack parametrizing weighted stable curves, and let M¯g,A[n]\\overline {M}_{g,A[n]} be its coarse moduli space. These spaces have been introduced by B. Hassett, as compactifications of Mg,n\\mathcal {M}_{g,n} and Mg,nM_{g,n}, respectively, by assigning rational weights A=(a1,…,an)A = (a_{1},\\dots ,a_{n}), 0>ai⩽10> a_{i} \\leqslant 1 to the markings. In particular, the classical Deligne-Mumford compactification arises for a1=⋯=an=1a_1 = \\dots = a_n = 1. In genus zero some of these spaces appear as intermediate steps of the blow-up construction of M¯0,n\\overline {M}_{0,n} developed by M. Kapranov, while in higher genus they may be related to the LMMP on M¯g,n\\overline {M}_{g,n}. We compute the automorphism groups of most of the Hassett spaces appearing in Kapranov’s blow-up construction. Furthermore, if g⩾1g\\geqslant 1 we compute the automorphism groups of all Hassett spaces. In particular, we prove that if g⩾1g\\geqslant 1 and 2g−2+n⩾32g-2+n\\geqslant 3, then the automorphism groups of both M¯g,A[n]\\overline {\\mathcal {M}}_{g,A[n]} and M¯g,A[n]\\overline {M}_{g,A[n]} are isomorphic to a subgroup of SnS_{n} whose elements are permutations preserving the weight data in a suitable sense.

In this paper, we study compactifications of the moduli of smooth del Pezzo surfaces using K-stability and the line arrangement. We construct K-moduli of log del Pezzo pairs with sum of lines as boundary divisors, and prove that for $d=2,3,4$ , these K-moduli of pairs are isomorphic to the K-moduli spaces of del Pezzo surfaces. For $d=1$ , we prove that they are different by exhibiting some walls.

Given an integer g 0 and a weight vector w ın Q^n (0, 1]^n satisfying 2g - 2 + w_i > 0 , let _g, w denote the moduli space of n -marked, w -stable tropical curves of genus g and volume one. We calculate the automorphism group Aut(_g, w) for g 1 and arbitrary w , and we calculate the group Aut(_0, w) when w is heavy/light. In both of these cases, we show that Aut(_g, w) Aut(K_w) , where K_w is the abstract simplicial complex on \\1, n\\ whose faces are subsets with w -weight at most 1. We show that these groups are precisely the finite direct products of symmetric groups. The space _g, w may also be identified with the dual complex of the divisor of singular curves in the algebraic Hassett space M_g, w . Following the work of Massarenti and Mella (2017) on the biregular automorphism group Aut(M_g, w) , we show that Aut(_g, w) is naturally identified with the subgroup of automorphisms which preserve the divisor of singular curves.

We construct explicit deformation quantizations of the noncompact complex surfaces Z_k:=Tot(O_P^1(-k)) and describe their effect on moduli spaces of vector bundles and instanton moduli spaces. We introduce the concept of rebel instantons, as being those which react badly to some quantizations, misbehaving by shooting off extra families of noncommutative instantons. We then show that the quantum instanton moduli space can be viewed as the étale space of a constructible sheaf over the classical instanton moduli space with support on rebel instantons.

One of the main themes of this long article is the study of projective varieties which are K(H,1)’s, i.e. classifying spaces BH for some discrete group H. After recalling the basic properties of such classifying spaces, an important class of such varieties is introduced, the one of Bagnera–de Franchis varieties, the quotients of an Abelian variety by the free action of a cyclic group. Moduli spaces of Abelian varieties and of algebraic curves enter into the picture as examples of rational K(H,1)’s, through Teichmüller theory. The main trhust of the paper is to show how in the case of K(H,1)’s the study of moduli spaces and deformation classes can be achieved through by now classical results concerning regularity of classifying maps. The Inoue type varieties of Bauer and Catanese are introduced and studied as a key example, and new results are shown. Motivated from this study, the moduli spaces of algebraic varieties, and especially of algebraic curves with a group of automorphisms of a given topological type are studied in detail, following new results by the author, Michael Lönne and Fabio Perroni. Finally, the action of the absolute Galois group on the moduli spaces of such K(H,1) varieties is studied. In the case of surfaces isogenous to a product, it is shown how this yields a faifhtul action on the set of connected components of the moduli space: for each Galois automorphism of order different from 2 there is an algebraic surface S such that the Galois conjugate surface of S has fundamental group not isomorphic to the one of S.

Let [...] be a compact connected Riemann surface of genus [...], and let [...] be the rank one Deligne-Hitchin moduli space associated to [...]. It is known that [...] is the twistor space for the hyper-Kähler structure on the moduli space of rank one holomorphic connections on [...]. We investigate the group [...] of all holomorphic automorphisms of [...]. The connected component of [...] containing the identity automorphism is computed. There is a natural element of [...]. We also compute the subgroup of [...] that fixes this second cohomology class. Since [...] admits an ample rational curve, the notion of algebraic dimension extends to it by a theorem of Verbitsky. We prove that [...] is Moishezon. [ProQuest: [...] denotes formulae omitted.]

Let X be a compact Riemann surface of genus g≥2 and M(G[sub.2] ) be the moduli space of polystable principal bundles over X , the structure group of which is the simple complex Lie group of exceptional type G[sub.2] . In this work, it is proved that the only automorphisms that M(G[sub.2] ) admits are those defined as the pull-back action of an automorphism of the base curve X . The strategy followed uses specific techniques that arise from the geometry of the gauge group G[sub.2] . In particular, some new results that provide relations between the stability, simplicity, and irreducibility of G[sub.2] -bundles over X have been proved in the paper. The inclusion of groups G[sub.2] ↪Spin(8,C) where G[sub.2] is viewed as the fixed point subgroup of an order of 3 automorphisms of Spin(8,C) that lifts the triality automorphism is also considered. Specifically, this inclusion induces the forgetful map of moduli spaces of principal bundles M(G[sub.2] )→M(Spin(8,C)). In the paper, it is also proved that the forgetful map is an embedding. Finally, some consequences are drawn from the results above on the geometry of M(G[sub.2] ) in relation to M(Spin(8,C)).