Mario Maican
We show the existence of geometric quotients for the spaces of certain classes of morphisms of sheaves on projective space, modulo the canonical action of the group of automorphisms.
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We show that certain semistable sheaves on the projective plane with linear Hilbert polynomial are cokernels of semistable morphisms of decomposable sheaves.We exhibit certain locally closed subvarieties of moduli spaces of semistable sheaves as quotients modulo nonreductive groups.
We decompose each moduli space of semistable sheaves on the complex projective plane with support of dimension one and degree four into locally closed subvarieties, each subvariety being the good or geometric quotient of a set of morphisms of locally free sheaves modulo a reductive or a nonreductive group. We find locally free resolutions of length one of all these sheaves and describe them.
Let X be a K3 surface with an involution g which has non-empty fixed locus X^g and acts non-trivially on a non-zero holomorphic 2-form. We shall construct all such pairs (X, g) in a canonical way, from some better known double coverings of log del Pezzo surfaces of index at most 2 or rational elliptic surfaces, and construct the only family of each of the three extremal cases where X^g contains 10 (maximum possible) curves. We also classify rational log Enriques surfaces of index 2. Our approach is more geometrical rather than lattice-theoretical (see Nikulin's paper for the latter approach).
The paper proves that if a reductive group scheme acts properly on a scheme then the geometric quotient exists as an algebraic space. As a consequence we obtain the existence of the moduli spcace of canonically polarized varieties over Spec Z.
GIT quotient of minimal dimensional Schubert variety modulo a subtorus
本文研究极小维度Schubert变种模子代数群的GIT商，证明其与迭代投影空间包的第r阶段总空间同构。
In this note, we prove that for any finite dimensional vector space $V$ over an algebraically closed field $k$, and for any finite subgroup $G$ of $GL(V)$ which is either solvable or is generated by pseudo reflections such that the $|G|$ is a unit in $k$, the projective variety $\mathbb P(V)/G$ is projectively normal with respect to the descent of $\mathcal O(1)^{\otimes |G|}$.
We give a stratification of the GIT quotient of the Grassmannian $G_{2,n}$ modulo the normaliser of a maximal torus of $SL_{n}(k)$ with respect to the ample generator of the Picard group of $G_{2,n}$. We also prove that the flag variety $GL_{n}(k)/B_{n}$ can be obtained as a GIT quotient of $GL_{n+1}(k)/B_{n+1}$ modulo a maximal torus of $SL_{n+1}(k)$ for a suitable choice of an ample line bundle on $GL_{n+1}(k)/B_{n+1}$.
We construct a birational equivalence between certain quotients of s-tuples of equidimensional linear subspaces of $C^n$ and some quotients of products of square matrices modulo diagonal conjugations. In particular, we prove the rationality of the quotient space of s-tuples of linear 2-planes in $C^n$ modulo the diagonal $\gl_n(C)$-action . Furthermore, we compute generators of the field of the rational invariants explicitly.
Keller introduced a notion of quotient of a differential graded category modulo a full differential graded subcategory which agrees with Verdier's notion of quotient of a triangulated category modulo a triangulated subcategory. This work is an attempt to further develop his theory. More than a half of the text is devoted to an overview of "well known" definitions and results. As a result, the e-print is essentially self-contained.
We study the Fulton-Macpherson operational Chow rings of good moduli spaces of properly stable, smooth, Artin stacks. Such spaces are étale locally isomorphic to geometric invariant theory quotients of affine schemes, and are therefore natural extensions of GIT quotients. Our main result is that, with rational coefficients, every operational class can be represented by a so-called topologically strong cycle on the corresponding stack. Moreover, this cycle is unique modulo rational equivalence on the stack. Using out methods, we prove that if $X$ is the good moduli space of a properly stable, s…
Groupes de Brauer algébriques modulo les constants d'espaces homogènes et leurs compactifications
本文证明连通线性代数群齐性空间的代数 Brauer 群对常数 Brauer 群的商到 $\mathrm H^1(K,\operatorname{Pic}\bar X)$ 的自然包含未必为同构，并将结论推广到光滑紧化。
GIT quotient of Schubert varieties modulo one dimensional torus
本文研究Schubert簇在一维单参数子群作用下的GIT半稳定性，给出判据并确定唯一最小维半稳定Schubert簇，且在PSL(n,C)情形将其商描述为矩阵射影空间。
The semi-invariant ring as the Cox ring of a GIT quotient
该研究针对满足特定线性化映射同构条件的GIT商，证明其Cox环同构于底空间中θ-稳定轨迹的半不变量环，该结论可应用于多类代数簇，并给出仿射平面n点Hilbert概形Cox环的显式计算。
Toric varieties modulo reflections
本文研究了拟环面模反射的结构，证明了晶格多面体在反射群作用下商同构于特定拟环面，并探讨了实拟环面在排列多面体情况下的收缩性质。
We study the GIT-quotient of the Cartesian power of projective space modulo the projective orthogonal group. A classical isomorphism of this group with the Inversive group of birational transformations of the projective space of one dimension less allows one to interpret these spaces as configuration spaces of complex or real spheres.
We find locally free resolutions of length one for all semi-stable sheaves supported on curves of multiplicity five in the complex projective plane. In some cases we also find geometric descriptions of these sheaves by means of extensions. We give natural stratifications for their moduli spaces and we describe the strata as certain quotients modulo linear algebraic groups. In most cases we give concrete descriptions of these quotients as fibre bundles.
Uniruledness of some low-dimensional ball quotients
本文利用复球上的反射模形式与 Gritsenko–Hulek 方法，证明若干3至5维球商单直纹，并给出 Hermitian 格及 P¹ 八点模空间等例子。
We study the moduli space of Gieseker semi-stable sheaves on the complex projective plane supported on sextic curves and having Euler characteristic one. We determine locally free resolutions of length one for all such sheaves. We decompose the moduli space into strata which occur naturally as quotients modulo actions of certain algebraic groups. In some cases we give concrete geometric descriptions of the strata.
Motivated by a problem originating in string theory, we study elliptic fibrations on K3 surfaces with large Picard number modulo isomorphism. We give methods to determine upper bounds for the number of inequivalent K3 surfaces sharing the same frame lattice. For any given Neron--Severi lattice $S_X$, such a bound on the `multiplicity' can be derived by investigating the quotient of the isometry group of $S_X$ by the automorphism group. The resulting bounds are strongest for large Picard numbers and multiplicities of unity do indeed occur for a number of K3 surfaces with Picard number 20. Under…
Let k be a field. Let K_0(V_k) denote the quotient of the free abelian group generated by the geometrically reduced varieties over k, modulo the relations of the form [X]=[X-Y]+[Y] whenever Y is a closed subvariety of X. Product of varieties makes K_0(V_k) into a ring. We prove that if the characteristic of k is zero, then K_0(V_k) is not a domain.
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