Deformation and Unobstructedness of Determinantal Schemes
Jan O. Kleppe, Rosa M. Miró-Roig
备注：
This version (v2) is a postprint (AAM) of an article published in the Memoirs of the American Mathematical Society, with very minor changes from version v1. The final authenticated version is available online at DOI: 10.1090/memo/1418. Remark 3.2 is, however, extended and the new extension in (3) and (4) is not included in the published paper. Remark 3.2(4) makes a correction to [33]
Let $Hilb ^{p(t)}(P^n)$ be the Hilbert scheme of closed subschemes of $P^n$ with Hilbert polynomial $p(t) \in Q[t]$, and let $W:= \overline{W(\underline{b};\underline{a};r)}$ be the closure of the locus in $Hilb ^{p(t)}(P^n)$ of determinantal schemes defined by the vanishing of the $(t-r+1)\times (t - r+1)$ minors of some matrix $\mathcal A$ of size $t\times (t+c-1)$ with $ij$-enty a homogeneous form of degree $a_j-b_i$ and with $r$ satisfying $\max\{1,2-c\} \le r < t$. $W$ is an irreducible algebraic set. First of all, we compute an upper $r$-independent bound for the dimension of $W$ in terms of $a_j$ and $b_i$ which is sharp for $r=1$. In the linear case ($a_j = 1, b_i=0$) and cases sufficiently close, we conjecture and to a certain degree prove that this bound is achieved for all $r$. Then, we study to what extent $W$ is a generically smooth component of $Hilb ^{p(t)}(P^n)$. Under some weak numerical assumptions on the integers $a_j$ and $b_i$ (or under some depth conditions) we conjecture and often prove that $W$ is a generically smooth component. Moreover, we also study the depth of the normal module of the homogeneous coordinate ring of $(X)\in W$ and of a closely related module. We conjecture, and in some cases prove, that their codepth is often 1 (resp. $r$). These results extend previous results on standard determinantal schemes to determinantal schemes; i.e. previous results of the authors on $W(\underline{b};\underline{a};1)$ to $W$ with $1\le r < t$ and $c\ge 2-r$. Finally, deformations of exterior powers of the cokernel of the map determined by $\mathcal A$ are studied and proven to be given as deformations of $X \subset P^n$ if $\dim X \ge 3$. The work contains many examples which illustrate the results obtained and a considerable number of open problems; some of them are collected as conjectures in the final section.
We consider compact Kählerian manifolds $X$ of even dimension 4 or more, endowed with a log-symplectic holomorphic Poisson structure $Π$ which is sufficiently general, in a precise linear sense, with respect to its (normal-crossing) degeneracy divisor $D(Π)$. We prove that $(X, Π)$ has unobsrtuced deformations, that the tangent space to its deformation space can be identified in terms of the mixed Hodge structure on $H^2$ of the open symplectic manifold $X\setminus D(Π)$, and in fact coincides with this $H^2$ provided the Hodge number $h^{2,0}_X=0$, and finally that the degeneracy locus $D(Π)$…
We give a new characterization of generalized Kähler structures in terms of their corresponding complex Dirac structures. We then give an alternative proof of Hitchin's partial unobstructedness for holomorphic Poisson structures. Our main application is to show that there is a corresponding unobstructedness result for arbitrary generalized Kähler structures. That is, we show that any generalized Kähler structure may be deformed in such a way that one of its underlying holomorphic Poisson structures remains fixed, while the other deforms via Hitchin's deformation. Finally, we indicate a close r…
Deformations of modules of maximal grade and the Hilbert scheme at determinantal schemes
设R是一个多项式环，M是R生成的分次R-模的极大分次（这意味着M的齐次表示矩阵\cA的极大子式生成的理想I_t（\cA）在R中有极大余维数）.设X：=Proj（R/I_t（\cA））在一个足够大的开子集上是光滑的，且dim X > 0.然后证明了在弱假设下M的局部分次形变函子与X \subset Proj（R）上的局部Hilbert（scheme）函子同构。在此假设下，我们得到了Hilbert格式在（X）上是光滑的，并给出了其局部环维数的显式表达式.作为推论，我们证明了R. M. Miró-Roig和作者指出，在表示矩阵中具有固定度数的标准行列式格式的轨迹的闭包是Hilbert格式的一般光滑分量V。当dim X > 0时，证明了他们关于V的维数的猜想.证明了在Proj（R）中X的正规层的上同调H^ i_{*}（{ \cN }_X）对0 < i < dim X-1为零.最后，对上述结果稍加修改，用多项式环的任意Cohen-Macaulay商代替R，仍为真。
We prove that a weak $\mathbb{Q}$-Fano $3$-fold with terminal singularities has unobstructed deformations. By using this result and computing some invariants of a terminal singularity, we provide two results on global deformation of a weak $\mathbb{Q}$-Fano $3$-fold. We also treat a stacky proof of the unobstructedness of deformations of a $\mathbb{Q}$-Fano $3$-fold.
This paper uses reconstruction algebras to construct simultaneous resolution of determinantal surfaces. The main new difference to the classical case is that, in addition to the quiver of the reconstruction algebra, certain noncommutative relations, namely those of the canonical algebra of Ringel, are required. All the relations of the reconstruction algebra except the canonical relation are then deformed, and these deformed relations together with variation of the GIT quotient achieve the simultaneous resolution.
We survey determinantal singularities, their deformations, and their topology. This class of singularities generalizes the well studied case of complete intersections in several different aspects, but exhibits a plethora of new phenomena such as for instance non-isolated singularities which are finitely determined, or smoothings with low connectivity; already the union of the coordinate axes in $(\mathbb{C}^3,0)$ is determinantal, but not a complete intersection. We start with the algebraic background and then continue by discussing the subtle interplay of unfoldings and deformations in this s…
We continue the development of the study of the equisingularity of isolated singularities, in the determinantal case. This version of the paper includes a substantial amount of new material (76% larger). The new material introduces the idea of the landscape of singularity, which includes the allowable deformations of the singularity and associated structure useful for equisingularity questions. Fixing a presentation matrix M of a determinantal singularity means viewing the singularity as a section via M of the set of matrices of a given or smaller rank. Varying M gives the allowable deformatio…
The generalized Lax conjecture is true for topological reasons related to compactness, convexity and determinantal deformations of increasing products of pointwise approximating linear forms
Let GradAlg(H) be the scheme parameterizing graded quotients of R=k[x_0,...,x_n] with Hilbert function H (it is a subscheme of the Hilbert scheme of P^n if we restrict to quotients of positive dimension, see definition below). A graded quotient A=R/I of codimension c is called standard determinantal if the ideal I can be generated by the t by t minors of a homogeneous t by (t+c-1) matrix (f_{ij}). Given integers a_0\le a_1\le ...\le a_{t+c-2} and b_1\le ...\le b_t, we denote by W_s(\underline{b};\underline{a}) the stratum of GradAlg(H) of determinantal rings where f_{ij} \in R are homogeneous …
We extend the notion of essential deformations from the case of the Iwasawa manifold, for which they were introduced recently by the first-named author, to the general case of page-$1$-$\partial\bar\partial$-manifolds that were jointly introduced very recently by all three authors. We go on to obtain an analogue of the unobstructedness theorem of Bogomolov, Tian and Todorov for page-$1$-$\partial\bar\partial$-manifolds. As applications of this discussion, we study the small deformations of certain Nakamura solvmanifolds and reinterpret the cases of the Iwasawa manifold and its $5$-dimensional …
The theorem of Hilbert- Burch provides a description of codimension two determinantal varieties and their deformations in terms of their presentation matrices. In this work we use this correspondence to study properties of determinantal varieties, based on methods of singularity theory. We establish the theory of singularities for nxp matrices extending previous results of Bruce and Tari and Frühbis-Krüger. The main result of this work is the description of equivalent conditions to G-finite determinacy of the presentation matrix of Cohen-Macaulay varieties of codimension 2. We apply the result…
