论文标题
决定性方案的变形和无障碍性
Deformation and Unobstructedness of Determinantal Schemes
论文作者
论文摘要
令$ hilb^{p（t）}（p^n）$为$ p^n $的希尔伯特方案，与hilbert polyenmial $ p（t）\ in q [t] $ in q [t] $，然后让$ w：= \ operline { ^{p（t）}（p ^n）确定方案的$由$（t-r+1）\ times（t-r+1）的消失所定义$ \ max \ {1,2-c \} \ le r <t $。 $ w $是不可约代代数集。首先，我们计算了$ w $ $ w $的$ w $和$ a_j $和$ b_i $的尺寸的上限，这是$ r = 1 $的尖锐。在线性案例（$ a_j = 1，b_i = 0 $）和案例足够接近，我们猜想，并在一定程度上证明了所有$ r $都可以实现此界限。然后，我们研究$ w $是$ hilb ^{p（t）}（p ^n）$的一般平滑组件。在整数上的一些弱数值假设下，我们猜想了，我们猜想，经常证明$ w $是一个普通平滑的组件。此外，我们还研究了$（x）\ in W $和密切相关的模块的均匀坐标环的正常模块的深度。我们猜想，在某些情况下证明，他们的代码专家通常是1（分别为$ r $）。这些结果将标准确定性方案的先前结果扩展到确定性方案。即作者在$ w（\ usepline {b}; \ usepline {a}; 1）$ to $ w $的先前结果，带有$ 1 \ le r <t $和$ c \ ge 2-r $。最后，研究并证明由$ \ Mathcal A $确定的地图的外部功能的变形被证明为$ x \ subset p^n $，如果$ \ dim x \ ge 3 $。该作品包含许多示例，这些例子说明了获得的结果和大量的开放问题；其中一些是在最后一部分中作为猜想收集的。
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.
