论文标题
原始多个方案
Primitive multiple schemes
论文作者
论文摘要
原始多个方案是一种复杂的Cohen-Macaulay方案$ y $，因此相关的还原方案$ x = y_ {red {red} $是平滑的，不可可用的，并且可以将$ y $在本地嵌入到平滑的尺寸$ \ dim（x）+1 $中的平滑尺寸。如果$ n $是$ y $的多样性，则有一个规范过滤$ x = x_1 \ subset x_2 \ subset \ cdots \ cdots \ subset x_n = y $，因此$ x_i $是一种primitive多重多重多数$ i $ $。最简单的示例是与$ x $上的Line Bundle $ l $相关的多重性$ n $的微不足道的原始多个方案：这是$ x $的$ n $ th $ x $的$ n $ th $ x $，如果零部分的线束$ l^*$嵌入。 令$ {\ bf z} _n = {spec}（c [t]/（t^n））$。原始多个多样性$ n $的原始方案是通过取出平滑品种$ x $的开放式封面$（u_i）$获得的，并通过使用$ u_ {ij} \ bf z} $ __ $ __ $ __ i的automorphisms粘贴$ u_i \ times {\ bf z} _n $。这导致研究了$ x \ times {\ bf z} _n $的非亚伯集团$ g_n $ g_n $的捆绑，这些$ x \ bf z} _n $使$ x $不变性，并研究其首次共同体。如果$ n \ geq 2 $，将$ x_n $扩展到$ n+1 $的原始多重多重方案，这在第二个同胞组$ h^2（x，e）的合适矢量束$ e $ e $上的$ x $上。 在本文中，我们研究了这些障碍和原始多个方案的参数化。例如，我们表明，如果$ x = p_m $带有$ m> = 3 $所有原始多个方案都是微不足道的。如果$ x = p_2 $，则只有两个非琐碎的原始多个方案，具有$ 2 $和$ 4 $的倍数，这不是准项目。另一方面，如果$ x $是曲线上的投影捆绑包，我们表明有无限序列$ x = x_1 \ subset x_2 \ subset \ cdots \ cdots \ subset x_n \ subset x_n \ subset x_ {n+1} \ 1} \ subset \ subset \ cdots \ cdots $ cdots $ cdots $ cdots $ cdots $ cdots $ cdots。
A primitive multiple scheme is a complex Cohen-Macaulay scheme $Y$ such that the associated reduced scheme $X=Y_{red}$ is smooth, irreducible, and that $Y$ can be locally embedded in a smooth variety of dimension $\dim(X)+1$. If $n$ is the multiplicity of $Y$, there is a canonical filtration $X=X_1\subset X_2\subset\cdots\subset X_n=Y$, such that $X_i$ is a primitive multiple scheme of multiplicity $i$. The simplest example is the trivial primitive multiple scheme of multiplicity $n$ associated to a line bundle $L$ on $X$: it is the $n$-th infinitesimal neighborhood of $X$, embedded if the line bundle $L^*$ by the zero section. Let ${\bf Z}_n={spec}(C[t]/(t^n))$. The primitive multiple schemes of multiplicity $n$ are obtained by taking an open cover $(U_i)$ of a smooth variety $X$ and by gluing the schemes $U_i\times{\bf Z}_n$ using automorphisms of $U_{ij}\times {\bf Z}_n$ that leave $U_{ij}$ invariant. This leads to the study of the sheaf of nonabelian groups $G_n$ of automorphisms of $X\times {\bf Z}_n$ that leave the $X$ invariant, and to the study of its first cohomology set. If $n\geq 2$ there is an obstruction to the extension of $X_n$ to a primitive multiple scheme of multiplicity $n+1$, which lies in the second cohomology group $H^2(X,E)$ of a suitable vector bundle $E$ on $X$. In this paper we study these obstructions and the parametrization of primitive multiple schemes. As an example we show that if $X=P_m$ with $m>=3$ all the primitive multiple schemes are trivial. If $X=P_2$, there are only two non trivial primitive multiple schemes, of multiplicities $2$ and $4$, which are not quasi-projective. On the other hand, if $X$ is a projective bundle over a curve, we show that there are infinite sequences $X=X_1\subset X_2\subset\cdots\subset X_n\subset X_{n+1}\subset\cdots$ of non trivial primitive multiple schemes.
