论文标题
原始多个方案上的向量束的模量
Moduli of vector bundles on primitive multiple schemes
论文作者
论文摘要
原始多个方案是一种cohen-macaulay方案$ y $，因此相关的减少方案$ x = y_ {red {red {red} $是平滑的，不可约的，并且$ y $可以本地嵌入在平滑的尺寸$ \ dim（x）+1 $的平滑尺寸中。如果$ 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 $的微不足道的原始多个方案：它是$ x $ $ x $的$ n $ th $ th $ x $的$ n $ th，嵌入了零部分的bundle $ l^*$。 本文的主要主题是矢量束的良好模量在原始多个方案上的构建和特性。假设$ y = x_n $是多重性$ n $，并且可以将其扩展到$ x_ {n+1} $乘以$ n+1 $的$ x_ {n+1} $，然后让$ m_n $ a $ x_n $上的矢量捆绑包。有了合适的假设，我们为$ x_ {n+1} $上的向量捆绑包构建了一个精细的模量$ m_ {n+1} $，其限制为$ x_n $属于$ m_n $。这是一个可扩展到$ x_ {n+1} $的子变量$ n_n \ subset m_n $的仿射包。总的来说，这个仿束不是平庸的。这特别适用于Picard组。 我们还提供了许多原始多个方案$ y $的新示例，以使二元捆绑$ω_y$很琐碎。
A primitive multiple scheme is a 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 in the line bundle $L^*$ by the zero section. The main subject of this paper is the construction and properties of fine moduli spaces of vector bundles on primitive multiple schemes. Suppose that $Y=X_n$ is of multiplicity $n$, and can be extended to $X_{n+1}$ of multiplicity $n+1$, and let $M_n$ a fine moduli space of vector bundles on $X_n$. With suitable hypotheses, we construct a fine moduli space $M_{n+1}$ for the vector bundles on $X_{n+1}$ whose restriction to $X_n$ belongs to $M_n$. It is an affine bundle over the subvariety $N_n\subset M_n$ of bundles that can be extended to $X_{n+1}$. In general this affine bundle is not banal. This applies in particular to Picard groups. We give also many new examples of primitive multiple schemes $Y$ such that the dualizing sheaf $ω_Y$ is trivial.
