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arXiv 1002.4382math.AG
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$SU_C(r)$的模子簇与双有理几何

Modular subvarieties and birational geometry of $SU_C(r)$

Michele Bolognesi, Sonia Brivio

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16 pages, corrected references
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摘要：

设$C$是代数光滑复亏格$g>1$曲线.本文研究了$C$上半稳定秩为r的具有0次行列式的向量丛的粗模空间$U_C(r,0)$及其由具有平凡行列式的向量丛给出的模子空间$SU_C(r)$的双有理结构.值得注意的是，我们证明了$U_C(r,0)$（resp. $SU_C(r)$）对于对称乘积$C^(rg)$（分别为）上的纤维化是双有理的。在$P^{(r-1)g}$上），其纤维是GIT前体$(P^{r-1})^{rg}//PGL(r)$。在低秩和亏格的情况下，我们的构造产生了包含在Coble超曲面中的经典模簇的家族。

英文摘要：

Let $C$ be an algebraic smooth complex genus $g>1$ curve. The object of this paper is the study of the birational structure of the coarse moduli space $U_C(r,0)$ of semi-stable rank r vector bundles on $C$ with degree 0 determinant and of its moduli subspace $SU_C(r)$ given by the vector bundles with trivial determinant. Notably we prove that $U_C(r,0)$ (resp. $SU_C(r)$) is birational to a fibration over the symmetric product $C^(rg)$ (resp. over $P^{(r-1)g}$) whose fibres are GIT quotients $(P^{r-1})^{rg}//PGL(r)$. In the cases of low rank and genus our construction produces families of classical modular varieties contained in the Coble hypersurfaces.

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