论文标题
立方三倍的theta除数作为模量空间
The desingularization of the theta divisor of a cubic threefold as a moduli space
论文作者
论文摘要
我们表明，gieseker稳定的带稳定的滑轮上的模量空间$ \叠加{m} _x（v）$，带有chern targue $ v =（3，-h，-h，-h^2/2，h^3/6）$的平滑立方三倍$ x $是光滑的，尺寸为四。此外，ABEL-JACOBI地图$ x $的中间Jacobian将其Bibation Bibation Bibation用$ x $θ$呈现，仅收缩了$ x \ subset \ subset \ overline \ edimelline {m} _x _x（v）$的副本，向单数点$ 0 \ inθ$ 0 \。 我们使用此结果为Cutic三倍的Torelli定理提供了新的证明，该版本可以从其Kuznetsov component $ \ operatorName {ku}（x）\ subset \ subset \ mathrm {d}^}^{\ mathrm {\ mathrm {b b}（x）$中回收。同样，这导致了对theta除数的奇异性的描述，以及Cubic三倍的古典Torelli定理的描述，即可以从其中间Jacobian中恢复$ x $。
We show that the moduli space $\overline{M}_X(v)$ of Gieseker stable sheaves on a smooth cubic threefold $X$ with Chern character $v = (3,-H,-H^2/2,H^3/6)$ is smooth and of dimension four. Moreover, the Abel-Jacobi map to the intermediate Jacobian of $X$ maps it birationally onto the theta divisor $Θ$, contracting only a copy of $X \subset \overline{M}_X(v)$ to the singular point $0 \in Θ$. We use this result to give a new proof of a categorical version of the Torelli theorem for cubic threefolds, which says that $X$ can be recovered from its Kuznetsov component $\operatorname{Ku}(X) \subset \mathrm{D}^{\mathrm{b}}(X)$. Similarly, this leads to a new proof of the description of the singularity of the theta divisor, and thus of the classical Torelli theorem for cubic threefolds, i.e., that $X$ can be recovered from its intermediate Jacobian.
