We use wall-crossing in the Bridgeland stability manifold to systematically study the birational geometry of the moduli space $M_σ(\mathbf{v})$ of $σ$-semistable objects of class $\mathbf{v}$ for a generic stability condition $σ$ on an arbitrary Enriques surface $X$. In particular, we show that for any other generic stability condition $τ$, the two moduli spaces $M_τ(\mathbf{v})$ and $M_σ(\mathbf{v})$ are birational. As a consequence, we show that for primitive $\mathbf{v}$ of odd rank $M_σ(\mathbf{v})$ is birational to a Hilbert scheme of points. Similarly, in even rank we show that $M_σ(\mathbf{v})$ is birational to a moduli space of torsion sheaves supported on a hyperelliptic curve when $\ell(\mathbf{v})=1$. As an added bonus of our work, we prove that the Donaldson-Mukai map $θ_{\mathbf{v},σ}:\mathbf{v}^\perp\to\mathrm{Pic}(M_σ(\mathbf{v}))$ is an isomorphism for these classes. Finally, we use our classification to fully describe the geometry of the only two examples of moduli of stable sheaves on $X$ that are uniruled (and thus not K-trivial).
MMP for Enriques pairs and singular Enriques varieties
Universität des Saarlandes（萨尔兰大学）；Université Paris-Saclay（巴黎萨克雷大学）；École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；Université de Lorraine（洛林大学）
A refined Derived Torelli Theorem for Enriques surfaces
证明了定义在特征不同于$2$的代数闭域上的两个一般Enriques曲面同构，如果它们的Kuznetsov分量等价.我们采用同样的技术给一个新的简单的证明猜想英格尔斯和库兹涅佐夫有关的导出范畴的爆破一般阿廷\textendash芒福德四次双固体和相关的恩里克斯表面。本文来自2018年5月2日至5日在图卢兹大学举办的\emph{Semiorthogonal decompositions, stability conditions and sheaves of categories}研讨会上的一个问题部分。
Verra fourfolds, twisted sheaves and the last involution
研究了K3曲面上扭层的某些模空间的几何性质。特别地，我们引入诱导自同构从一个K3表面上的扭曲层的模空间在这个K3表面上。作为应用，我们证明了具有不变格$U(2)\oplus D_4(-1)$或$U(2)\oplus E_8(-2)$的非辛对合的不可约$K3^{[2]}$型全纯辛流形的模空间的单合理性.这补充了[Mongardi and Wandel 2015]，[Bossiere et al 2016]中获得的结果，以及[arXiv：1603.00403]中关于使用Verra四重曲线上$(1,1)$二次曲线的Hilbert方案构建IHS四重曲线几何的结果。作为一个副产品，我们发现，IHS四倍的$K3^{[2]}$型皮卡德格$U(2)\oplus E_8(-2)$自然包含非节Enriques曲面。
