Towards a specialization map modulo semi-orthogonal decompositions

大辛或正交模l单值

Big symplectic or orthogonal monodromy modulo l

设k是一个特征不为2的域，L是k中几乎所有可逆有理素数的集合。设我们有一个簇X/k和严格相容的可构造F_ell层系统{M_ell -> X：ell in L}。若系统是正交或辛自对偶的，则M_ell的几何单值群是F_ell上一个等距群G_ell的子群，若M_ell包含导子群DG_ell=[G_ell，G_ell]，则称M_ell的几何单值群为大单值群.证明了一个定理，给出了M_ell具有大单值性的充分条件.我们应用该定理所产生的显式系统的中间上同调的家庭的超椭圆曲线和椭圆曲面，以显示的monodromy是一致大，因为我们改变ell和系统。

三维标志模对合的可分解猜想

The Crepant Resolution Conjecture for 3-dimensional flags modulo an involution

在向量空间V上固定一个非退化双线性型后，我们通过取一个标志到它的正交补来定义V中标志流形F的对合。当V为3维时，我们检验了J. Bryan和T. Graber认为：[F / Z_2]的亏格零（轨道折叠）Gromov-Witten势函数与商格式F / Z_2的临界分辨率Y的亏格零Gromov-Witten势函数在设定量子参数为-1、变量作线性变换和解析连续系数后一致（直到不稳定项）。临界分解Y（希尔伯特方案Hilb^2 P^2中的超曲面）是P^2上一个新的秩为2的向量丛的投影化。

On residual categories for Grassmannians

We define and discuss some general properties of residual categories of Lefschetz decompositions in triangulated categories. In the case of the derived category of coherent sheaves on the Grassmannian $\text{G}(k,n)$ we conjecture that the residual category associated with Fonarev's Lefschetz exceptional collection is generated by a completely orthogonal exceptional collection. We prove this conjecture for $k = p$, a prime number, modulo completeness of Fonarev's collection (and for $p = 3$ we check this completeness).

正交与酉 Shimura 簇自然嵌入下的 Ekedahl-Oort 分层

Ekedahl-Oort strata under natural embeddings of orthogonal and unitary Shimura varieties

研究 GSpin Shimura 簇与 Rapoport-Smithling-Zhang 酉 Shimura 簇在好约化之间的自然嵌入下，Ekedahl-Oort 分层的像，并计算这些 Shimura 簇的 p-秩和 a-数等离散不变量。

Most secant varieties of tangential varieties to Veronese varieties are nondefective

We prove a conjecture stated by Catalisano, Geramita, and Gimigliano in 2002, which claims that the secant varieties of tangential varieties to the $d$th Veronese embedding of the projective $n$-space $\mathbb{P}^n$ have the expected dimension, modulo a few well-known exceptions. As Bernardi, Catalisano, Gimigliano, and Idá demonstrated that the proof of this conjecture may be reduced to the case of cubics, i.e., $d=3$, the main contribution of this work is the resolution of this base case. The proposed proof proceeds by induction on the dimension $n$ of the projective space via a specializati…

Global Frobenius Liftability I

We formulate a conjecture characterizing smooth projective varieties in positive characteristic whose Frobenius morphism can be lifted modulo $p^2$ - we expect that such varieties, after a finite étale cover, admit a toric fibration over an ordinary abelian variety. We prove that this assertion implies a conjecture of Occhetta and Wiśniewski, which states that in characteristic zero a smooth image of a projective toric variety is a toric variety. To this end we analyse the behaviour of toric varieties in families showing some generization and specialization results. Furthermore, we prove a pos…