旋量螺旋度簇

Higher secants of spinor varieties

Let $S_h$ be the even pure spinors variety of a complex vector space $V$ of even dimension $2h$ endowed with a non degenerate quadratic form $Q$ and let $σ_k(S_h) $ be the $k$-secant variety of $S_h$. We decribe a probabilistic algorithm which computes the complex dimension of $σ_k(S_h) $. Then, by using an inductive argument, we get our main result: $σ_3(S_h) $ has the expected dimension except when $h\in \{7,8\} $. Also we provide theoretical arguments which prove that $S_7$ has a defective 3-secant variety and $S_8$ has defective 3-secant and 4-secant varieties.

自旋簇的割线簇的可辨识性与奇异轨迹

Identifiability and singular locus of secant varieties to spinor varieties

本文分析自旋簇割线簇的Spin(V)-轨道结构，解决可辨识性与切向可辨识性问题，确定第二Terracini轨迹，并证明奇异轨迹包含两个最低维轨道且位于切簇中。

双旋量 Calabi-Yau 簇

Double spinor Calabi-Yau varieties

本文研究十六维半旋量表示射影化中十维旋量簇的两个一般平移的交集，证明所得 Calabi-Yau 五重簇 X 与其射影对偶交集 Y 虽非双有理等价，但满足导出等价和 L-等价。

Blow-ups of $\mathbb{P}^{n-3}$ at $n$ points and spinor varieties

Work of Dolgachev and Castravet-Tevelev establishes a bijection between the $2^{n-1}$ weights of the half-spin representations of $\mathfrak{so}_{2n}$ and the generators of the Cox ring of the variety $X_n$ which is obtained by blowing up $\mathbb{P}^{n-3}$ at $n$ points. We derive a geometric explanation for this bijection, by embedding ${\rm Cox}(X_n)$ into the even spinor variety (the homogeneous space of the even half-spin representation). The Cox ring of the blow-up $X_n$ is recovered geometrically by intersecting torus translates of the even spinor variety. These are higher-dimensional g…

Fourfolds of Weil type and the spinor map

Recent papers by Markman and O'Grady give, besides their main results on the Hodge conjecture and on hyperkaehler varieties, surprising and explicit descriptions of families of abelian fourfolds of Weil type with trivial discriminant. They also provide a new perspective on the well-known fact that these abelian varieties are Kuga Satake varieties for certain weight two Hodge structures of rank six. In this paper we give a pedestrian introduction to these results. The spinor map, which is defined using a half-spin representation of SO(8), is used intensively. For simplicity, we use basic repres…

Lagrangian Grassmannians and Spinor Varieties in Characteristic Two

The vector space of symmetric matrices of size $n$ has a natural map to a projective space of dimension $2^n-1$ given by the principal minors. This map extends to the Lagrangian Grassmannian ${\rm LG}(n,2n)$ and over the complex numbers the image is defined, as a set, by quartic equations. In case the characteristic of the field is two, it was observed that, for $n=3,4$, the image is defined by quadrics. In this paper we show that this is the case for any $n$ and that moreover the image is the spinor variety associated to ${\rm Spin}(2n+1)$. Since some of the motivating examples are of interes…

Rapoport-Zink spaces for spinor groups

We develop a theory of Hodge type Rapoport-Zink formal schemes, which uniformize certain formal completions of the canonical integral models of Shimura varieties of Hodge type at primes of good reduction. We then apply the general theory to the special case of Shimura varieties associated with groups of spinor similitudes, and, in the basic case, determine explicitly the reduced scheme underlying the Rapoport-Zink formal scheme.

无质量粒子的运动学簇

Kinematic Varieties for Massless Particles

本文研究了编码无质量粒子运动学数据的代数簇，利用克利福德代数推导旋量括号坐标，并通过多项式约束描述具有对称和反对称切片的张量。