超平面上的多项式常数与超二次曲面的CR映射

Projective dimensions of hyperplane arrangements

We establish a general theory for projective dimensions of the logarithmic derivation modules of hyperplane arrangements. That includes the addition-deletion and restriction theorem, Yoshinaga-type result, and the division theorem for projective dimensions of hyperplane arrangements. They are generalizations of the free arrangement cases, that can be regarded as the special case of our result when the projective dimension is zero. The keys to prove them are several new methods to determine the surjectivity of the Euler and the Ziegler restriction maps, that is combinatorial when the projective…

Lefschetz Theorem Using Real Morse Theory

We prove the Lefschetz hyperplane section theorem using a simpler machinery by making the observation that we can compose the Lefschetz Pencil with a Real Morse function to get a map from the variety to $\mathbb{R}$ which is "close" to being a Real Morse function. The proof uses a new method unlike the conventional one which uses vanishing cycles, thimbles and monodromy. We prove the genus formula for plane curves using Morse theory, Lefschetz pencil and Bézout's theorem. And then we prove the Riemann Hurwitz formula for ramified maps between curves by employing techniques from deformation the…

通过割线映射研究 Hirzebruch 曲面二重对称积的双有理几何

Birational geometry of the twofold symmetric product of a Hirzebruch surface via secant maps

本文利用割线与切线构成的 Grassmann 子簇刻画有理正规卷轴长度 2 Hilbert 概形的初等收缩，计算其次数与 $a=1$ 奇点，并阐明 $r=4$ 时相关 Fano 三维簇及 GIT 稳定性。

Quantum K theory for flag varieties

Forgetting a subspace from a partial flag yields another partial flag composed of fewer subspaces. This induces a forgetful map $π: X \to X'$ between the corresponding flag varieties. We prove here that, for a degree large enough, the variety associated with degree d stable maps sending their marked points within Schubert varieties $X_i$ of $X$ is a rationally connected fibration over its image, which parametrizes degree $π_* d$ stable maps sending their marked points within the Schubert varieties $π(X_i)$ of $X'$. The Euler characteristic of these varieties are quantum $K$-invariants. Our res…

Enriques 曲面上的 Gauss-Prym 映射

Gauss-Prym maps on Enriques surfaces

本文证明了在满足特定极化条件的非节点 Enriques 曲面上，第 $k$ 阶高斯映射与 Gauss-Prym 映射的满射性，并给出了 $k=1$ 时的充分条件。

环面极映射与示性类

Toric polar maps and characteristic classes

该研究证明复射影空间中超曲面的环面极映射多重度与特定开集的Chern-Schwartz-MacPherson类系数符号一致，给出平面曲线映射次数的局部不变量公式，并构造了任意维数的双有理环面极映射的不可约超曲面族。

作为射影空间（交换）有理自映射族的Landen变换

Landen transforms as families of (commuting) rational self-maps of projective space

经典的（m，k）-Landen变换F_{m,k}是有理函数C（z）域的自映射，它是通过将一个有理函数在m次单位根上进行加权平均而得到的。通过将d次有理映射集与P^{2d+1}的仿射开子集相对应，证明了F_{m,0}诱导出P^{2d+1}的代数次数为m的支配有理自映射R_{d,m,0}，并且当0 < k < m时，变换F_{m,k}诱导出P^{2d+1}中某个超平面的代数次数为m的支配有理自映射R_{d,m,k}.我们证明了在所有情况下R_{d,m,k}都能很好地推广到Spec（Z）上P^{2d+1}的映射，并且{ R_{d,m,0}：m \ge 0}是可交换的映射族.

Constructing Reducible Brill--Noether Curves II

In this paper, we study maps from reducible curves $f : C \cup_ΓD \to \mathbb{P}^r$. We restrict our attention to two cases: first, when $f|_D$ factors through a hyperplane $H$ and $f|_C$ is transverse to $H$; and second, when $r = 3$. Degeneration to stable maps of this type have played a crucial role in works of Hartshorne, Ballico, and others, on special cases of the maximal rank conjecture. However, the general problem of studying when such stable maps with specified combinatorial types exist remains open. Here, we give criteria for such Brill--Noether curves of this first type to exist, o…