We study moduli of ``self-associated'' sets of points in ${\bf P}^n$ for small $n$. In particular, we show that for $n=5$ a general such set arises as a hyperplane section of the Lagrangean Grassmanian $LG(5,10) \subset {\bf P}^{15}$ (this was conjectured by Eisenbud-Popescu in {\it Geometry of the Gale transform}, J. Algebra 230); for $n=6$, a general such set arises as a hyperplane section of the Grassmanian $G(2,6) \subset {\bf P}^{14}$. We also make a conjecture for the next case $n=7$. Our results are analogues of Mukai's characterization of general canonically embedded curves in ${\bf P}^6$ and ${\bf P}^7$, resp.
This paper designs an alogrithm to compute the minimal combinations of finite sets in Euclidean spaces, and applys the algorithm of study the moment maps and geometric invariant stability of hypersurfaces. The classical example of cubic curves is repeated by the algorithm. Furhtermore the alogrithm works for cubic surfaces. For given affinely indepdent subsets of monomials, the algorithm can output the unique unstable points of the Morse strata if it exists. Also there is a discussion on the affinely dependent sets of monomials.
Motivated by a question from V. Arnold about self-dual curves in projective spaces, we study {\cal M}_{m,n,k}: the moduli space of m-self-dual n-gons in {\mathbb P}^k. This paper lays out an explicit construction of self-dual polygons, and for specific cases of n and m, provides the dimension of {\cal M}_{m,n,k}. We include a conjecture about the Pentagram map in higher dimensions that generalizes Clebsch's theorem, which states that every pentagon in \mathbb{RP}^2 is invariant under the Pentagram map.
We call a set of points $Z\subset{\mathbb P}^{3}_{\mathbb C}$ an $(a,b)$-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point $P$ to a plane is a complete intersection of curves of degrees $a$ and $b$. Examples which we call grids have been known since 2011. The only nongrid nondegenerate examples previously known had $ab=12, 16, 20, 24, 30, 36, 42, 48, 54$ or $60$. Here, for any $4 \leq a \leq b$, we construct nongrid nondegenerate $(a,b)$-geproci sets in a systematic way. We also show that the only such example with $a=3$ is a $(3,4)$-geproci…
KU Leuven（荷语鲁汶大学）；Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé（里尔大学、法国国家科学研究中心、联合研究单位8524-保罗·潘勒韦实验室）；Chalmers University of Technology（查尔姆斯理工大学）；University of Cambridge（剑桥大学）；University of Münster（明斯特大学）
City University of Hong Kong（香港城市大学）；Univ. de Buenos Aires & CONICET（布宜诺斯艾利斯大学与阿根廷国家科学研究委员会）；City College and the Graduate Center of CUNY（纽约市立学院与纽约市立大学研究生中心）
On the height boundedness of periodic and preperiodic points of dominant rational self-maps on projective varieties
Osaka Metropolitan University（大阪公立大学）；NTT Institute for Fundamental Mathematics, NTT Communication Science Laboratories, NTT, Inc.（日本电信电话株式会社基础数学研究所、通信科学实验室）
Analytic and Algebraic Oka-1 Approximation for Smooth Projective Morphisms with Rationally Connected Fibers
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学重点实验室）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
We study the complexity of birational self-maps of a projective threefold $X$ by looking at the birational type of surfaces contracted. These surfaces are birational to the product of the projective line with a smooth projective curve. We prove that the genus of the curves occuring is unbounded if and only if $X$ is birational to a conic bundle or a fibration into cubic surfaces. Similarly, we prove that the gonality of the curves is unbounded if and only if $X$ is birational to a conic bundle.
This paper deals with holomorphic self-maps of the complex projective plane and the algebraic relations among the eigenvalues of the derivatives at the fixed points. These eigenvalues are constrained by certain index theorems such as the holomorphic Lefschetz fixed-point theorem. A simple dimensional argument suggests there must exist even more algebraic relations that the ones currently known. In this work we analyze the case of quadratic self-maps having an invariant line and obtain all such relations. We also prove that a generic quadratic self-map with an invariant line is completely deter…
In this paper, we consider the limit $ \lim_{n \to \infty} \sum_{v\in S} λ_{Y,v}(f^{n}(x))/h_{H}(f^{n}(x)) $ where $f \colon X \longrightarrow X$ is a surjective self-morphism on a smooth projective variety $X$ over a number field, $S$ is a finite set of places, $ λ_{Y,v}$ is a local height function associated with a proper closed subscheme $Y \subset X$, and $h_{H}$ is an ample height function on $X$. We give a geometric condition which ensures that the limit is zero, unconditionally when $\dim Y=0$ and assuming Vojta's conjecture when $\dim Y\geq1$. In particular, we prove (one is unconditio…
The twistor space of the moduli space of solutions of Hitchin's self-duality equations can be identified with the Deligne-Hitchin moduli space of $λ$-connections. We use real projective structures on Riemann surfaces to prove the existence of new components of real holomorphic sections of the Deligne-Hitchin moduli space. Applying the twistorial construction we show the existence of new hyper-Kähler manifolds associated to any compact Riemann surface of genus $g\geq2$. These hyper-Kähler manifolds can be considered as moduli spaces of (certain) singular solutions of the self-duality equations.
