Functionality for isomorphism classes of curves and hypersurfaces
Max-Planck Institute of Molecular Cell Biology and Genetics（马克斯·普朗克细胞生物学与遗传学研究所）；Univ Rennes, CNRS, IRMAR - UMR 6625（雷恩大学、法国国家科学研究中心、IRMAR - UMR 6625）；Universität Ulm（乌尔姆大学）；Université Côte d’Azur（蔚蓝海岸大学）
Thomas Bouchet, Reynald Lercier, Jeroen Sijsling, Christophe Ritzenthaler
本文提出基于不变量理论的算法，解决 genus 2、3 和 4 曲线的几何问题，并包含新的理论结果。
摘要：
我们描述了基于不变量理论的算法，用于解决曲线的几何问题，主要涉及 genus 2、3 和 4 的情况。还包含了基于第一作者博士论文的新理论结果。
英文摘要：
We describe algorithms based on invariant theory to solve problems on the geometry of curves, mainly those of genus 2, 3 and 4. New theoretical results building on the first author's PhD thesis are also included.
Some of the 95 families of weighted K3 hypersurfaces have been known to have the isometric lattice polarizations. It is shown that weighted K3 hypersurfaces in such families are to one-to-one correspond by explicitly constructing the monomial birational morphisms among the weighted projective spaces. All the weight systems having the isometric Picard lattices commonly possess an anticanonical sublinear system, being confirmed that the Picard lattice of the sublinear system we obtained is the same as those of the complete linear systems.
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We study an analytically irreducible algebroid germ (X, 0) of complex singularity by considering the filtrations of its analytic algebra, and their associated graded rings, induced by the divisorial valuations associated to the irreducible components of the exceptional divisor of the normalized blow-up of the normalization (Y, 0) of (X, 0), centered at the point 0 of Y. If (X, 0) is a quasi-ordinary hypersurface singularity, we obtain that the associated graded ring is an algebra of finite type over the field of complex numbers, namely the coordinate ring of a non necessarily normal affine tor…
Let $X$ be a compact connected Riemann surface of genus $g$, with $g\geq 2$, and ${\cal M}_ξ$ a smooth moduli space of fixed determinant semistable vector bundles of rank $n$, with $n\geq 2$, over $X$. Take a smooth anticanonical divisor $D$ on ${\cal M}_ξ$. So $D$ is a Calabi-Yau variety. We compute the number of moduli of $D$, namely $\dim H^1(D, T_D)$, to be $3g-4 + \dim H^0({\cal M}_ξ, K^{-1}_{{\cal M}_ξ})$. Denote by $\cal N$ the moduli space of all such pairs $(X',D')$, namely $D'$ is a smooth anticanonical divisor on a smooth moduli space of semistable vector bundles over the Riemann su…
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We describe a vector bundle $\sE$ on a smooth $n$-dimensional ACM variety in terms of its cohomological invariants $H^i_*(\sE)$, $1\leq i \leq n-1$, and certain graded modules of "socle elements" built from $\sE$. In this way we give a generalization of the Horrocks correspondence. We prove existence theorems where we construct vector bundles from these invariants and uniqueness theorems where we show that these data determine a bundle up to isomorphisms. The cases of the quadric hypersurface in $\mathbb P^{n+1}$ and the Veronese surface in $\mathbb P^5$ are considered in more detail.
In this paper we describe the geometry of the 2m-dimensional Fano manifold G parametrizing (m-1)-planes in a smooth complete intersection Z of two quadric hypersurfaces in the complex projective space P^{2m+2}, for m>0. We show that there are exactly 2^{2m+2} distinct isomorphisms in codimension one between G and the blow-up of P^{2m} at 2m+3 general points, parametrized by the 2^{2m+2} distinct m-planes contained in Z, and describe these rational maps explicitly. We also describe the cones of nef, movable and effective divisors of G, as well as their dual cones of curves. Finally, we determin…
