Implicitizing rational hypersurfaces using approximation complexes

Implicitization of rational hypersurfaces via linear syzygies: a practical overview

We unveil in concrete terms the general machinery of the syzygy-based algorithms for the implicitization of rational surfaces in terms of the monomials in the polynomials defining the parametrization, following and expanding our joint article with M. Dohm. These algebraic techniques, based on the theory of approximation complexes due to J. Herzog, A, Simis and W. Vasconcelos, were introduced for the implicitization problem by J.-P. Jouanolou, L. Busé, and M. Chardin. Their work was inspired by the practical method of moving curves, proposed by T. Sederberg and F. Chen, translated into the lang…

库默表面与二次线复形在特征二的情况

Kummer surfaces and quadratic line complexes in characteristic two

本文探讨了在特征二条件下二次线复形与库默表面之间的关系及其经典理论的类比。

On the closed image of a rational map and the implicitization problem

In this paper, we investigate some topics around the closed image $S$ of a rational map $λ$ given by some homogeneous elements $f_1,...,f_n$ of the same degree in a graded algebra $A$. We first compute the degree of this closed image in case $λ$ is generically finite and $f_1,...,f_n$ define isolated base points in $\Proj(A)$. We then relate the definition ideal of $S$ to the symmetric and the Rees algebras of the ideal $I=(f_1,...,f_n) \subset A$, and prove some new acyclicity criteria for the associated approximation complexes. Finally, we use these results to obtain the implicit equation of…

Intersection spaces, perverse sheaves and type IIB string theory

The method of intersection spaces associates rational Poincaré complexes to singular stratified spaces. For a conifold transition, the resulting cohomology theory yields the correct count of all present massless 3-branes in type IIB string theory, while intersection cohomology yields the correct count of massless 2-branes in type IIA theory. For complex projective hypersurfaces with an isolated singularity, we show that the cohomology of intersection spaces is the hypercohomology of a perverse sheaf, the intersection space complex, on the hypersurface. Moreover, the intersection space complex …

正加权齐次除子的扭曲对数复形

Twisted logarithmic complexes of positively weighted homogeneous divisors

本文研究正加权齐次除子的扭曲对数复形的拟同构条件，证明LCT与Bernstein-Sato多项式根及一阶算子生成零化子等价，并给出超平面排列情形的简单证明。

Implicitization of rational maps

Motivated by the interest in computing explicit formulas for resultants and discriminants initiated by Bézout, Cayley and Sylvester in the eighteenth and nineteenth centuries, and emphasized in the latest years due to the increase of computing power, we focus on the implicitization of hypersurfaces in several contexts. Our approach is based on the use of linear syzygies by means of approximation complexes, following [Busé Jouanolou 03], where they develop the theory for a rational map $f:P^{n-1}\dashrightarrow P^n$. Approximation complexes were first introduced by Herzog, Simis and Vasconcelos…

Algorithmic Computation of de Rham Cohomology of Complements of Complex Affine Varieties

Let $X=\C^n$. In this paper we present an algorithm that computes the de Rham cohomology groups $H^i_{dR}(U,\C)$ where $U$ is the complement of an arbitrary Zariski-closed set $Y$ in $X$. Our algorithm is a merger of the algorithm given by T.~Oaku and N.~Takayama (\cite{O-T2}), who considered the case where $Y$ is a hypersurface, and our methods from \cite{W-1} for the computation of local cohomology. We further extend the algorithm to compute de Rham cohomology groups with support $H^i_{dR,Z}(U,\C)$ where again $U$ is an arbitrary Zariski-open subset of $X$ and $Z$ is an arbitrary Zariski-clo…