The implicitization problem for $ϕ: P^n --> (P^1)^{n+1}$

Implicitization using approximation complexes

We present a method for the implicitization problem that goes back to the work of Sederberg and Chen. The formalism we use, with approximation complexes as a key ingredient, is due to Jean-Pierre Jouanolou and was explained in details in this context in his joint work with Laurent Buse. Most of this note is dedicated to presenting the method, the geometric ideas behind it and the tools from commutative algebra that are needed. In the last section, we give the most advanced results we know related to this approach.

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…

计算多分次多项式映射的隐式化

Computing Implicitizations of Multi-Graded Polynomial Maps

本文提出一种基于多分次和线性代数的隐式化计算方法，在 Gröbner 基方法失效时能高效计算核的生成元，并已在系统发育学示例中验证，算法可并行化。

Koszul duality for Lie algebroids

This paper studies the role of dg-Lie algebroids in derived deformation theory. More precisely, we provide an equivalence between the homotopy theories of formal moduli problems and dg-Lie algebroids over a commutative dg-algebra of characteristic zero. At the level of linear objects, we show that the category of representations of a dg-Lie algebroid is an extension of the category of quasi-coherent sheaves on the corresponding formal moduli problem. We describe this extension geometrically in terms of pro-coherent sheaves.

张量积曲面与分层斜导子

Tensor product surfaces and graded syzygies

本文研究了由双分层理想I_U具有单分层斜导子时的张量积曲面的隐式方程问题，通过扩展Duarte-Schenck和作者之前的工作，解决隐式化难题。

形式模问题的一种新方法

A new approach to formal moduli problems

本文利用operadic calculus引入了处理无穷小形变问题的新框架，构建了代数与其Koszul对偶代数间的伴随关系，并给出同调准则，为Lurie--Pridham定理等提供了新证明且能直接生成点集模型。