Mu-Lin Li
Using virtual residue, which is a generalization of Grothendieck residue, we generalized Cayley- Bacharach Theorem to the cases with positive dimensions.
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The Cayley-Bacharach Theorem states that all cubic curves through eight given points in the plane also pass through a unique ninth point. We write that point as an explicit rational function in the other eight.
Given a 0-dimensional scheme $\mathbb{X}$ in a projective space $\mathbb{P}^n_K$ over a field $K$, we characterize the Cayley-Bacharach property of $\mathbb{X}$ in terms of the algebraic structure of the Dedekind different of its homogeneous coordinate ring. Moreover, we characterize Cayley-Bacharach schemes by Dedekind's formula for the conductor and the complementary module, we study schemes with minimal Dedekind different using the trace of the complementary module, and we prove various results about almost Gorenstein and nearly Gorenstein schemes.
In this paper, we prove the Noether theorem with the multiplicities described by PD operators. Despite the known analogue versions in this case the provided conditions are necessary and sufficient. We also prove the Cayley-Bacharach theorem with PD multiplicities. As far as we know this is the first generalization of this theorem in the case of multiple intersections.
The Cayley-Bacharach property and the Levinson-Ullery conjecture
本文证明了Levinson-Ullery猜想在d=4且r≥1的情况下的成立，研究了有限点集在射影空间中的几何配置性质。
In this paper, we examine linear conditions on finite sets of points in projective space implied by the Cayley-Bacharach condition. In particular, by bounding the number of points satisfying the Cayley-Bacharach condition, we force them to lie on unions of low-dimensional linear spaces. These results are motivated by investigations into degrees of irrationality of complete intersections, which are controlled by minimum-degree rational maps to projective space. As an application of our main theorem, we describe the fibers of such maps for certain complete intersections of codimension two.
We prove a Cayley-Bacharach-type theorem for points in projective space $\mathbb{P}^n$ that lie on a complete intersection of $n$ hypersurfaces. This is made possible by new bounds on the growth of the Hilbert function of almost complete intersections.
Let $M$ be a matroid satisfying a matroidal analogue of the Cayley-Bacharach condition. Given a number $k \ge 2$, we show that there is no nontrivial bound on ranks of a $k$-tuple of flats covering the underlying set of $M$. This addresses a question of Levinson-Ullery motivated by earlier results which show that bounding the number of points satisfying the Cayley-Bacharach condition forces them to lie on low-dimensional linear subspaces. We also explore the general question what matroids satisfy the matroidal Cayley-Bacharach condition of a given degree and its relation to the geometry of gen…
In this paper, we study the geometry of points in complex projective space that satisfy the Cayley-Bacharach condition with respect to the complete linear system of hypersurfaces of given degree. In particular, we improve a result by Lopez and Pirola and we show that, if $k\geq 1$ and $Γ=\{P_1,\dots,P_d\}\subset \mathbb{P}^n$ is a set of distinct points satisfying the Cayley-Bacharach condition with respect to $|\mathcal{O}_{\mathbb{P}^n}(k)|$, with $d\leq h(k-h+3)-1$ and $3\leq h\leq 5$, then $Γ$ lies on a curve of degree $h-1$. Then we apply this result to the study of linear series on curve…
In recent work, J. Hansen uses cohomological methods to find a lower bound for the minimum distance of an evaluation code determined by a reduced complete intersection in the projective plane. In this paper, we generalize Hansen's results from P^2 to P^m; we also show that the hypotheses in Hansen's work may be weakened. The proof is succinct and follows by combining the Cayley-Bacharach theorem and bounds on evaluation codes obtained from reduced zero-schemes.
Grid-free linear hypergraphs via Cayley-Bacharach
本文通过Cayley-Bacharach构造了无r×r网格的线性超图，解决了r≥3时的超图构造问题。
Error terms for the motives of discriminant complements and a Cayley-Bacharach theorem
本文在简化假设下证明了关于Cayley-Bacharach集的相关猜想，推导其位于低次曲线的结论，并以此估计光滑超曲面空间归一化motive的误差项，同时证明高维下motive的收敛性等判别式补空间相关结果。
We consider the relationship between a matroidal analogue of the degree $a$ Cayley-Bacharach property (finite sets of points failing to impose independent conditions on degree $a$ hypersurfaces) and geometric properties of matroids. If the matroid polytopes in question are nestohedra, we show that the minimal degree matroidal Cayley-Bacharach property denoted $MCB(a)$ is determined by the structure of the building sets used to construct them. This analysis also applies for other degrees $a$. Also, it does not seem to affect the combinatorial equivalence class of the matroid polytope. However, …
Griffiths and Harris showed in 1978 that if E is a rank n vector bundle on a smooth projective variety of dimension n, and if s is a section of E vanishing simply on a finite set Z, then any section of (K_X + det E) vanishing at all but one of the points of Z must also vanish on the remaining one. This generalizes the classical theorem of Cayley-Bacharach, which appears when E is a direct sum of line bundles on projective space. In a recent paper, Mu-Lin Li proposed an extension allowing for the possibility that the zero-locus of s has positive dimensional components, but his result requires a…
Multivariate Lagrange Interpolation and an Application of Cayley-Bacharach Theorem For it
本文深入研究了多元拉格朗日插值，提出了充分交的概念，利用Bezout定理等证明了沿充分交代数流形的多项式空间维数，并基于Mysovskikh定理给出了适定节点集存在性的刻画条件，最后通过推广的Cayley-Bacharach定理解决了低维代数流形上的插值问题。
Covering gonality of symmetric products of curves and Cayley-Bacharach condition on Grassmannians
本文研究亏格至少为 k+1 的曲线的 k 重对称积的覆盖 gonality，证明当 k=2,3,4 时等于曲线本身的 gonality，关键工具是 Grassmann 流形上的 Cayley-Bacharach 条件及其线性子空间几何。
Using a new point of view inspired by hyperplane arrangements, we generalize the converse to Pascal's Theorem, sometimes called the Braikenridge-Maclaurin Theorem. In particular, we show that if 2k lines meet a given line, colored green, in k triple points and if we color the remaining lines so that each triple point lies on a red and blue line then the points of intersection of the red and blue lines lying off the green line lie on a unique curve of degree k-1. We also use these ideas to extend a second generalization of the Braikenridge-Maclaurin Theorem, due to Möbius. Finally we use Terrac…
Given an ACM set $\mathbb{X}$ of points in a multiprojective space $\mathbb{P}^m\times\mathbb{P}^n$ over a field of characteristic zero, we are interested in studying the Kähler different and the Cayley-Bacharach property for $\mathbb{X}$. In $\mathbb{P}^1\times\mathbb{P}^1$, the Cayley-Bacharach property agrees with the complete intersection property and it is characterized by using the Kähler different. However, this result fails to hold in $\mathbb{P}^m\times\mathbb{P}^n$ for $n>1$ or $m>1$. In this paper we start an investigation of the Kähler different and its Hilbert function and then pr…
Using Cayley trick, we define the notions of mixed toric residues and mixed Hessians associated with $r$ Laurent polynomials $f_1,...,f_r$.We conjecture that the values of mixed toric residues on the mixed Hessians are determined by mixed volumes of the Newton polytopes of $f_1,...,f_r$. Using mixed toric residues, we generalize our Toric Residue Mirror Conjecture to the case of Calabi-Yau complete intersections in Gorenstein toric Fano varieties obtained from nef-partitions of reflexive polytopes.
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