arXiv subjects
Publications and source records attributed to Yuze Luan.
We work on two conjectures on tensor eigenvalue multiplicities. By using the language of algebraic geometry we give stronger, more refined versions of the conjectures and we prove them in many new cases, notably for all $2\times 2\times\dots\times 2$ tensors. We also establish a connection between the rank of a tensor and the multiplicities of the zero eigenvalue.
We construct a stratification of the punctual Hilbert scheme of points on a non-reduced and nodal plane curve, $x^uy^v=0$. Each stratum is indexed by a new combinatorial object we define: a weak diagonal partition. The approach is based on introducing filtrations on ideals, together with a valuation adapted to the non-reduced structure, which allows us to analyze generators and their degrees of freedom in a systematic way. In particular, each stratum is affine when $u=1,2$; and each stratum is isomorphic to an algebraic torus times an affine space, $(\mathbb{C}^*)^{m_1} \times \mathbb{C}^{m_2}$, when $u=v,v-1,v-2$. We consequently compute the Poincar\'e polynomials of the punctual Hilbert scheme of points on curves $x^uy^v=0$ when $u=1,2,v-2,v-1,v$. As an application, we prove the colored Oblomkov-Rasmussen-Shende conjecture for the Hopf link for $u=1, v$ arbitrary, showing that the Poincar\'e polynomial is the row-colored link homology up to change of variables.
We classify the irreducible components of the Hilbert scheme of $n$ points on non-reduced algebraic plane curves, and give a formula for the multiplicities of the irreducible components. The irreducible components are indexed by partitions of $n$; all have dimension $n$; and their multiplicities are given as a polynomial of the parts of the corresponding partitions.