Related papers: Triangle singularities, ADE-chains, and weighted p…

We show a surprising link between singularity theory and the invariant subspace problem of nilpotent operators as recently studied by C. M. Ringel and M. Schmidmeier, a problem with a longstanding history going back to G. Birkhoff. The link…

The present paper focuses on the study of the stable category of vector bundles for the weighted projective lines of weight triple. We find some important triangles in this category and use them to construct tilting objects with tubular…

For a weighted projective line, the stable category of its vector bundles modulo lines bundles has a natural triangulated structure. We prove that, for any positive integers $p, q, r$ and $r'$ with $r'\leq r$, there is an explicit…

This paper deals with the triangle singularity defined by the \linebreak equation $f=X_1^{p_1}+X_2^{p_2}+X_3^{p_3}$ for weight triple $(p_1,p_2,p_3)$, as well as the category of coherent sheaves over the weighted projective line…

Using cluster tilting theory, we investigate tilting objects in the stable category of vector bundles on a weighted projective line of weight type $(2, 2, 2, 2)$. More precisely, a tilting object consisting of rank-two bundles is…

We construct a tilting object for the stable category of vector bundles on a weighted projective line X of type (2,2,2,2;\lambda), consisting of five rank two bundles and one rank three bundle, whose endomorphism algebra is a canonical…

This paper classifies all the tilting bundles in the category of coherent sheaves on the weighted projective line of weight type $(2, 2, n)$, and investigates the abelianness of the "missing part" from the category of coherent sheaves to…

In this paper, we provide a new method for constructing tilting objects in a triangulated category via recollements. The $p$-cycle approach to exceptional curve processes significant advantages in constructing recollements and ladders, due…

The present paper focuses on the study of t-stabilities on a triangulated category in the sense of Gorodentsev, Kuleshov and Rudakov. We give an equivalent description for the finest t-stability on a piecewise hereditary triangulated…

We associate to a regular system of weights a weighted projective line over an algebraically closed field of characteristic zero in two different ways. One is defined as a quotient stack via a hypersurface singularity for a regular system…

Let $\mathbb{X}$ be a weighted projective line and $\mathcal{C}_\mathbb{X}$ the associated cluster category. It is known that $\mathcal{C}_\mathbb{X}$ can be realized as a generalized cluster category of quiver with potential. In this note,…

We shall show that the stable categories of graded Cohen-Macaulay modules over quotient singularities have tilting objects. In particular, these categories are triangle equivalent to derived categories of finite dimensional algebras. Our…

We prove the existence of extremal, non-csc, K\"ahler metrics on certain unstable projectivised vector bundles $\P (E) \to M$ over a cscK-manifold $M$ with discrete holomorphic automorphism group, in certain adiabatic K\"ahler classes. In…

We show that the projectivization of the exceptional rank 2 vector bundle on an arbitrary smooth V14 Fano threefold after a certain natural flop turns into the projectivization of an instanton vector bundle on a smooth cubic threefold. And…

Let $\Cal E$ be a very ample vector bundle of rank two on a smooth complex projective threefold $X$. An inequality about the third Segre class of $\Cal E$ is provided when $K_X+\det \Cal E$ is nef but not big, and when a suitable positive…

We study the singularity category of the Brieskorn-Pham singularity $R=k[X_1, \dots, X_4]/(\sum_{i=1}^{4} X_i^{p_i})$, associated with the Geigle-Lenzing projective space $\mathbb{X}$ of weight quadruple $(p_1,\dots, p_4)$, by investigating…

In this paper we give positive answers for two open questions on extension bundles over weighted projective lines, raised by Kussin, Lenzing and Meltzer in the paper ``Triangle singularities, ADE-chains and weighted projective lines''.

Let $M=P(E)$ be the complex manifold underlying the total space of the projectivization of a holomorphic vector bundle $E \to \Sigma$ over a compact complex curve $\Sigma$ of genus $\ge 2$. Building on ideas of Fujiki, we prove that $M$…

For a higher Nakayama algebra $A$ in the sense of Jasso-K\"{u}lshammer, we show that the singularity category of $A$ is triangulated equivalent to the stable module category of a self-injective higher Nakayama algebra. This generalizes a…

Working over an algebraically closed field $k$ of any characteristic, we determine the matrix factorizations for the --- suitably graded --- triangle singularities $f=x^a+y^b+z^c$ of domestic type, that is, we assume that $(a,b,c)$ are…