A Construction of the Rational Function Sheaves on Elliptic Curves

Abstract The authors introduce an effective method to construct the rational function sheaf κ on an elliptic curve E, and further study the relationship between κ and any coherent sheaf on E. Finally, it is shown that the category of all coherent sheaves of finite length on E is completely characterized by κ.     

The double Ringel-Hall algebra on a hereditary abelian finitary length category

In this paper, we study the category H (ρ) of semi-stable coherent sheaves of a fixed slope ρ over a weighted projective curve. This category has nice properties: it is a hereditary abelian finitary length category. We will define the Ringel-Hall algebra of H (ρ) and relate it to generalized Kac-Moody Lie algebras. Finally we obtain the Kac type theorem to describe the indecomposable objects in this category, i.e. the indecomposable semi-stable sheaves.     

Tilting objects on tubular weighted projective lines: A cluster tilting approach

Using the cluster tilting theory,we investigate the 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 constructed via the cluster tilting mutation.Moreover,the cluster tilting approach also provides a new method to classify the endomorphism algebras of the tilting objects in the category of coherent sheaves and the associated bounded derived category.     

Nadel-type multiplier ideal sheaves on complex spaces with singularities

In this article, we introduce multiplier ideal sheaves on complex spaces with singularities(not necessarily normal) via Ohsawa’s extension measure, as a special case of which, it turns out to be the socalled Mather-Jacobian multiplier ideals in the algebro-geometric setting. Inspired by Nadel’s coherence and Guan-Zhou’s strong openness properties of the multiplier ideal sheaves, we discuss similar properties for the generalized multiplier ideal sheaves. As applications, we obtain a reasonable ge...     

Some Characterizations of Prfer v-Multiplication Rings

In this paper, we study Prfer v-multiplication rings(PVMRs) and give some new characterizations of PVMRs. Moreover, we show that a Marot ring R is a PVMR if and only if every w-ideal of R is complete.     

Scattering Diagrams,Sheaves, and Curves

We review the recent proof of the N. Takahashi's conjecture on genus 0 Gromov–Witten invariants of(P~2, E), where E is a smooth cubic curve in the complex projective plane P~2. The main idea is the use of the algebraic notion of scattering diagram as a bridge between the world of Gromov–Witten invariants of(P~2, E) and the world of moduli spaces of coherent sheaves on P~2. Using this bridge, the N. Takahashi's conjecture can be translated into a manageable question about moduli spaces of coherent sheaves on P~2. This survey is based on a three hours lecture series given as part of the Beijing–Zurich moduli workshop in Beijing, 9–12 September 2019.     

Abelianness of the “missing part” from a sheaf category to a module category

This paper investigates the structure of the"missing part"from the category of coherent sheaves over a weighted projective line of weight type(2,2,n)to the category of finitely generated right modules on the associated canonical algebra.By constructing a t-structure in the stable category of the vector bundle category,we show that the"missing part"is equivalent to the heart of the t-structure,hence it is abelian.Moreover,it is equivalent to the category of finitely generated modules on the path algebra of type An-1.     

利用Prüfer变换求解微分方程

针对一类二阶非线性常微分方程,利用Prüfer变换将其约化为特殊的一阶常微分方程组,从而使其求解过程得以简化.实例说明应用Prüfer变换求解一类偏微分方程边值问题的技巧.     

GENERIC OSCILLATIONS OF FIRST ORDER IMPULSIVE DELAY DIFFERENTIAL EQUATIONS

In this paper, we will study generic oscillation and generic nonoscillation of first order impulsive delay differential equations. A necessary and sufficient condition and some sufficient conditions are obtained for both phenomena based on the root of characteristic equation.     

L2 Extension and Effectiveness of Lp Strong Openness Property

In this note, we present an L² extension approach to the effectiveness result of Lpstrong openness property of multiplier ideal sheaves.     

Adic Attractor and Topological Entropy

1 IntroductionLet M be a manifold (possibly with boundary), and F : M → M be continuous. Call a closed invariant set A (?) M an adic attractor of f if it attracts almost all points (in the sense of Lebesgue measure) and the restriction f|A is topologically conjugate to an adic system. Such an attractor A is called n-adic if the restriction f|A can be topologically conjugate the n-adic system.     

The geometry of the moduli space of one-dimensional sheaves

Let Md be the moduli space of stable sheaves on P2with Hilbert polynomial dm+1.In this paper,we determine the effective and the nef cone of the space Md by natural geometric divisors.Main idea is to use the wall-crossing on the space of Bridgeland stability conditions and to compute the intersection numbers of divisors with curves by using the Grothendieck-Riemann-Roch theorem.We also present the stable base locus decomposition of the space M6.As a byproduct,we obtain the Betti numbers of the moduli spaces,which confirm the prediction in physics.     

Multivariate discriminant and iterated resultant

In this paper,we study the relationship between iterated resultant and multivariate discriminant.We show that,for generic form f(x_n) with even degree d,if the polynomial is squarefreed after each iteration,the multivariate discriminant △(f) is a factor of the squarefreed iterated resultant.In fact,we find a factor Hp(f,[x_1,...,x_n]) of the squarefreed iterated resultant,and prove that the multivariate discriminant △(f) is a factor of Hp(f,[x_1,...,x_n]).Moreover,we conjecture that Hp(f,[x_1,...,x_n]) = △(f) holds for generic form/,and show that it is true for generic trivariate form f(x,y,z).     

TRAVELING WAVES IN A BIOLOGICAL REACTION-DIFFUSION MODEL WITH STRONG GENERIC DELAY KERNEL AND NON-LOCAL EFFECT

In this paper,we consider the reaction diffusion equations with strong generic delay kernel and non-local effect,which models the microbial growth in a flow reactor.The existence of traveling waves is established for this model.More precisely,using the geometric singular perturbation theory,we show that traveling wave solutions exist provided that the delay is sufficiently small with the strong generic delay kernel.     

Minimal silting modules and ring extensions

Ring epimorphisms often induce silting modules and cosilting modules, termed minimal silting or minimal cosilting. The aim of this paper is twofold. Firstly, we determine the minimal tilting and minimal cotilting modules over a tame hereditary algebra. In particular, we show that a large cotilting module is minimal if and only if it has an adic module as a direct summand. Secondly, we discuss the behavior of minimality under ring extensions. We show that minimal cosilting modules over a commutat...     

Multiplier ideal sheaves in complex and algebraic geometry

The application of the method of multiplier ideal sheaves to effective problems in algebraic geometry is briefly discussed. Then its application to the deformational invari-ance of plurigenera for general compact algebraic manifolds is presented and discussed. Finally its application to the conjecture of the finite generation of the canonical ring is explored, and the use of complex algebraic geometry in complex Neumann estimates is discussed.     

Categorification of integrable representations of quantum groups

We categorify the highest weight integrable representations and their tensor products of a symmetric quantum Kac–Moody algebra. As a byproduct, we obtain a geometric realization of Lusztig's canonical bases of these representations as well as a new positivity result. The main ingredient in the underlying geometric construction is a class of micro-local perverse sheaves on quiver varieties.     

Bilinear forms with trace functions over arbitrary sets and applications to Sato-Tate

We prove non-trivial upper bounds for general bilinear forms with trace functions of bountiful sheaves,where the supports of two variables can be arbitrary subsets in F_p of suitable sizes.This essentially recovers the Polya-Vinogradov range,and also applies to symmetric powers of Kloosterman sums and Frobenius traces of elliptic curves.In the case of hyper-Kloosterman sums,we can beat the Pólya-Vinogradov barrier by combining additive combinatorics with a deep result of Kowalski,Miche...     

The Finite-time Ruin Probability for the Jump-Diffusion Model with Constant Interest Force

In this paper, we consider the finite time ruin probability for the jump-diffusion Poisson process. Under the assurnptions that the claimsizes are subexponentially distributed and that the interest force is constant, we obtain an asymptotic formula for the finite-time ruin probability. The results we obtain extends the corresponding results of Kliippelberg and Stadtmüller and Tang.     

Exponential martingale for compound Poisson process with latent variable and its applications

In this article, we construct an exponential martingale for the compound Poisson process with latent variableWith the help of this exponential martingale, we provide an asymptotic behavior of the coherent entropic risk measure for the compound Poisson process and a deviation inequality for the ruin probability of the partly shifted risk process.