Dawei Chen
A ribbon is a double structure on P^1. The geometry of a ribbon is closely related to that of a smooth curve. In this note we consider linear series on ribbons. Our main result is an explicit determinantal description for the locus W^{r}_{2n} of degree 2n line bundles with at least (r+1)-dimensional sections on a ribbon. We also discuss some results of Clifford and Brill-Noether type.
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Syzygies of canonical ribbons on higher genus curves
研究高亏格曲线上典范ribbon的合冲，证明一般ribbon的两种Clifford指标相等，指出分裂ribbon的指标相等与亏格猜想、Green猜想相关，并将其简化为辅助合冲模Koszul上同调群的消失问题。
Formal punctured ribbons and two-dimensional local fields
本文研究曲线上的形式带，证明其连同附加几何数据与二维局部域的特定子空间存在一一对应。
Extensions of curves with high degree with respect to the genus
本文分类了次数d满足4g-4 ≤ d ≤ 4g+4的线性正规曲面，并应用于多典范曲线和亏格3曲线的扩张理论，计算了相关高斯映射的余秩，证明了所有这类曲线上的ribbon是可积的，从而存在通用扩张。
We propose a definition of tropical linear series that isolates some of the essential combinatorial properties of tropicalizations of not-necessarily-complete linear series on algebraic curves. The definition combines the Baker-Norine notion of rank with the notion of tropical independence and has the property that the restriction of a tropical linear series of rank r to a connected subgraph is a tropical linear series of rank r. We show that tropical linear series of rank 1 are finitely generated as tropical modules and state a number of open problems related to algebraic, combinatorial, and …
In this paper we describe all complete and very ample linear series on 4-gonal smooth curves.
Continuous Linear Series
本文引入连续线性级数这一新对象，用精细模空间参数化线性级数到两光滑分支横截相交奇异曲线的退化，该模空间作为Hilbert商构成Osserman及Eisenbud-Harris极限线性级数模空间的新紧化。
In this note, we study the volume of arithmetic linear series with base conditions. As an application, we consider the problem of Zariski decompositions on arithmetic varieties.
We introduce a formalism of descent of moduli spaces, and use it to produce limit linear series moduli spaces for families of curves in which the components of fibers may have monodromy. We then construct a universal stack of limit linear series over the stack of semistable curves of compact type, and produce new results on existence of real curves with few real linear series.
The paper has been withdrawn because the proof of part (b) of the main theorem is incomplete.
We describe the space of Eisenbud-Harris limit linear series on a chain of elliptic curves and compare it with the theory of divisors on tropical chains. Either model allows to compute some invariants of Brill-Noether theory using combinatorial methods. We introduce effective limit linear series.
Simple limit linear series for curves of compact type with three irreducible components
本文研究了具有三个不可约分支的紧型曲线上的简单 Osserman 极限线性系，发现其简单性不再普遍成立，并通过分配性质刻画了简单基的存在条件，同时在正合极限线性系中给出了简单极限线性系的刻画及相关不等式。
Relating Different Definitions of Linear Series on Tropical Curves
研究热带曲线上线性系列不同定义间关系，引入结构化和局部弱递归热带线性系列，证明相关蕴含关系，得到组合极限线性系列等价刻画，建立扩展并构造反例，还研究了置换阵列作为局部组合数据的情况。
Slicing up multigraded linear series
研究多分次线性序列的角化态射及其乘积态射的单射条件，并应用于重构代数上的模和商栈的等变Hilbert与Quot概形。
We compute the equivariant fundamental class of the orbit closure of a linear series on the projective line. We also describe the boundary of the orbit closure and how the orbits specialise in one parameter families.
In this paper, we prove a version of the arithmetic Bertini theorem asserting that there exists a strictly small and generically smooth section of a given arithmetically free graded arithmetic linear series.
We study (a generalization of) the notion of linked determinantal loci recently introduced by the second author, showing that as with classical determinantal loci, they are Cohen-Macaulay whenever they have the expected codimension. We apply this to prove Cohen-Macaulayness and flatness for moduli spaces of limit linear series, and to prove a comparison result between the scheme structures of Eisenbud-Harris limit linear series and the spaces of limit linear series recently constructed by the second author. This comparison result is crucial in order to study the geometry of Brill-Noether loci …
Our aim in this work is to study exact Osserman limit linear series on curves of compact type $X$ with three irreducible components. This case is quite different from the case of two irreducible components studied by Osserman. For instance, for curves of compact type with two irreducible components, every refined Eisenbud-Harris limit linear series has a unique exact extension. But, for the case of three irreducible components, this property is no longer true. We find a condition characterizing when a given refined Eisenbud-Harris limit linear series has a unique exact extension. To do this, i…
In math.AG/0407496, a new construction of limit linear series is presented which functorializes and compactifies the original construction of Eisenbud and Harris, using a new space called the linked Grassmannian. The boundary of the compactification consists of crude limit series, and maps with positive-dimensional fibers to crude limit series of Eisenbud and Harris. In this paper, we carry out a careful analysis of the linked Grassmannian to obtain an upper bound on the dimension of the fibers of the map on crude limit series, thereby concluding an upper bound on the dimension of the locus of…
We generalize the theory of Newton-Okounkov bodies of big divisors to the case of graded linear series. One of the results is the generalization of slice formulas and the existence of generic Newton-Okounkov bodies for birational graded linear series. We also give a characterization of graded linear series which have full volume in terms of their base locus.
Eisenbud and Harris introduced the theory of limit linear series, and constructed a space parametrizing their limit linear series. Recently, Osserman introduced a new space which compactifies the Eisenbud-Harris construction. In the Eisenbud-Harris space, the set of refined limit linear series is always dense on a general reducible curve. Osserman asks when the same is true for his space. In this paper, we answer his question by characterizing the situations when the crude limit linear series contain an open subset of his space.
We present an algorithm for detecting basepoints of linear series of curves in the plane. Moreover, we give an algorithm for constructing a linear series of curves in the plane for given basepoints. The underlying method of these algorithms is the classical procedure of blowing up points in the plane. We motivate the algorithmic version of this procedure with several applications.
We give an analytic description for the volume of a graded linear series.
Limit linear series: combinatorial theory
本文在度量图上基于超立方体秩函数和斜率结构发展了极限线性系的纯组合理论，完成了秩一情形的完整分类并探讨了与热带代数及组合代数几何的联系。
By using classical invariant theory approach a formulas for computation of the Poincare series of the kernel of linear locally nilpotent derivations is found.
A linear series on a curve C in $P^3$ is "primary" when it does not contain the series cut by planes. We provide a lower bound for the degree of these series, in terms of deg(C), g(C) and of the number $s = min{i: h^0(I_C(i))\neq 0}$; as a corollary, we obtain bounds for the gonality of C. Examples show that our bound is sharp. Extensions to the case of general linear series and to the case of curves in higher projective spaces are considered.
We study linear series on a general curve of genus $g$, whose images are exceptional with regard to their secant planes. Working in the framework of an extension of Brill-Noether theory to pairs of linear series, we prove that a general curve has no linear series with exceptional secant planes, in a very precise sense, whenever the total number of series is finite. Next, we partially solve the problem of computing the number of linear series with exceptional secant planes in a one-parameter family in terms of tautological classes associated with the family, by evaluating our hypothetical formu…
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