Thomas Bauer, Alex Kuronya, Tomasz Szemberg
The purpose of this paper is to investigate the behaviour of certain asymptotic invariants of line bundles on projective surfaces. In particular, we describe the volume of line bundles and their destabilizing numbers.
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Let $X$ be a nonsingular complex projective surface. The Weyl and Zariski chambers give two interesting decompositions of the big cone of $X$. We study these two decompositions and determine when a Weyl chamber is contained in the interior of a Zariski chamber and vice versa. We also determine when a Weyl chamber can intersect non-trivially with a Zariski chamber.
The big cone of every K3 surface admits two natural chamber decompositions: the decomposition into Zariski chambers, and the decomposition into simple Weyl chambers. In the present paper we compare these two decompositions and we study their mutual relationship: First, we give a numerical criterion for the two decompositions to coincide. Secondly, we study the mutual inclusions of Zariski and simple Weyl chambers. Finally, we establish the fact that -- even though the decompositions themselves may differ -- the number of Zariski chambers always equals the number of simple Weyl chambers.
Zariski chambers provide a natural decomposition of the big cone of an algebraic surface into rational locally polyhedral subcones that are interesting from the point of view of linear series. In the present paper we present an algorithm that allows to effectively determine Zariski chambers when the negative curves on the surface are known. We show how the algorithm can be used to compute the number of chambers on Del Pezzo surfaces.
The big cone of every smooth projective surface $X$ admits the natural decomposition into Zariski chambers. The purpose of this note is to give a simple criterion for the interiors of all Zariski chambers on $X$ to be numerically determined Weyl chambers. Such a criterion generalizes the results of Bauer-Funke on K3 surfaces to arbitrary smooth projective surfaces. In the last section, we study the relation between decompositions of the big cone and elliptic fibrations on Enriques surfaces.
Zariski chambers are natural pieces into which the big cone of an algebraic surface decomposes. They have so far been studied both from a geometric and from a combinatorial perspective. In the present paper we complement the picture with a metric point of view by studying a suitable notion of chamber sizes. Our first result gives a precise condition for the nef cone volume to be finite and provides a method for computing it inductively. Our second result determines the volumes of arbitrary Zariski chambers from nef cone volumes of blow-downs. We illustrate the applicability of this method by e…
Boucksom-Zariski and Weyl chambers on irreducible holomorphic symplectic manifolds
受 Bauer 等人启发，本文为射影不可约全纯辛流形的大锥建立腔室分解，并研究其与 Weyl 腔室及体积函数的关系，最终确定伪有效锥结构。
We introduce tropical dual numbers as an extension of tropical semiring. By this innovation, one can work with honest ideals, instead of congruences, and recover the Euclidean topology on affine tropical spaces similar to Zariski's approach in classical algebraic geometry. The bend loci of an ideal over tropical dual numbers coincides with the bend loci of a congruence over tropical semiring and this enables tropical dual numbers to serve as an algebraic structure for tropical geometry. Tropical Zariski topology on affine tropical spaces whose closed sets are non-linear loci of ideals over tro…
We present an improved algorithm for the computation of Zariski chambers on algebraic surfaces. The new algorithm significantly outperforms the so far available method and allows therefore to treat surfaces of high Picard number, where huge chamber numbers occur. As an application, we efficiently compute the number of chambers supported by the lines on the Segre-Schur quartic.
On the distribution of mixed Hodge loci
研究混合Hodge域在Zariski拓扑下的稠密性及其在大monodromy条件下的空集性质，改进了Brosnan-Pearlstein-Schnell的经典结果。
Two decades ago, as part of their work of generic vanishing theorems, Green-Lazarsfeld showed that over a compact Kahler manifold $X$, the cohomology jump loci in the $Pic^τ(X)$ are all translates of subtori. In this paper, we generalize this result to higher dimensional vector bundles. It is showed by Nadel that locally the moduli space of vector bundles with vanishing chern classes is canonically isomorphic to a quadratic cone in the Zariski tangent space of a point. We prove that under the isomorphism, the cohomology jump loci are defined by linear equations.
An Okounkov body is a convex subset of Euclidean space associated to a divisor on a smooth projective variety with respect to an admissible flag. In this paper, we recover the asymptotic base loci from the Okounkov bodies by studying various asymptotic invariants such as the asymptotic valuations and the moving Seshadri constants. Consequently, we obtain the nefness and ampleness criteria of divisors in terms of the Okounkov bodies. Furthermore, we compute the divisorial Zariski decomposition by the Okounkov bodies, and find upper and lower bounds for moving Seshadri constants given by the siz…
We give a concrete expression of a minimal singular metric of a big line bundle on a compact Kähler manifold which is the total space of a toric bundle over a complex torus. In this class of manifolds, Nakayama constructed examples which have line bundles admitting no Zariski-decomposition even after any proper modifications. As an application, we discuss the Zariski-closedness of non-nef loci and the openness conjecture of Demailly and Kollár in this class.
We determine several classes of smooth complex projective surfaces on which Zariski decomposition can be combined with vanishing theorems to yield cohomology formulae for all line bundles. The obtained formulae express cohomologies in terms of divisor class intersections, and are adapted to the decomposition of the effective cone into Zariski chambers. In particular, we show this occurs on generalised del Pezzo surfaces, toric surfaces, and K3 surfaces. In the second part we use these surface results to derive formulae for all line bundle cohomology on a simple class of elliptically fibered Ca…
Asymptotic base loci on hyper-Kähler manifolds
本文研究射影 hyper-Kähler 流形上大除子的渐近基轨迹，利用数值刻画和分解方法确定了 $k$-丰富除子锥的对偶，并将有效锥分解为 Mori 型腔室，肯定回答了 Sam Payne 的问题。
We study the cone of Moriwaki divisors on \bar{M}_g by means of augmented base loci. Using a result of Moriwaki, we prove that an R-divisor D satisfies the strict Moriwaki inequalities if and only if the augmented base locus of D is contained in the boundary of \bar{M}_g. Then we draw some interesting consequences on the Zariski decomposition of divisors on \bar{M}_g, on the minimal model program of \bar{M}_g and on the log canonical models \bar{M}_g(α).
We investigate the minimal singularities of metrics on a big line bundle $L$ over a projective manifold when the stable base locus $Y$ of $L$ is a submanifold of codimension $r\geq 1$. Under some assumptions on the normal bundle and a neighborhood of $Y$, we give a explicit description of the minimal singularity of metrics on $L$. We apply this result to study a higher (co-)dimensional analogue of Zariski's example, in which the line bundle $L$ is not semi-ample, however it is nef and big.
On the unirationality of conic bundles with discriminant of degree eight
本文研究了判别次数为八的锥面的不可理化性，通过构造有理多截面和变形理论分析，揭示了其在不同域上的不可理化性质。
Projective metric geometry of tropical nuclei: gap matrices, event loci, and order chambers
本文研究热带核的射影度量几何，通过间隙矩阵和事件集揭示核的结构，结合射影度量和多面体分解，建立核与最优分配理论的联系。
In very rough terms, the main theorem is that the set, which consists of semistable vector bundles with trivial rational Chern classes and nontrivial kth cohomology on a smooth complex projective variety, is a degeneration of a union of abelian varieties. More precisely, we consider the subset of the moduli space of Higgs bundles satisfying the analogous cohomological condition. We show that this set is Zariski closed and that if Sigma is the normalization of an irreducible component containing a stable point, then a connected component of a general fiber of the restriction of the Hitchin map …
Moduli spaces of stable bundles on ruled 3-folds and Brill-Noether problems
本文重新定义墙与室理论，用于研究高维代数簇上秩2稳定向量丛的模空间，并应用于规则三维流形，证明某些Brill-Noether轨迹非空。
A central question in the study of line arrangements in the complex projective plane $\mathbb{CP}^2$ is: when does the combinatorial data of the arrangement determine its topological properties? In the present work, we introduce a topological invariant of complexified real line arrangements, the chamber weight. This invariant is based on the weight counting over the points of the arrangement dual configuration, located in particular chambers of the real projective plane $\mathbb{RP}^2$, dealing only with geometrical properties. Using this dual point of view, we construct several examples of co…
Geometric Invariant Theory of Peterson Varieties
本文研究彼得森簇的几何不变量理论，利用理查森分层明确其GIT商的稳定与半稳定轨迹，确定腔分解与跨壁态射，还刻画正则性并证明商光滑的充要条件，揭示彼得森簇奇异几何与GIT商变化的关联。
In this paper, we prove a wall-crossing formula for $ε$-stable quasimaps to GIT quotients conjectured by Ciocan-Fontanine and Kim, for all targets in all genera, including the orbifold case. We prove that stability conditions in adjacent chambers give equivalent invariants, provided that both chambers are stable. In the case of genus-zero quasimaps with one marked point, we compute the invariants in the left-most stable chamber in terms of the small $I$-function. Using this we prove that the quasimap $J$-functions are on the Lagrangian cone of the Gromov--Witten theory. The proofs are based on…
Profinite completions and cohomology jump loci
本文证明 profinite 完备化在特定假设下决定 CW 复形的上同调跳跃轨迹及言语子群分次商，并应用于超平面排列，揭示其不区分算术 Zariski 对且不决定排列群。
Integral points of bounded height on a certain toric variety
本文研究某一环面上高度有界整点的计数渐近公式，构造 Peyre 常数 α 的类似物，并揭示其与整点 Zariski 稠密性新障碍的关系。
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