Arend Bayer
Using results of Gathmann, we prove the following theorem: If a smooth projective variety X has generically semisimple (p,p)-quantum cohomology, then the same is true for the blow-up of X at any number of points. This a successful test for a modified version of Dubrovin's conjecture from the ICM 1998.
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Let X be a compact toric surface. There exists a sequence of torus equivariant blow-ups of X such that the blown-up toric surface obtained admits a cscK metric.
We prove that the sum of the $α$-invariants of two different Kollár components of a Kawamata log terminal singularity is less than $1$.
We extend an argument of Stoppa to make some prgress towards a proof that Kähler-Einstein manifolds are "b-stable". We point out some algebro-geometric questions, involving finite generation, that arise.
An asymptotic formula for the Tian-Paul CM-line of a flat family blown-up at a flat closed sub-scheme is given. As an application we prove that the blow-up of a polarized manifold along a (relatively) Chow-unstable submanifold admits no (extremal) constant scalar curvature Kahler metrics in classes making the exceptional divisors sufficiently small. Moreover a geometric characterization of relatively Chow-unstable configuration of points in the projective space is given. From this we get new examples of classes admitting no extremal Kahler metric also in the case of the projective plane blown-…
In this paper, using the gluing formula of Gromov-Witten invariants under symplectic cutting, due to Li and Ruan, we studied the Gromov-Witten invariants of blow-ups at a smooth point or along a smooth curve. We established some relations between Gromov-Witten invariants of M and its blow-ups at a smooth point or along a smooth curve.
Refined Chern characteristic classes of blow-ups
该研究针对紧复流形的精细化陈特征类，证明其爆破公式，建立对应无分母黎曼-罗赫定理，并以岩泽流形为例分析其精细化陈特征类在爆破变换下的行为。
We propose another proof of the geometric class field theory for curves by considering blow-ups of symmetric products of curves.
The purely log terminal blow-ups of three-dimensional terminal toric singularities are described.
We consider the problem to determine which blow-ups along subvarieties in products of two projective spaces are log Fano. By describing the nef cones of such blow-ups with special centers, we give a partial classification result. For each example, explicit boundary divisors are also given.
Divisors with minimal discrepancy over cA points are classified
We answer a question of Keel: How much can one improve a non-toroidal ideal sheaf by a sequence of toroidal blow-ups? Version2: Corollary 5 was proved earlier by Tevelev and Ulirsch.
In this paper we determine which blow-ups $X$ of $\mathbb{P}^n$ at general points are log Fano, that is, when there exists an effective $\mathbb{Q}$-divisor $Δ$ such that $-(K_X+Δ)$ is ample and the pair $(X,Δ)$ is klt. For these blow-ups, we produce explicit boundary divisors $Δ$ making $X$ log Fano.
Higher Nash blow-ups and Nobile theorem
本文研究T. Yasuda提出的高阶纳什爆破及经典诺比尔定理的高阶版本，给出分次情形下高阶诺比尔定理的无特征证明，以及特征零情形下次二阶纳什爆破的证明。
Logarithmic resolution of singularities in characteristic 0 using weighted blow-ups
本文研究特征零域上的奇点消解问题，核心方法是使用加权爆破，构造出例外因子呈简单正常交叉的对数消解。
In the first part of the paper, we give an explicit algorithm to compute the (genus zero) Gromov-Witten invariants of blow-ups of an arbitrary convex projective variety in some points if one knows the Gromov-Witten invariants of the original variety. In the second part, we specialize to blow-ups of P^r and show that many invariants of these blow-ups can be interpreted as numbers of rational curves on P^r having specified global multiplicities or tangent directions in the blown-up points. We give various numerical examples, including a new easy way to determine the famous multiplicity d^{-3} fo…
We use degeneration formula to study the change of stable pair invariants of 3-folds under blow-ups and obtain some closed blow-up formulae. Related results on Donaldson-Thomas invariants are also discussed. Our results give positive evidence for GW/DT/P correspondence, and also give partial correspondence for varieties not necessarily toric or complete intersections.
In this paper we prove the existence of purely log terminal blow-up for Kawamata log terminal singularity and obtain the criterion for a singularity to be weakly exceptional in terms of the exceptional divisor of plt blow-up.
Positivity of line bundles on general blow ups of Hirzebruch surfaces
本文研究Hirzebruch曲面一般爆胀上线丛的正性，针对丰富性、整体生成性、非常丰富性和k-非常丰富性分别给出充分的数值条件。
We give an explicit presentation with generators and relations of the quantum cohomology ring of the blow-up of a projective space along a linear subspace.
Inverting log blow-ups in log geometry
本文通过局部逆构对数胀开引入可除对数空间范畴，证明对数光滑可除对数空间的闭浸入存在胀开，为对数概形母题六函子形式化提供基础。
This paper is a revision of the author's old preprint "Exactness, integrality, and log modifications". We will prove that any quasi-compact morphism of fs log schemes can be modified locally on the base to an integral morphism by base change by fs log blow-ups.
In this paper we study the pseudoeffective cones of blow-ups of Grassmannians at sets of points. For small numbers of points, the cones are often spanned by proper transforms of Schubert classes. In some special cases, we provide sharp bounds for when the Schubert classes fail to span and we describe the resulting geometry.
We use a family of algebraic foliations given by A. Lins Neto to provide new evidences to a conjecture, related to the Harbourne-Hirschowitz's one and implying the Nagata's conjecture, which concerns the structure of the Mori cone of blow-ups of $\mathbb{P}^2$ at very general points.
We study k-very ampleness of line bundles on blow-ups of hyperelliptic surfaces at r very general points. We obtain a numerical condition on the number of points for which a line bundle on the blow-up of a hyperelliptic surface at these r points gives an embedding of order k.
We suggest an effective procedure to calculate numerical invariants for rank two bundles over blown-up surfaces. We study the moduli spaces M_j of bundles on the blown-up plane splitting over the exceptional divisor as O(j)+O(-j). We use the numerical invariants to give a topological decomposition of M_j.
We prove the irredcibility (and the rational connectedness) of the moduli spaces of (free) morphisms from a projective line to a successive blowing-up of a product of projective spaces if a suitable numerical condition on morphisms is provided (and also if all centers of the blowing-up are rationally connected).
Cones of effective cycles on blow ups of projective spaces along rational curves
该研究针对射影空间沿光滑有理曲线的爆破上的有效闭链锥问题，明确确定了有理正规曲线爆破上的除子、一维及二维闭链锥，低维下结果更优，其割线簇消解几何为核心，有效闭链计算具独立价值。
Virtual Cartier divisors and blow-ups
本文基于“虚拟有效Cartier除子”概念证明正则浸入子概形胀开的泛性质，并在导出代数几何中构造拟光滑闭浸入的胀开，推进了胀开相关理论研究。
In this paper we classify weak Fano varieties that can be obtained by blowing-up general points in prime Fano varieties. We also classify spherical blow-ups of Grassmannians in general points, and we compute their effective cone. These blow-ups are, in particular, Mori dream spaces. Furthermore, we compute the stable base locus decomposition of the blow-up of a Grassmannian in one point, and we show how it is determined by linear systems of hyperplanes containing the osculating spaces of the Grassmannian at the blown-up point, and by the rational normal curves in the Grassmannian passing throu…
We establish a blow-up formula for Hodge cohomology of locally free sheaves on smooth proper varieties over an algebraically closed field of positive characteristic. For this, we introduce a notion of relative Hodge sheaves and study their behavior under blow-ups along smooth centers. In particular, as an application, we study the blow-up invariance of the $E_2$-degeneracy of the Hochschild--Kostant--Rosenberg spectral sequence for smooth proper varieties.
The aim of this note is to investigate the relation between two types of non-singular projective varieties of Picard rank 2, namely the Projective bundles over Projective spaces and certain Blow-up of Projective spaces.
We study Fano manifolds of pseudoindex greater than one and dimension greater than five, which are blow-ups of smooth varieties along smooth centers of dimension equal to the pseudoindex of the manifold. We obtain a classification of the possible cones of curves of these manifolds, and we prove that there is only one such manifold without a fiber type elementary contraction.
The aim of this paper is twofold. Firstly, we determine which blow-ups of products of projective spaces at general points are varieties of Fano type, and give boundary divisors making these spaces log Fano pairs. Secondly, we describe generators of the Cox rings of some cases.
Sequences of point blow-ups over perfect fields from a combinatorial point of view
本文用加权有向图和 d 元相交形式刻画完美域上的点爆发序列及其复合，并通过相容划分证明二者在组合与代数等价类之间均存在双射。
In this paper we classify all the compactifications of affine homology $3$-cells into the blow-ups of the projective $3$-space along smooth curves such that the log canonical divisors are linearly trivial. As a result, we prove that each embedded affine $3$-fold is isomorphic to the affine $3$-space except one example.
We extend the formula for the Chern classes of blow-ups of algebraic varieties due to Porteous and Lascu-Scott, and of symplectic and complex manifolds due to Geiges and Pasquotto, to the blow-ups of almost complex manifolds. Our approach is based on a concrete partition for the tangent bundle of a blow-up. The use of topological K-theory of vector bundles simplifies the classical approaches.
Anti-flips of the blow-ups of the projective spaces at torus invariant points
本文通过反翻转显式构造光滑环面法诺簇，使其同构于射影空间在余维一环面不变点处的爆破。
In order to study wall crossing formula of Donaldson type invariants on the blown-up plane, Nakajima-Yoshioka constructed a sequence of blow-up/blow-down diagrams connecting the moduli space of torsion free framed sheaves on projective plane, and that on its blow-up. In this paper, we prove that Nakajima-Yoshioka's diagram realizes the minimal model program. Furthermore, we obtain a fully-faithful embedding between the derived categories of these moduli spaces.
Given a sequence of point blow-ups of smooth n-dimensional projective varieties $Z_{i}$ defined over an algebraically closed field k, $Z_{s}\rightarrow Z_{s-1}\rightarrow ...\rightarrow Z_{1}\rightarrow Z_{0}$, we give two presentations of the Chow ring of its sky $A^{\bullet}(Z_{s})$. The first one using the classes of the total transforms of the exceptional components as generators and the second one using the classes of the strict transforms ones. We prove that the skies of two sequences of point blow-ups of the same length have isomorphic Chow rings. Finally we give a characterization of f…
Polarized cylinders on blow-ups of weighted projective planes
研究加权射影平面吹胀得到的有理表面中的偏极化圆柱，分析其作为加权超曲面或拟光滑完全交集的自然出现形式。
We study the projective spectrum of the Rees algebra of a module, and characterize it by a universal property. As applications, we give descriptions of universal flatifications of modules and of birational projective morphisms.
Resolving plane curves using stack-theoretic blow-ups
本文针对正特征域上曲线奇点消解中因加权爆破导致切空间维度增加而难以进行归纳的问题，提出利用规范构造的多重加权爆破方法克服了这一挑战。
Bigness of tangent bundles of blow-ups of ruled surfaces
研究直纹曲面\(S = \mathbb{P}_C(\mathcal{E})\)在点爆破下切丛丰沛性的保持情况，通过第一塞格雷不变量\(s_1(\mathcal{E})\)给出爆破点数的精确界。
Raphaël Rouquier introduced an invariant of triangulated categories which is known as Rouquier dimension. Orlov conjectured that for any smooth quasi-projective variety $X$ the Rouquier dimension of $D^b_{\mathrm{coh}}(X)$ is equal to $\mathrm{dim}\, X$. In this note we show that some blow-ups of projective spaces satisfy Orlov's conjecture. This includes a blow-up of $\mathbb{P}^2$ in nine arbitrary distinct points, or a blow-up of three distinct points lying on an exceptional divisor of a blow-up of $\mathbb{P}^3$ in a line. In particular, our method gives an alternative proof of Orlov's con…
We prove that the redundant blow-up preserves the finite generation of the Cox ring of a rational surface under a suitable assumption, and we study the birational structure of Mori dream rational surfaces via redundant blow-ups. It turns out that the redundant blow-up completely characterizes birational morphisms of Mori dream rational surfaces with anticanonical Iitaka dimension $0$. As an application, we construct new Mori dream rational surfaces with anticanonical Iitaka dimension $0$ and $-\infty$ of arbitrarily large Picard number.
Linear systems on blow-ups of Hirzebruch surfaces
本文提出并证明了Hirzebruch曲面上关于线性系统的猜想，适用于特定点数的吹胀情况。
We discuss automorphisms and pseudo-automorphisms on blowups of complex projective space with an eye to finding ones with interesting dynamical behavior.
A conjecture, related to the Nagata conjecture and the Segre-Harbourne-Gimigliano-Hirschowitz conjecture, states that every integral curve with negative self-intersection on the blow-up of $¶^2$ at a set of points in very general position is a (-1)-curve. In this paper we verify this conjecture for all curves whose image on $¶^2$ is a curve with a singularity of multiplicity two at one of the centers of the blow-up.
The integral Chow ring of weighted blow-ups
该研究给出加权爆破的整 Chow 环计算公式，同时得到加权射影栈丛的整 Chow 环、加权爆破的 Gysin 同态公式与分裂原理的推广，并在附录中算出加权爆破的陈类。
The paper has been withdrawn.
We completely classify redundant blow-ups appearing in the theory of rational surfaces with big anticanonical divisor due to Sakai. In particular, we construct a rational surface with big anticanonical divisor which is not a minimal resolution of a del Pezzo surface with only rational singularities, which gives a negative answer to a question raised in a paper by Testa, Várilly-Alvarado, and Velasco.
Given an arbitrary projective birational morphism of varieties, we provide a natural and explicit way of constructing relative compactifications of the maps induced on the main components of the jet schemes. In the case the morphism is the Nash blow-up of a variety, such relative compactifications are shown to be given by the Nash blow-ups of the main components of the jet schemes.
In this paper we apply Shokurov's inductive method to study terminal and canonical singularities. As an easy consequence of the Minimal Model Program we show that for any three-dimensional log terminal singularity there exists some special, so called, plt blow-up. We discuss properties of them and construct some examples.
We prove that certain Severi varieties of nodal curves of positive genus on general blow-ups of the twofold symmetric product of a general elliptic curve are non-empty and smooth of the expected dimension. This result, besides its intrinsic value, is an important preliminary step for the proof of nonemptiness of Severi varieties on general Enriques surfaces in arXiv:2109.10735.
Rationality of Seshadri constants on blow-ups of ruled surfaces
本文研究直纹曲面爆破上的 Seshadri 常数，提出分类负自交曲线的猜想，并在假设该猜想成立时构造出具有无理 Seshadri 常数的丰富线丛。
On the quantum cohomology of blow-ups of four-dimensional quadrics
本文研究四维二次曲面吹起的量子上同调，提出并验证与 Galkin 下界猜想相关的猜想，同时证明 Conjecture $\mathcal{O}$ 适用于两类吹起情形。
We consider weak Fano manifolds with small contractions obtained by blowing up successively curves and subvarieties of codimension 2 in products of projective spaces. We give a classification result for a special case. In the process of proof, we describe explicitly the structure of nef cones and compute the self intersection numbers of anti-canonical divisors for such weak Fano manifolds.
Torus actions, weighted blow-ups, and desingularization of plane curves
本文通过加权吹胀构造平面曲线的脱 singularization，利用环面作用和 equivariant 中心进行归纳论证，无需限制于完美基域。
Let $C \subset \mathbb{P}^2$ be an irreducible and reduced curve of degree $e > 0$. Let $X$ be the blow up of $\mathbb{P}^2$ at $r$ distinct smooth points $p_1,\ldots,p_r \in C$. We study line bundles on $X$ and establish conditions for ampleness and $k$-very ampleness.
Given a submanifold Z inside X, let Y be the blow-up of X along Z. When the normal bundle of Z in X is convex with a minor assumption, we prove that genus-zero GW-invariants of Y with cohomology insertions from X, are identical to GW-invariants of X. Under the same hypothesis, a vanishing theorem is also proved. An example to which these two theorems apply is when the normal bundle is generated by global sections. These two main theorems do not hold for arbitrary blow-ups, and counter-examples are included.
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