Blow-ups of three-dimensional terminal singularities: cA case

高阶纳什爆破与诺比尔定理

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。

利用加权爆破在特征0中实现奇点的对数消解

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。

Gromov-Witten invariants of blow-ups

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…

Stable pair invariants under blow-ups

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.

Hirzebruch曲面一般爆胀上线丛的正性

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。

On the quantum cohomology of blow-ups of projective spaces along linear subspaces

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.

对数几何中对数胀开的逆构

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。

Integral morphisms and log blow-ups

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.

Effective cycles on blow-ups of Grassmannians

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.

Lins Neto's examples of foliations and the Mori cone of blow-ups of $P^2$

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.

A note on k-very ampleness of line bundles on general blow-ups of hyperelliptic surfaces

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.

Numerical invariants for bundles on blow-ups

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.

射影空间沿有理曲线的爆破上的有效闭链锥

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。

虚拟Cartier除子与胀开

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。

Spherical blow-ups of Grassmannians and Mori Dream Spaces

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…

Hodge cohomology on blow-ups along subvarieties

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.

Projective bundles and blow-ups of Projective spaces

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.

Fano manifolds and blow-ups of low-dimensional subvarieties

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.

Log Fano structures and Cox rings of blow-ups of products of projective spaces

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.

从组合观点看完美域上的点爆发序列

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。

Compactifications of affine homology $3$-cells into blow-ups of the projective $3$-space with trivial log canonical divisors

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.

射影空间在环面不变点处爆破的反翻转

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。

Birational geometry of moduli spaces of perverse coherent sheaves on blow-ups

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.

The Chow ring of a sequence of point blow-ups

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…

偏极化圆柱在加权射影平面吹胀表面中的研究

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。

Total blow-ups of modules and universal flatifications

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.

使用栈论爆破消解平面曲线

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。

直纹曲面爆破的切丛的丰沛性

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。

Rouquier dimension of some blow-ups

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…

Cox rings of rational surfaces and redundant blow-ups

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.

理性曲面上的线性系统

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。

Negative curves on very general blow-ups of P^2

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.

加权爆破的整 Chow 环

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。

Redundant blow-ups of rational surfaces with big anticanonical divisor

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.

Nash blow-ups of jet schemes

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.

Blow-ups of canonical singularities

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.

Severi varieties on blow--ups of the symmetric square of an elliptic curve

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.

直纹曲面爆破上 Seshadri 常数的有理性

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。

论四维二次曲面吹起（blow-ups）的量子上同调

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。

On weak Fano manifolds with small contractions obtained by blow-ups of a product of projective spaces

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.

环面作用、加权吹胀与平面曲线的脱 singularization

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。

Positivity of line bundles on special blow ups of $\mathbb{P}^2$

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.

Gromov-Witten invariants of blow-ups along submanifolds with convex normal bundles

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.

A splitting criterion for vector bundles on blowing ups of the plane

Let $f_s: X_s \to {\bf {P}}^2$ be the blowing-up of $s$ distinct points and $E$ a vector bundle on $X_s$. Here we give a cohomological criterio which is equivalent to $E \cong f_s^\ast (A)$ with $A$ a direct sum of line bundles. We also some cohomological characterizations of very particular rank 2 vector bundles on ${\bf {P}}^2$.

Monads for framed torsion-free sheaves on multi-blow-ups of the projective plane

We construct monads for framed torsion-free sheaves on blow-ups of the complex projective plane at finitely many distinct points. Using these monads we prove that the moduli space of such sheaves is a smooth algebraic variety. Moreover we construct monads for families of such sheaves parameterized by a noetherian scheme S of finite type. A universal monad on the moduli space is introduced and used to prove that the moduli space is fine.

三维环面奇点的爆破

Blow-ups of three-dimensional toric singularities

该研究刻画了三维终端环面奇点的纯对数终端爆破，以及满足特定条件的三维除子收缩的结构，为代数几何中环面奇点的研究提供了相关理论结果。

Mori Dream Spaces and blow-ups of weighted projective spaces

For every $n\geq 3$, we find a sufficient condition for the blow-up of a weighted projective space $\mathbb{P}(a,b,c,d_1,\cdots,d_{n-2})$ at the identity point not to be a Mori Dream Space. We exhibit several infinite sequences of weights satisfying this condition in all dimensions $n\geq 3$.

Positivity of line bundles on general blow ups of $\mathbb{P}^2$

Let $X$ be the blow up of $\mathbb{P}^2$ at $r$ general points $p_1,\ldots,p_r \in \mathbb{P}^2$. We study line bundles on $X$ given by plane curves of degree $d$ passing through $p_i$ with multiplicity $m_i$. We establish conditions for ampleness, very ampleness and global generation of such line bundles.

Hirzebruch曲面爆破上的幻影子范畴

Phantom subcategories on blow-ups of Hirzebruch surfaces

本文证明Hirzebruch曲面在一般点爆破后存在幻影子范畴，覆盖所有n≥3的情况。

Examples of non--effective rays at the boundary of the Mori cone of blow--ups of the plane

In this paper we prove that no multiple of the linear system of plane curves of degree $d\geq 4$ with a point of multiplicity $d-m$ (with $2 \leq m \leq d$) and $m(2d-m)$ simple general points is effective.

Towards the weighted bounded negativity conjecture for blow-ups of algebraic surfaces

In the present paper, we focus on a weighted version of the Bounded Negativity Conjecture which predicts that for every smooth projective surface in characteristic zero the self-intersection numbers of reduced and irreducible curves are bounded from below by a global constant. We gather evidence for this conjecture by showing various bounds on the self-intersection number of curves in an algebraic surface. We focus our attention on blow-ups of algebraic surfaces, which have so far been neglected.

固定域曲线计数与射影空间吹胀的曲线计数

Fixed-domain curve counts for blow-ups of projective space

本文研究了射影空间吹胀中固定复结构的曲线计数问题，通过几何和虚拟计数方法分析其一致性和计算方法。

关于射影丛爆破的K-稳定性

On the K-stability of blow-ups of projective bundles

研究法诺簇上射影线丛爆破的K-稳定性，证明当除子B与2L成比例时，爆破的K-半稳定和K-多稳定等价于对数法诺对的相应性质，并给出系数；当比例非2时爆破K-不稳定。

爆破与曲面的量子谱

Blow-ups and the quantum spectrum of surfaces

本文研究光滑射影曲面在爆破下量子谱的变化，证明其量子谱渐近地由极小模型的量子谱与对应例外除子的有限附加点组成，从而证实了Kontsevich猜想。