Bumsig Kim, Yongnam Lee, Kyungho Oh
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).
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Let $H$ and $H'$ be two ample line bundles over a smooth projective surface $X$, and $M(H)$ (resp. $M(H')$) the coarse moduli scheme of $H$-semistable (resp. $H'$-semistable) sheaves of fixed type $(r,c_1,c_2)$. We construct a sequence of blowing-ups which describes how $M(H)$ differs from $M(H')$ not only when $r=2$ but also when $r$ is arbitrary. Means we here utilize are elementary transforms and the notion of a sheaf with flag.
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 ${\bf {P}}^2$.
Let $H$ and $H'$ be two ample line bundles over a nonsingular projective surface $X$, and $M(H)$ (resp. $M(H')$) the coarse moduli scheme of $H$-semistable (resp. $H'$-semistable) sheaves of fixed type $(r=2,c_1,c_2)$. In a moduli-theoretic way that comes from elementary transforms, we connect $M(H)$ and $M(H')$ by a sequence of blowing-ups when walls separating $H$ and $H'$ are not necessarily good. As an application, we also consider the polarization change problem of Donaldson polynomials.
Equivalent conjectures on blowing-ups of $\mathbb P^2$
该研究刻画了代数曲面上nef且big除子的渐近特殊性，证明射影平面极一般位置点爆破上线性系统的SHGH猜想等价于nef类非特殊，并给出了$r < 2^n$时nef除子渐近非特殊的结论。
In this paper, we first introduce geometric operations for linear categories, and as a consequence generalize Orlov's blow up formula [O04] to possibly singular local complete intersection centres. Second, we introduce refined blowing up of linear category along base--locus, and show that this operation is dual to taking linear section. Finally, as an application we produce examples of Calabi--Yau manifolds which admits Calabi--Yau categories fibrations over projective spaces.
A codimension one singular holomorphic foliation is Newton non-degenerate if it satisfies the classical conditions of Kouchnirenko and Oka, in terms of its Newton polyhedra system. In this paper we prove that a foliation is Newton non-degenerate if and only if it admits a logarithmic reduction of singularities of a combinatorial nature.
On blowing up the weighted projective plane
本文研究加权射影平面在一般点处的爆裂，给出判断结果是否为Mori dream曲面的判据与算法，并计算若干情形的Cox环，同时探讨其在 $\overline{M}_{0,n}$ 研究中的应用。
We compute the Cox rings of the blow-ups $\mathrm{Bl}_Δ(X'\times X')$ and $\mathrm{Bl}_Δ(\mathbb P_1^n)$ where $X'$ is a product of projective spaces and $Δ$ is the (generalised) diagonal.
We study the question of whether the blow-ups of toric surfaces of Picard number one at the identity point of the torus are Mori Dream Spaces. For some of these toric surfaces, the question whether the blow-up is a Mori Dream Space is equivalent to countably many planar interpolation problems. We state a conjecture which generalizes a theorem of Gonzalez and Karu. We give new examples and non-examples of Mori Dream Spaces among these blow-ups.
We study categories of matrix factorizations. These categories are defined for any regular function on a suitable regular scheme. Our paper has two parts. In the first part we develop the foundations; for example we discuss derived direct and inverse image functors and dg enhancements. In the second part we prove that the category of matrix factorizations on the blowing-up of a suitable regular scheme X along a regular closed subscheme Y has a semi-orthogonal decomposition into admissible subcategories in terms of matrix factorizations on Y and X. This is the analog of a well-known theorem for…
The Picard number of a Fano manifold X obtained by blowing up a curve in a smooth projective variety is known to be at most 5, in any dimension greater than or equal to 4. We show that the Picard number attains to the maximal if and only if X is the blow-up of the projective space whose center consists of two points, the strict transform of the line joining them and a linear subspace or a quadric of codimension 2. This result is obtained as a consequence of a classification of special types of Fano manifolds.
We extend the $\hat U$-theorem for projective varieties proved in arXiv:1601.00340 and arXiv:1607.04181 to projective schemes over $\mathbb C$. The condition "ss=s" required in arXiv:1601.00340 and arXiv:1607.04181 is reformulated algebraically for our setting. We also describe a blowing up for the condition "ss=s". Different from the sequence of blowing ups in arXiv:1607.04181 , we can get "ss=s" in one blowing up.
On blowing up minimal toric surfaces
本文证明了Picard秩2的极小环曲面爆破的Cox环有限生成，并给出了正规射影Q-因子曲面Cox环有限生成的充要条件。
We study the blow-ups X of P3 along a proj. normal curve C. We look for very ample divisor classes on X of low degree, and we study the ideal of the embedding of X. Some result is generalized to higher dimensions.
Hilbert schemes on blowing ups and the free Boson
本文通过中岛-吉冈构造，将曲面上爆破的秩1反常凝聚层模空间及希尔伯特概型的上同调分别等同于原曲面希尔伯特概型上同调与费米子/玻色子福克空间的张量积，并给出无限维代数作用的几何对应。
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.
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.
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 characterise smooth curves in P^3 whose blow-up produces a threefold with anticanonical divisor big and nef. These are curves C of degree d and genus g lying on a smooth quartic, such that (i) $4d-30 \le g\le 14$ or $(g,d) = (19,12)$, (ii) there is no 5-secant line, 9-secant conic, nor 13-secant twisted cubic to C. This generalises the classical similar situation for the blow-up of points in P^2. We describe then Sarkisov links constructed from these blow-ups, and are able to prove the existence of Sarkisov links which were previously only known as numerical possibilities.
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.
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.
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.
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.
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.
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。
We discuss some properties of the extremal rays of the cone of effective curves of surfaces that are obtained by blowing up the projective plane at points in very general position. The main motivation is to rectify an incorrect interpretation, in terms of the geometry of this cone, of the Segre--Harbourne--Gimigliano--Hirschowitz conjecture. Even though the arguments are based on elementary computations, the point of view leads to observe some other properties of the Mori cone, among which the fact that in the case of ten or more points the cone is not countably generated.
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。
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.
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 $n\geq 3$.
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…
Blowing-ups of primitive multiple schemes
本文研究原始多重概型沿光滑子概型和好子概型的爆破性质，证明爆破结果仍为同重数的原始多重概型，并在射影平面单点爆破情形下分类了所有底层为 $\widetilde{\rm P}_2$ 的 K3-carpets。
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。
Framed instanton pairs on the blow-up of the projective 3-space at a point
本文研究了射影3空间在一点处吹胀得到的Fano 3-fold上的Huybrechts和Lehn标架层，给出了3维中此类对的新例子，并证明了其模空间是fine、拟射影且无阻碍的。
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.
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。
General position on Severi--Brauer surfaces
本文将射影平面有限点集爆破得到del Pezzo曲面的条件推广到任意域上的Severi-Brauer曲面，利用Galois下降、交理论和组合论证给出了爆破中心的显式算术与几何条件。
Let $\mathcal T_{s,p,n}$ be the canonical blow-up of the Grassmann manifold $G(p,n)$ constructed by blowing up the Plücker coordinate subspaces associated with the parameter $s$. We prove that the higher cohomology groups of the tangent bundle of $\mathcal T_{s,p,n}$ vanish. As an application, $\mathcal T_{s,p,n}$ is locally rigid in the sense of Kodaira-Spencer.
In this paper, we study the cones of higher codimension (pseudo)effective cycles on point blow-ups of projective space. We determine bounds on the number of points for which these cones are generated by the classes of linear cycles, and for which these cones are finitely generated. Surprisingly, we discover that for (very) general points, the higher codimension cones behave better than the cones of divisors. For example, for the blow-up $X_r^n$ of $\mathbb P^n$, $n>4$, at $r$ very general points, the cone of divisors is not finitely generated as soon as $r> n+3$, whereas the cone of curves is …
Flexibility of affine cones over blow-ups of weighted projective planes
该代数几何研究针对加权射影平面吹胀所得曲面的仿射锥，分情况证明其灵活性或一般灵活性，明确了灵活性与吹胀点数的关系。
A remark on weighted Bounded Negativity for blow-ups of the projective plane
本文受加权有界负性猜想启发，证明了射影平面在 n 个点处爆破上除有限多条约化不可约曲线外均满足一个涉及 C^2、C.L 和 n 的显式下界不等式，部分改进了 Laface 和 Pokora 的结果。
Canonical blow-ups of Grassmannians II
本文通过爆破显式单项式理想构造了Grassmann流形G(2,n)的Lafforgue空间，推广了Faltings的结果，并给出了一族具有良好紧化的高复杂度齐性簇，展示了非球面情形下的归途紧化概念。
Let $X$ be a fixed projective scheme which is flat over a base scheme $S$. The association taking a quasi-projective $S$-scheme $Y$ to the scheme parametrizing $S$-morphisms from $X$ to $Y$ is functorial. We prove that this functor preserves limits, and both open and closed immersions. As an application, we determine a partition of schemes parametrizing rational curves on the blow-ups of projective spaces at finitely many points. We compute the dimensions of its components containing rational curves outside the exceptional divisor and the ones strictly contained in it. Furthermore, we provide …
We define graded, quasi-coherent $\mathcal{O}_S$-algebras over a given base derived scheme $S$, and show that these are equivalent to derived $\mathbb{G}_{m,S}$-schemes which are affine over $S$. We then use this $\mathbb{G}_{m,S}$-action to define the projective spectrum $\mathrm{Proj} (\mathcal{B})$ of a graded algebra $\mathcal{B}$ as a quotient stack, show that $\mathrm{Proj} (\mathcal{B})$ is representable by a derived scheme over $S$, and describe the functor of points of $\mathrm{Proj} (\mathcal{B})$ in terms of line bundles. The theory of graded algebras and projective spectra is then …
We construct an explicit compactification for the space of parametrized rational curves in a Grassmanian by a nonsingular projective variety such that the boundary is a divisor with simple normal crossings. This compactification is obtained by blowing up a Quot scheme successively along appropriate subschemes.
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…
Seshadri constants on some blow-ups of projective spaces
本文研究射影空间沿一般点和直线爆破所得簇上的 $l$-极丰富线丛，计算低维情形的 nef 锥及若干丰富线丛的 Seshadri 常数。
Consider the blow-up $X$ of $\mathbb{P}^3$ at 6 points in very general position and the 15 lines through the 6 points. We construct an infinite-order pseudo-automorphism $ϕ_X$ on $X$, induced by the complete linear system of a divisor of degree 13. The effective cone of $X$ has infinitely many extremal rays and hence, $X$ is not a Mori Dream Space. The threefold $X$ has a unique anticanonical section which is a Jacobian K3 Kummer surface $S$ of Picard number 17. The restriction of $ϕ_X$ on $S$ realizes one of Keum's 192 infinite-order automorphisms of Jacobian K3 Kummer surfaces. In general, w…
We give further counterexamples to the conjectural construction of Bridgeland stability on threefolds due to Bayer, Macrì, and Toda. This includes smooth projective threefolds containing a divisor that contracts to a point, and Weierstraß elliptic Calabi-Yau threefolds. Furthermore, we show that if the original conjecture, or a minor modification of it, holds on a smooth projective threefold, then the space of stability conditions is non-empty on the blow up at an arbitrary point. More precisely, there are stability conditions on the blow up for which all skyscraper sheaves are semistable.
Let $\text{Bl}_{\mathbb{P}^1} \mathbb{P}^n$ be a Kähler manifold obtained by blowing up a complex projective space $\mathbb{P}^n$ along a line $\mathbb{P}^1$. We prove that $\text{Bl}_{\mathbb{P}^1} \mathbb{P}^n$ does not admit constant scalar curvature Kähler metrics in any rational Kähler class, but admits extremal metrics, with an explicit formula in action-angle coordinates, in Kähler classes that are close to the pullback of the Fubini--Study class.
Let X be a smooth projective surface. Here we study the postulation of a general union Z of fat points of X, when most of the connected components of Z have multiplicity 2. This problem is related to the existence of "good" families of curves on X, with prescribed singularities, most of them being nodes, and to the cohomology of suitable line bundles on blowing ups of X. More precise statements are obtained in the case X=P^2.
K-stability and space sextic curves of genus three
本文研究沿亏格3光滑六次曲线吹张三维射影空间得到的法诺三维簇，构造了其新K-稳定实例，并刻画了所有可忠实作用于其上的有限群。
On K-stability of $\mathbb{P}^3$ blown up along a quintic elliptic curve
本文研究沿五次光滑椭圆曲线吹胀三维射影空间所得光滑 Fano 三维流形的 K-稳定性，刻画了此类流形的稳定性性质。
Let f: X -> Z be a local, projective, divisorial contraction between normal varieties of dimension n with Q-factorial singularities. Let $Y \subset X$ be a f-ample Cartier divisor and assume that f|Y: Y -> W has a structure of a weighted blow-up. We prove that f: X -> Z, as well, has a structure of weighted blow-up. As an application we consider a local projective contraction f: X -> Z from a variety X with terminal Q-factorial singularities, which contracts a prime divisor E to an isolated Q-factorial singularity $P\in Z$, such that $-(K_X + (n-3)L)$ is f-ample, for a f-ample Cartier divisor …
Autoequivalences of Blow-Ups of Minimal Surfaces
本文研究极小曲面在有限个非常一般点处爆破的自等价与球面对象，证明此类簇只有标准自等价、无非平凡Fourier-Mukai伙伴且不存在球面对象，并刻画了正Kodaira维数情形下的球面对象。
Birational transformations on irreducible compact Hermitian symmetric spaces
本文研究了不可约紧致赫尔米特对称空间上的双有理变换，通过构造吹胀和吹降操作将其转化为射影空间，并解决了Landsberg和Manivel提出的双有理映射问题。
The Chow ring of a sequence of point and rational curve blow-ups
本文研究从 P^3 出发的点与有理曲线爆发序列，给出终态周环的显式表示，并证明相同邻近关系也可能导致不同构周环，进而刻画例外分支间允许的邻近类型。
The space of smooth rational curves of degree $d$ in a projective variety $X$ has compactifications by taking closures in the Hilbert scheme, the moduli space of stable sheaves or the moduli space of stable maps respectively. In this paper we compare these compactifications by explicit blow-ups and -downs when $X$ is a projective homogeneous variety and $d\leq 3$. Using the comparison result, we calculate the Betti numbers of the compactifications when $X$ is a Grassmannian variety.
Kollár gave a series of examples of rational surfaces of Picard number $1$ with ample canonical divisor having cyclic singularities. In this paper, we construct several series of new examples in a geometric way, i.e., by blowing up several times inside a configuration of curves on the projective plane and then by contracting chains of rational curves. One series of our examples have the same singularities as Kollár's examples.
Moduli space of complete stable pairs
本文研究了平滑射影簇上完备稳定对的模空间构造，展示了其与稳定对模空间和Quot方案的关系，并以$\mathbb P^1$为例说明其结构。
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.
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