Edoardo Ballico, Francesco Malaspina
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 ${\bf {P}}^2$.
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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).
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 $H$ 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。
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.
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…
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.
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 establish a blowing down criterion in the context of birational symplectic geometry in dimension 6.
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.
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。
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.
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。
Blowing-up Hermitian Yang--Mills connections
本文研究Kähler流形爆破后拉回向量丛的埃尔米特杨-米尔斯联络，在温和假设下给出了该丛存在相应联络序列的充要数值判据，并证明联络收敛于分次对象的拉回联络。
Work of Dolgachev and Castravet-Tevelev establishes a bijection between the $2^{n-1}$ weights of the half-spin representations of $\mathfrak{so}_{2n}$ and the generators of the Cox ring of the variety $X_n$ which is obtained by blowing up $\mathbb{P}^{n-3}$ at $n$ points. We derive a geometric explanation for this bijection, by embedding ${\rm Cox}(X_n)$ into the even spinor variety (the homogeneous space of the even half-spin representation). The Cox ring of the blow-up $X_n$ is recovered geometrically by intersecting torus translates of the even spinor variety. These are higher-dimensional g…
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.
Let X be an analytic vector field defined in a real analytic manifold of dimension three. We prove that all the singularities of X can be made elementary by a finite number of blowing-ups in the ambient space. New version: Some misprints have been corrected and the text have been slightly reorganized. The final version appears in Acta Mathematica -Volume 197, Number 2, 167-289.
Let $(S,L_{S})$ be a polarized abelian surface, and let $M = c \cdot π^*L_S - α\cdot \sum_{i=1}^r E_i$ be a line bundle on ${\rm Bl}_{r}(S)$, where $π:{\rm Bl}_{r}(S) \rightarrow S$ is the blow-up of $S$ at $r$ general points with exceptional divisors $E_{1},\dots,E_{r}$. In this paper, we provide a criterion for $k$-very ampleness of $M$. Also, we deal with the case when $S$ is an arbitrary surface of Picard number one with a numerically trivial canonical divisor.
Multiplicity-One Cox Rings of Toric Point Blow-Ups
该研究针对环面点爆破的Cox环，给出Rees重数1生成的有限判据，将其应用于伪加权射影空间与射影环面曲面，得到重数1生成的等价条件。
Let $k$ be a field of characteristic zero. By using Hironaka's desingularisation theorem, we prove an extension criterion for a functor defined on nonsingular k-schemes and taking values on a category of complexes. Roughly speaking, the criterion shows that if such a functor satisfies the standard exact sequence of a blowing-up, then the functor can be extended to all separated k-schemes of finite type. The result is applied to the Grothendieck's theory of motives, to the Hodge-De Rham filtered complex of an analytic space, and to the rational homotopy of k-schemes in algebraic De Rham theory.
In this paper we give a criterion for an isolated, hypersurface singularity of dimension $n\ (\geq 2)$ to have the canonical modification by means of a suitable weighted blow-up. Then we give a counter example to the following conjecture by Reid-Watanabe: For a 3-dimensional, isolated, non-canonical, log-canonical singularity $(X,x)$ of embedded dimension 4, there exists an embedding $(X,x)\subset ({\bC}^4, 0)$ and a weight ${\bw}=(w_0,w_1,\ldots ,w_n)$, such that the ${\bw}$-blow-up gives the canonical modification of $(X,x)$.
We prove a fast computable criterion that expresses non-flatness in terms of torsion: Let R be a regular algebra of finite type over a field K of characteristic zero and let F be a module finitely generated over an R-algebra of finite type. Given a maximal ideal m in R, let S be the coordinate ring of the blowing-up of Spec(R) at the closed point m. Then F is flat over R localized in m if and only if the tensor product of F with S over R is a torsion-free module over R localized in m. If K is the field of reals or complex numbers, we give a stronger criterion - without the regularity assumptio…
Lifting negative curves from finite fields and Seshadri constants at ten points
本文通过有限域计算与混合特征形变，在十个一般点处构造 nef 除子，将 Seshadri 常数下界改进至 8658/27379。
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