Alexander Polishchuk
We prove a new criterion for the homogeneous coordinate ring of a finite set of points in ${\Bbb P}^n$ to be Koszul. Like the well known criterion due to Kempf it involves only incidence conditions on linear spans of subsets of a given set. We also give a sufficient condition for the Koszul property to be preserved when passing to a subset of a finite set of points in ${\Bbb P}^n$.
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A new section on projections of coherent sheaves from a projective space to a lower-dimensional projective space has been added. Also some of the notation has been altered to bring it into line with the joint paper with Eisenbud and Schreyer.
We are going to show that the sheafication of graded Koszul modules $% K_Γ$ over $Γ_{n}=K[ x_{0},x_{1}...x_{n}] $ form an important subcategory $\overset{\wedge}{K}_Γ$ of the coherents sheaves on projective space, $Coh(P^{n}).$ One reason is that any coherent sheave over $P^{n}$ belongs to $\overset{\wedge}{K}_Γ$up to shift. More importantly, the category $K_Γ$ allows a concept of almost split sequence obtained by exploiting Koszul duality between graded Koszul modules over $Γ$ and over the exterior algebra $Λ.$ This is then used to develop a kind of relative Auslander-Reiten theory for the ca…
Koszul complexes and spectra of projective hypersurfaces with isolated singularities
该研究推广了非奇异射影超曲面的相关经典结论，利用Koszul上同调与极点阶谱序列，推导了孤立奇点加权齐次射影超曲面的BS多项式根与Jacobian环挠部的关系。
We construct relative and global Euler sequences of a module. We apply it to prove some acyclicity results of the Koszul complex of a module and to compute the cohomology of the sheaves of (relative and absolute) differential $p$-forms of a projective bundle. In particular we generalize Bott's formula for the projective space to a projective bundle over a scheme of characteristic zero.
Cohomology Vanishing, Koszul Cohomology and Multigraded Regularity on Projective Varieties
本文在多重分次Castelnuovo–Mumford正则性框架下，证明射影簇的通用上同调消失定理，推广前期工作，刻画射影空间乘积线丛的极小正则性，重证Green定理并计算超椭圆曲线典范像的Betti表。
Koszul Duality for Coherent Sheaves
研究为无限维 Koszul 代数建立 Koszul 对偶性，应用于二次单项式代数等两类代数，还应用于代数几何，得到交换诺特 Koszul 代数定义的射影概型的有界导出范畴的 Koszul 对偶描述及 BGG 型对应。
Let $R$ be a noetherian ring and $M$ a finite $R$-module. With a linear form $χ$ on $M$ one associates the Koszul complex $K(χ)$. If $M$ is a free module, then the homology of $K(χ)$ is well-understood, and in particular it is grade sensitive with respect to $\Imχ$. In this note we investigate the case of a module $M$ of projective dimension 1 (more precisely, $M$ has a free resolution of length 1) for which the first non-vanishing Fitting ideal $\I_M$ has the maximally possible grade $r+1$, $r=\rank M$. Then $h=\grade \Imχ\le r+1$ for all linear forms $χ$ on $M$, and it turns out that $H_{r-i…
Let $X$ be a smooth projective Calabi-Yau variety and $L$ a Koszul line bundle on $X$. We show that for Betti numbers of a maximal Cohen-Macaulay module over the homogeneous coordinate ring $A$ of $X$ there are formulas similar to the formulas for cohomology number. This similarity is realized via the box-product resolution of the diagonal $Δ_X \subset X \times X$.
Vanishing of weight one syzygies of projective varieties
本文探究射影簇上权1 Koszul上同调的消失条件，结合(M_q)与(N_p)性质推导相关推论，深化了对极小自由消解的认知。
Let R be a commutative ring and I an ideal in R which is locally generated by a regular sequence of length d. Then, each projective R/I-module V has an R-projective resolution P. of length d. In this paper, we compute the homology of the n-th Koszul complex associated with the homomorphism P_1 --> P_0 for all n, if d = 1. This computation yields a new proof of the classical Adams-Riemann-Roch formula for regular closed immersions which does not use the deformation to the normal cone any longer. Furthermore, if d = 2, we compute the homology of the complex N Sym^2 K(P.) where K and N denote the…
We prove three results on pure resolutions of vector bundles on projective spaces. First, we show that there are simple vector bundles of rank n on Pn with arbitrary homological dimension. We then analyze the pure resolutions given by the sheafification of the Koszul complex of a certain algebra and by the sheafification of the minimal free resolution of a compressed Gorenstein Artinian graded algebra, proving that their syzygies are simple vector bundles. Our main tool is a result originally established by Brambilla, for which we give an alternative proof using representations of quivers.
Rational Normal Curves in Weighted Projective Space
该研究将有理正规曲线的经典同调理论推广至加权射影空间，建立了行列式性以及二次生成与 Koszul 性质的非标准形式。
The minimal free resolution of the coordinate ring of a complete intersection in projective space is a Koszul complex on a regular sequence. In the product of projective spaces $\mathbb{P}^1 \times \mathbb{P}^1$, we investigate which sets of points have a virtual resolution that is a Koszul complex on a regular sequence. This paper provides conditions on sets of points; some of which guarantee the points have this property, and some of which guarantee the points do not have this property.
Koszul Monoids in Quasi-abelian Categories
本文在具有足够平坦投射对象的双完备闭对称幺半拟阿贝尔范畴中，以无元素方式定义 Koszul 与二次幺半群，并借助左心嵌入证明其分次模导出范畴子范畴的对偶等价。
Koszul property of Ulrich bundles and rationality of moduli spaces of stable bundles on Del Pezzo surfaces
本文研究 Del Pezzo 曲面上 Ulrich 丛的 Koszul 性质与迭代 syzygy 丛的斜率半稳定性，并证明无穷多个稳定丛模空间具有有理性，为 Costa 和 Miró-Roig 猜想提供新证据。
In this paper, we classify the configurations of the singular points which appear on the quotients of the projective plane by the $1$-foliations of degree $-1$ in characteristic $2$.
We consider the graded space $R$ of syzygies for the coordinate algebra $A$ of projective variety $X=G/P$ embedded into projective space as an orbit of the highest weight vector of an irreducible representation of semisimple complex Lie group $G$. We show that $R$ is isomorphic to the Lie algebra cohomology $H=H^\bdot(\Lt,\CC)$, where $\Lt$ is graded Lie subalgebra of the graded Lie s-algebra $L=L_1\oplus\Lt$ Koszul dual to $A$. We prove that the isomorphism identifies the natural associative algebra structures on $R$ and $H$ coming from their Koszul and Chevalley DGA resolutions respectively.…
Higher resonance schemes and Koszul modules of simplicial complexes
本文研究有限单纯复形外Stanley-Reisner代数的共振概型与高阶Koszul模，证明共振概型既约，计算Hilbert级数并给出正则性与投射维数界，推广直角Artin群Chen秩公式。
A configuration of 7 points in RP2 is called typical if it has no collinear triples and no coconic sextuples of points. We show that there exist 14 deformation classes of such configurations. This yields classification of real Aronhold sets.
In this note we exhibit the so-called Harbourne constants which capture and measure the Bounded Negativity on various birational models of an algebraic surface. We show an estimation for Harbourne constants for conic configurations on the complex projective plane with transversal intersection points.
We study the GIT-quotient of the Cartesian power of projective space modulo the projective orthogonal group. A classical isomorphism of this group with the Inversive group of birational transformations of the projective space of one dimension less allows one to interpret these spaces as configuration spaces of complex or real spheres.
This note is motivated by the Question 16 of http://cubics.wikidot.com: Which configurations of 15 points in the projective 3-space arise as eigenpoints of a cubic surface? We prove that a general eigenscheme in the projective n-space is the complete intersection of two suitable smooth determinantal curves on a smooth determinantal surface. Moreover, we prove that the converse result holds if n=3, providing an answer in any degree to the cited question. Finally, we show that any general set of points in the projective 3-space can be enlarged to an eigenscheme of a partially symmetric tensor.
The spaces of point configurations on the projective line up to the action of $\mathrm{SL}(2,\mathbb K)$ and its maximal torus are canonically compactified by the Grothdieck-Knudsen and Losev-Manin moduli spaces $\overline M_{0,n}$ and $\overline L_n$ respectively. We examine the configuration space up to the action of the maximal unipotent group $\mathbb G_a\subseteq \mathrm{SL}(2,\mathbb K)$ and define an analogous compactification. For this we first assign a canonical quotient to the action of a unipotent group on a projective variety. Moreover, we show that similar to $\overline M_{0,n}$ a…
We describe the Chow rings of moduli spaces of ordered configurations of points on the projective line for arbitrary (sufficiently generic) stabilities. As an application, we exhibit such a moduli space admitting two small desingularizations with non-isomorphic cohomology rings.
Clifford spaces of empty intersections of quadrics
针对射影空间中交为空的二次曲面线性空间，提出克利福德空间概念，证明其导出范畴由可扩展为1周期螺旋的完全例外集生成，等价于对应分次代数的非交换射影谱，并详细讨论极大与极小两种特殊情况的性质。
We construct configuration spaces for cyclic covers of the projective line that admit extra automorphisms and we describe the locus of curves with given automorphism group. As an application we provide examples of arbitrary high genus that are defined over their field of moduli and are not hyperelliptic.
Counting point configurations in projective space
本文提出了一种在射影空间中计数具有固定相对位置点配置的方法，并建立了与不同维度射影空间相关的递推关系。
Creating quantum projective spaces by deforming q-symmetric algebras
通过图示计算变形 q-symmetric 代数构造大量 Koszul、Calabi-Yau 量子射影空间，证明无阻碍性，并首次显式计算验证二次 Poisson 结构的 Kontsevich 量子化收敛性。
Critical configurations for three projective views
本文用代数方法研究三个投影相机在运动恢复结构中的临界构型，证明其均位于二次曲面交线上，并给出构成临界构型的准确分类。
Curve configurations in the projective plane and their characteristic numbers
本文研究射影平面中曲线构型的特征数与陈斜率，证明极值直线构型具有相同的渐近不变量，并给出某类光滑曲线构型特征数的上界$8/3$。
We establish an equivalence between the stable category of coherent sheaves (satisfying a mild restriction) on a projective space and the homotopy category of a certain class of minimal complexes of free modules over the exterior algebra Koszul dual to the homogeneous coordinate algebra of the projective space. We also relate these complexes to the Tate resolutions of the respective sheaves. In this way, we extend from vector bundles to coherent sheaves the results of Coandă and Trautmann [Trans. AMS 385 (2005)], which interpret in terms of the BGG correspondence the results of Trautmann [Math…
We study families of linear spaces in projective space whose union is a proper subvariety X of the expected dimension. We establish relations between configurations of focal points and existence or non-existence of a fixed tangent space to X along a general element of the family. We apply our results to the classification of ruled 3-dimensional varieties.
We study the iterated blow-up X of projective space along an arbitrary collection of linear subspaces. By replacing the universal torsor with an $\mathbb{A}^1$-homotopy equivalent model, built from $\mathbb{A}^1$-fiber bundles not just algebraic line bundles, we construct an "algebraic uniformization": X is a quotient of affine space by a solvable group action. This provides a clean dictionary, using a single coordinate system, between the algebra and geometry of hypersurfaces: effective divisors are characterized via toric and invariant-theoretic techniques. In particular, the Cox ring is an …
Critical configurations for two projective views, a new approach
本文研究运动恢复结构中的临界配置问题，采用代数方法证明两个射影相机的临界配置均位于二次曲面上，并精确分类了相关二次曲面及非唯一重建结果间的关系。
Projective rigidity of point-line configurations in the plane
本文提出“射影刚性矩阵”框架研究平面上点线构型的保关联运动，并引入对称自适应矩阵分析对称构型及其保对称运动。
The relation type of point configurations in the projective plane
研究射影平面中有限约化点集理想的关系类型，主要关注非线性呈现的点构型，证明至多10个点的构型关系类型为1或3，并给出11个一般位置点构型关系类型为5的首次实例。
We consider the space $F_n$ of configurations of $n$ points in $P^2$ satisfying the condition that no three of the points lie on a line. For $n = 4, 5, 6$, we compute $H^*(F_n; \mathbb{Q})$ as an $S_n$-representation. The cases $n = 5, 6$ are computed via the Grothendieck--Lefschetz trace formula in étale cohomology and certain "twisted" point counts for analogous spaces over $\mathbb{F}_q$.
We survey and expand on the work of Segal, Milgram and the author on the topology of spaces of maps of positive genus curves into $n$-th complex projective space, $n\geq 1$ (in both the holomorphic and continuous categories). Both based and unbased maps are studied and in particular we compute the fundamental groups of the spaces in question. The relevant case when $n=1$ is given by a non-trivial extension which we fully determine.
The module of Valabrega-Valla of the Jacobian ideal of a reduced projective variety $V$ is the torsion of the Aluffi algebra. One considers the problem of its vanishing in the case of where $V$ is a reduced set of points in the projective plane. It is shown that the module is nonzero for several cases of a special configuration class therein -- called $(s-r)$-{fold collinear configuration}. A complete classification of types is given for $5$ and $6$ points in regard to this problem.
The space of marked n distinct points on the complex projective line up to projective transformations will be called a configuration space in this paper. There are two families of complex hyperbolic structures on the configuration space constructed by Deligne-Mostow and Thurston. We first confirm that these families are the same. Then in view of the deformation theory for real hyperbolic cone 3-manifolds, we review the families for small n.
In the present note, we focus on certain properties of special curves that might be used in the theory of multi-point Seshadri constants for ample line bundles on the complex projective plane. In particular, we provide three Ein-Lazarsfeld-Xu-type lemmas for plane curves and a lower bound on the multi-point Seshadri constant of $\mathcal{O}_{\mathbb{P}^{2}}(1)$ under the assumption that the chosen points are not very general. In the second part, we focus on certain arrangements of points in the plane which are given by line arrangements. We show that in some cases the multi-point Seshadri cons…
Artin-Schreier geproci configurations in projective spaces of arbitrary dimension
研究人员引入𝔽_N-Artin-Schreier构型，刻画ℙ³中该构型为(q,N)-geproci的条件（q≤N），并通过提升构造得到任意n≥3时ℙⁿ中的geproci集。
We prove that the factorization homologies of a scheme with coefficients in truncated polynomial algebras compute the cohomologies of its generalized configuration spaces. Using Koszul duality between commutative algebras and Lie algebras, we obtain new expressions for the cohomologies of the latter. As a consequence, we obtain a uniform and conceptual approach for treating homological stability, homological densities, and arithmetic densities of generalized configuration spaces. Our results categorify, generalize, and in fact provide a conceptual understanding of the coincidences appearing in…
The Klein and Wiman configurations are highly symmetric configurations of lines in the projective plane arising from complex reflection groups. One noteworthy property of these configurations is that all the singularities of the configuration have multiplicity at least three. In this paper we study the surface X obtained by blowing up the projective plane in the singular points of one of these line configurations. We study invariant curves on X in detail, with a particular emphasis on curves of negative self-intersection. We use the representation theory of the stabilizers of the singular poin…
B. Totaro showed \cite{totaro} that the rational cohomology of configuration spaces of smooth complex projective varieties is isomorphic as an algebra to the $E_2$ term of the Leray spectral sequence corresponding to the open embedding of the configuration space into the Cartesian power. In this note we show that the isomorphism can be chosen to be compatible with the mixed Hodge structures. In particular, we prove that the mixed Hodge structures on the configuration spaces of smooth complex projective varieties are direct sums of pure Hodge structures.
We are concerned with the problem of the stability of the syzygy bundles associated to base point free vector spaces of forms of the same degree d on the projective space of dimension n. We deduce directly, from Mark Green's vanishing theorem for Koszul cohomology, that any such bundle is stable if his rank is sufficiently high. With a similar argument, we prove the semistability of a certain syzygy bundle on a general complete intersection of hypersurfaces of degree d in the projective space. This answers a question of H. Flenner (1984). We then give an elementary proof of H. Brenner's criter…
Welschinger invariants of the real projective plane can be computed via the enumeration of enriched graphs, called marked floor diagrams. By a purely combinatorial study of these objects, we prove a Caporaso-Harris type formula which allows one to compute Welschinger invariants for configurations of points with any number of complex conjugated points.
Relative uniform Yau--Tian--Donaldson correspondence for projective bundles over a curve
本文研究了曲面投影包的相对均匀 Yau-Tian-Donaldson 对应关系，通过构造兼容测试配置，证明了相对均匀稳定性与极值度量存在性的等价性。
Associated to an arrangement of projective hyperplanes A is the module D(A), which consists of derivations tangent to A. We study D(A) when A is a configuration of lines in the projective plane. In this setting, we relate the deletion/restriction construction used in the study of hyperplane arrangements to elementary modifications of bundles. This allows us to obtain bounds on the Castelnuovo-Mumford regularity of D(A). We also give simple combinatorial conditions for the associated bundle to be stable, and describe its jump lines. These regularity bounds and stability considerations impose co…
Finding the so-called characteristic numbers of the complex projective plane ${\mathbb C}P^2$ is a classical problem of enumerative geometry posed by Zeuthen more than a century ago. For a given $d$ and $g$ one has to find the number of degree $d$ genus $g$ curves that pass through a certain generic configuration of points and at the same time are tangent to a certain generic configuration of lines. The total number of points and lines in these two configurations is $3d-1+g$ so that the answer is a finite integer number. In this paper we translate this classical problem to the corresponding en…
A venerable problem in combinatorics and geometry asks whether a given incidence relation may be realized by a configuration of points and lines. The classic version of this would ask for algebraic lines over some field or possibly real pseudolines: embedded circles (isotopic to $RP^1$) in the real projective plane. In this paper we investigate whether a configuration is realized by a collection of $2$-spheres embedded, in the symplectic, smooth, or topological (locally flat) categories, in the complex projective plane. We find obstructions to realizability in the topological category, which a…
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