SEARCH · Searcharxiv
Search indexed arXiv papers on artificial intelligence, large language models, computer vision and robotics. Read source abstracts and follow links to arXiv.
Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.
The quantum cohomology algebra of a projective manifold X is the cohomology H(X,Q) endowed with a different algebra structure, which takes into account the geometry of rational curves in X. We show that this algebra takes a remarkably simple form for complete intersections when the dimension is large enough w.r.t. the degree. As a reward we get a number of surprising enumerative formulas relating lines, conics and twisted cubics on X.
We study the motive of the moduli spaces of semistable rank two vector bundles over an algebraic curve. When the degree is odd the moduli space is a smooth projective variety, we obtain the absolute Hodge motive of this, and in particular the Hodge-Poincare polynomial. When the degree is even the moduli space is a singular projective variety, we compute pure Euler characteristics and show that only two weights can occur in each cohomology group, we also see that its cohomology is pure up to a certain degree. As a by-product we obtain the isogeny type of some intermediate jacobians of the moduli spaces.
A notion of duality of weight systems which corresponds to Batyrev's toric mirror symmetry is given. Explicit duality on the (1,1)-cohomology of K3 surfaces which are minimal models of toric hypersurfaces is constructed using monomial divisor mirror map of Aspinwall-Greene-Morrison. It is shown that Arnold's strange duality for exceptional unimodal singularities reduces to this duality.
Computation of the fundamental group of the complement in the complex plane of the branch curve S , of a generic projection of the Veronese surface to the plane is presented. This paper is a continuation of our previous papers: Braid Group Technique I - IV. In I and II we developed algorithms to compute braid monodromy of a brunch curve, provided there exist a degeneration of the surface to union of planes in a configuration where the associated branch curve is partial to a line arrangement dual to generic. In III we constructed a degeneration of the Veronese surface of order 3 to union of planes with the desired property and in IV we used I -III to compute the braid monodromy of the associated branch curve. Here we use the Van-Kampen method to compute the fundamental group of the complement from a braid monodromy factorization and some extra properties of the factorization , namely invariant properties, also proven in IV. The group is presented using a certain quotient of the Braid group defined by transversal half-twists.
The purpose of this paper is to give a linear and effective height inequality for algebraic points on curves over functional fields. Our height inequality can be viewed as the logarithmic canonical class inequality of a punctured curve over a functional field (a fibered surface minus a section).
If there is one polygon inscribed into some smooth conic and circumscribed about another one, then there are infinitely many such polygons. This is Poncelet's theorem. The aim of this note is to collect some (mostly classical) versions of this theorem, namely: - Weyr's Poncelet theorem in $P_3$ (1870), - Emch's theorem on circular series (1901), - Gerbaldi's formula for the number of Poncelet pairs (1919), - the Money-Coutts theorem on three circles (1971), - the zig-zag theorem (1974), - a (probably new) Poncelet theorem on three conics, - a Poncelet formula for quadrics of revolution.
Let $g : X \to Y$ be the contraction of an extremal ray of a smooth projective 4-fold $X$ such that $\dim Y=3$. Then $g$ may have a finite number of 2-dimensional fibers. We shall classify those fibers. Especially we shall prove that any two points of such a fiber is joined by a chain of rational curves of length at most 2 with respect to $-K_X$, and that $|-K_X|$ is $g\text{-free}$.
A Beilinson completion algebra (BCA) A is a complete semilocal algebra over a perfect field k, whose residue fields are high dimensional local fields. In addition A is a semi-topological algebra. The completion of the structure sheaf of an algebraic k-variety along a saturated chain of points is the prototypical example of a BCA. We single out two kinds of homomorphisms between BCAs: morphisms and intensifications. The first kind includes residually finite local homomorphisms, whereas the second kind is a sort of localization. We prove that every BCA A has a dual module K(A), and these dual modules are contravariant w.r.t. morphisms and covariant w.r.t. intensifications. For any semi-topological A-module M we define its dual Dual_{A} M := Hom_{A}^{cont}(M, K(A)). This duality operation has the remarkable property of respecting differential operators: given a continuous DO D : M --> N, there is a dual DO Dual_{A}(D) : Dual_{A} N --> Dual_{A} M. The results above are used (in a subsequent paper) to construct the Grothendieck residue complex K_{X}^{.} on any finite type k-scheme X, and to derive many of its properties.
We give a short proof of the following result: Let $X$ be a complex surface of general type. If the canonical divisor of the minimal model of $X$ has selfintersection $= 1$, then $X$ is not diffeomorphic to a rational surface. Our proof is the natural extension of the argument given in our paper in Inventiones Mathematicae (1989) for the case when $X$ is minimal. This argument also gives information about the non--existence of certain smooth embeddings of $2$--spheres in $X$, if $X$ has geometric genus zero.
We give an elementary algebraic proof of some asymptotic estimates (called by Demailly asymptotic Morse inequalities) for the dimensions of cohomology groups of the difference of two ample line bundles on a smooth complex projective variety of any dimension.
The paper proves that if a reductive group scheme acts properly on a scheme then the geometric quotient exists as an algebraic space. As a consequence we obtain the existence of the moduli spcace of canonically polarized varieties over Spec Z.
We construct a moduli scheme for semistable pre-$\D$-modules with prescribed singularities and numerical data on a smooth projective variety. These pre-$\D$-modules are to be viewed as regular holonomic $\D$-modules with `level structure'. We also construct a moduli scheme for perverse sheaves on the variety with prescribed singularities and other numerical data, and represent the de Rham functor (which gives the Riemann-Hilbert correspondence) by an analytic morphism between the two moduli schemes.
Generalizing a classical lemma of Castelnuovo, we characterize rational normal curves (resp. linearly normal elliptic curves) as curves $C\subset \PP^n$ such that the number of linearly independent hypersurfaces $Z\supset C$ of given degree~$m$ is maximal (resp. next to maximal).
In this short note, a new computation of the degree of the locus of 3-nodal plane curves in the linear system of degree d plane curves is given. The answer is expressed as a tautological class on a blow-up of the Hilbert scheme of 3 points in the plane. The class is evaluated by the Bott residue formula.
In this paper we study the problem of describing the integral subschemes within a fixed even linkage class $Ł$ of subschemes in $\Pn$ of codimension two. In the case that $Ł$ is not the class of arithmetically Cohen-Macaulay subschemes, we associate to any $X \in Ł$ two invariants $θ_X$ and $η_X$. When taken with the height $h_X$, each of these invariants determines the location of $X$ in $Ł$, thought of as a poset under domination. In terms of these invariants, necessary conditions are given for integral subschemes. The necessary conditions are almost sufficient in the sense that if a subscheme $X$ satisfies the necessary conditions and dominates an integral subscheme $Y$, then $X$ can be deformed with constant cohomology through subschemes in $Ł$ to an integral subscheme. In particular, if an even linkage class has a minimal element which is integral, then the conditions are both necessary and sufficient.
We explain in detail the correspondence between algebraic connections over CP^{1}, logarithmic at X = { x_{1},...,x_{n} } \subset CP^{1}, and flat bundles over CP^{1}-X with integer weighted filtrations near each x_{j}. Included is a gauge fixing theorem for logarithmic connections. (Thus far, one could work over any Riemann surface.) We prove a bound on the splitting type of a semi-stable logarithmic connection over CP^{1}. Using this we extend and simplify some results on the Riemann-Hilbert-Problem, which asks for a logarithmic connection on a holomorphically trivial bundle over CP^{1}, extending a given flat bundle over CP^{1}-X. The work is self contained and elementary, using only basic knowledge of Gauge Theory and the Birkhoff-Grothendieck-Theorem.
The relationship between stable holomorphic vector bundles on a compact complex surface and the same such objects on a blowup of the surface is investigated, where "stability" is with respect to a Gauduchon metric on the surface and naturally derived such metrics on the blowup. The main results are: descriptions of holomorphic vector bundles on a blowup; conditions relating (semi)-stability of these to that of their direct images on the surface; sheaf-theoretic constructions for "stabilising" unstable bundles and desingularising moduli of stable bundles; an analysis of the behaviour of Hermitian-Einstein connections on bundles over blowups as the underlying Gauduchon metric degenerates; the definition of a topology on equivalence classes of stable bundles on blowups over a surface and a proof (using results of a concurrent preprint) that this topology is compact in many cases---i.e., a new compactification for moduli of stable bundles on the original surface.