Hilbert Scheme Of Curves Research Articles

Geometry of canonical genus 4 curves

AbstractWe apply the machinery of Bridgeland stability conditions on derived categories of coherent sheaves to describe the geometry of classical moduli spaces associated with canonical genus 4 space curves via an effective control over its wall‐crossing. This article provides the first description of a moduli space of Pandharipande–Thomas stable pairs that is used as an intermediate step toward the description of the associated Hilbert scheme, which in turn is the first example where the components of a classical moduli space were completely determined via wall‐crossing. We give a full list of irreducible components of the space of stable pairs, along with a birational description of each component, and a partial list for the Hilbert scheme. There are several long standing open problems regarding classical sheaf theoretic moduli spaces, and the present work will shed light on further studies of such moduli spaces such as Hilbert schemes of curves and moduli of stable pairs that are very hard to tackle without the wall‐crossing techniques.

Curve counting and DT/PT correspondence for Calabi-Yau 4-folds

Recently, Cao-Maulik-Toda defined stable pair invariants of a compact Calabi-Yau 4-fold X. Their invariants are conjecturally related to the Gopakumar-Vafa type invariants of X defined using Gromov-Witten theory by Klemm-Pandharipande. In this paper, we consider curve counting invariants of X using Hilbert schemes of curves and conjecture a DT/PT correspondence which relates these to stable pair invariants of X.After providing evidence in the compact case, we define analogous invariants for toric Calabi-Yau 4-folds. We formulate a vertex formalism for both theories and conjecture a relation between the (fully equivariant) DT/PT vertex, which we check in several cases. This relation implies a DT/PT correspondence for toric Calabi-Yau 4-folds with primary insertions.

Moduli of sheaves supported on Quartic space curves

As a continuation of the work of Freiermuth and Trautmann, we study the geometry of the moduli space of stable sheaves on $\mathbb{P}^3$ with Hilbert polynomial $4m+1$. The moduli space has three irreducible components whose generic elements are, respectively, sheaves supported on rational quartic curves, on elliptic quartic curves, or on planar quartic curves. The main idea of the proof is to relate the moduli space with the Hilbert scheme of curves by wall crossing. We present all stable sheaves contained in the intersections of the three irreducible components. We also classify stable sheaves by means of their free resolutions.

Families of Smooth Rational Curves of Small Degree on the Fano Variety of Degree 5 of Main Series

In this paper we consider some families of smooth rational curves of degree 2, 3 and 4 on a smooth Fano threefold X which is a linear section of the Grassmanian G(1, 4) under the Pl¨ucker embedding. We prove that these families are irreducible. The proof of the irreducibility of the families of curves of degree d is based on the study of degeneration of a rational curve of degree d into a curve which decomposes into an irreducible rational curve of degree d−1 and a projective line intersecting transversally at a point. We prove that the Hilbert scheme of curves of degree d on X is smooth at the point corresponding to such a reducible curve. Then calculations in the framework of deformation theory show that such a curve varies into a smooth rational curve of degree d. Thus, the set of reducible curves of degree d of the above type lies in the closure of a unique component of the Hilbert scheme of smooth rational curves of degree d on X. From this fact and the irreducibility of the Hilbert scheme of smooth rational curves of degree d on the Grassmannian G(1, 4) one deduces the irreducibility of the Hilbert scheme of smooth rational curves of degree d on a general Fano threefold X.

Hilbert schemes and stable pairs: GIT and derived category wall crossings

Nous montrons que le schéma de Hilbert de courbes et l'espace de modules de Le Potier de paires stables à support à une dimension, ont une construction GIT commune. Les deux espaces correspondents aux chambres de par et d'autre d'un mur dans l'espace de linéarisations GIT. Nous expliquons pourquoi cela ne suffit pas pour prouver la « conjecture de croisement de murs DT/PT » qui relie les invariants dérivés de ces espaces de modules quand la variété sous-jacente est un 3-fold. Nous donnons, ensuite, une introduction simple à une petite partie de la théorie de Joyce sur les croisements de murs de ce type, et nous nous en servons pour donner une brève démonstration d'une identité reliant les caractéristiques d'Euler de ces espaces de modules. Quand le 3-fold est de type Calabi-Yau, l'identité est le pendant, pour la caractéristique d'Euler, de la conjecture de croisement de murs DT/PT, mais dans le cas général elle s'avère être différente de celle-ci, comme nous l'expliquons.

On the Irreducibility of the Hilbert Scheme of Curves in ℙ 5

Denote by ℐ d,g,r the Hilbert scheme parametrizing smooth irreducible complex curves of degree d and genus g embedded in ℙ r . Severi (1921) claimed that ℐ d,g,r is irreducible if d ≥ g + r. Ein proved that the conjecture is true for r = 3 and 4, and in general that ℐ d,g,r is irreducible if (Ein, 1986, 1987). As it is known, for r ≥ 6 the conjecture is incorrect and r = 5 remains the only unsettled case. Here I prove that ℐ d,g,5 is irreducible, if , which doesn't yet resolve Severi's conjecture for r = 5, but expands the known irreducibility range.

Moduli of reflexive sheaves on smooth projective 3-folds

We compute the expected dimension of the moduli space of torsion-free rank 2 sheaves at a point corresponding to a stable reflexive sheaf, and give conditions for the existence of a perfect tangent-obstruction complex on a class of smooth projective threefolds; this class includes Fano and Calabi–Yau threefolds. We also explore both local and global relationships between moduli spaces of reflexive rank 2 sheaves and the Hilbert scheme of curves.

Murphy’s law in algebraic geometry: Badly-behaved deformation spaces

We consider the question: “How bad can the deformation space of an object be?” The answer seems to be: “Unless there is some a priori reason otherwise, the deformation space may be as bad as possible.” We show this for a number of important moduli spaces. More precisely, every singularity of finite type over ℤ (up to smooth parameters) appears on: the Hilbert scheme of curves in projective space; and the moduli spaces of smooth projective general-type surfaces (or higher-dimensional varieties), plane curves with nodes and cusps, stable sheaves, isolated threefold singularities, and more. The objects themselves are not pathological, and are in fact as nice as can be: the curves are smooth, the surfaces are automorphism-free and have very ample canonical bundle, the stable sheaves are torsion-free of rank 1, the singularities are normal and Cohen-Macaulay, etc. This justifies Mumford’s philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise. Thus one can construct a smooth curve in projective space whose deformation space has any given number of components, each with any given singularity type, with any given non-reduced behavior. Similarly one can give a surface over $\mathbb{F}_{p}$ that lifts to ℤ/p 7 but not ℤ/p 8. (Of course the results hold in the holomorphic category as well.) It is usually difficult to compute deformation spaces directly from obstruction theories. We circumvent this by relating them to more tractable deformation spaces via smooth morphisms. The essential starting point is Mnev’s universality theorem.

THE HILBERT COMPACTIFICATION OF THE UNIVERSAL MODULI SPACE OF SEMISTABLE VECTOR BUNDLES OVER SMOOTH CURVES

We construct the Hilbert compactification of the universal moduli space of semistable vector bundles over smooth curves. The Hilbert compactification is the GIT quotient of some open part of an appropriate Hilbert scheme of curves in a Graβmannian. It has all the properties asked for by Teixidor.

A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on $K3$ fibrations

We briefly review the formal picture in which a Calabi-Yau $n$-fold is the complex analogue of an oriented real $n$-manifold, and a Fano with a fixed smooth anticanonical divisor is the analogue of a manifold with boundary, motivating a holomorphic Casson invariant counting bundles on a Calabi-Yau 3-fold. We develop the deformation theory necessary to obtain the virtual moduli cycles of \cite{LT}, \cite{BF} in moduli spaces of stable sheaves whose higher obstruction groups vanish. This gives, for instance, virtual moduli cycles in Hilbert schemes of curves in $\Pee^3$, and Donaldson-- and Gromov-Witten-- like invariants of Fano 3-folds. It also allows us to define the holomorphic Casson invariant of a Calabi-Yau 3-fold $X$, prove it is deformation invariant, and compute it explicitly in some examples. Then we calculate moduli spaces of sheaves on a general $K3$ fibration $X$, enabling us to compute the invariant for some ranks and Chern classes, and equate it to Gromov-Witten invariants of the ``Mukai-dual'' 3-fold for others. As an example the invariant is shown to distinguish Gross' diffeomorphic 3-folds. Finally the Mukai-dual 3-fold is shown to be Calabi-Yau and its cohomology is related to that of $X$.

On the Hilbert scheme of curves in higher-dimensional projective space

In this paper we prove that, for anyn≥3, there exist infinitely manyr∈N and for each of them a smooth, connected curveC r in ℙ r such thatC r lies on exactlyn irreducible components of the Hilbert scheme Hilb(ℙ r ). This is proven by reducing the problem to an analogous statement for the moduli of surfaces of general type.

Concerning the existence of nice components in the Hilbert scheme of curves in n for n = 4 and 5.

Article Concerning the existence of nice components in the Hilbert scheme of curves in n for n = 4 and 5. was published on January 1, 1996 in the journal Journal für die reine und angewandte Mathematik (volume 1996, issue 475).

Some irreducible Hilbert schemes

number. (Thus H a parametrizes subschemes of IF' r consisting of finitely many points.) There are some examples of irreducible Hilbert schemes of curves of low genus and degree (cp. [5]), but the only classes of irreducible Hilbert schemes which are mentioned in the literature seem to be those of plane curves embedded in some P , and of hypersurfaces (cp. [1], Thm. 1.4 and [7], Prop. 3). The aim of the paper is to find a class of examples, which in some sense is a mixture of these two classes. In order to state the result we have to recall a theorem of F.S. Macaulay, according to which the Hilbert scheme H e is non-empty if and only if