Components Of Hilbert Schemes Research Articles

Galois closures and elementary components of Hilbert schemes of points

Galois closures and elementary components of Hilbert schemes of points

Small elementary components of Hilbert schemes of points

AbstractWe answer an open problem posed by Iarrobino,Hilbert scheme of points: Overview of last ten years. Proceedings of Symposia in Pure Mathematics, 46 (American Mathematical Society, Providence, RI, 1987), 297–320: Is there a component of the punctual Hilbert scheme [Grothendieck, Techniques de construction et théorèmes d'existence en géométrie algébrique. IV. Les schémas de Hilbert',in Séminaire Bourbaki, 6 (Societe Mathematique de France, Paris, 1995), 221, 249–276]$\operatorname {\mathrm {Hilb}}^d({\mathscr {O}}_{\mathbb {A}^n,p})$with dimension less than$(n-1)(d-1)$? For each$n\geq 4$, we construct an infinite class of elementary components in$\operatorname {\mathrm {Hilb}}^d(\mathbb {A}^n)$producing such examples. Our techniques also allow us to construct an explicit example of a local Artinian ring [Iarrobino and Kanev,Power sums, Gorenstein algebras, and determinantal loci(Springer-Verlag, Berlin, 1999), 221–226] of the formwith trivial negative tangents, vanishing nonnegative obstruction space, and socle-dimension$2$.

The Hilbert scheme of hyperelliptic Jacobians and moduli of Picard sheaves

Let $C$ be a hyperelliptic curve embedded in its Jacobian $J$ via an Abel-Jacobi map. We compute the scheme structure of the Hilbert scheme component of $\textrm{Hilb}_J$ containing the Abel-Jacobi curve as a point. We relate the result to the ramification (and to the fibres) of the Torelli morphism $\mathcal M_g\rightarrow \mathcal A_g$ along the hyperelliptic locus. As an application, we determine the scheme structure of the moduli space of Picard sheaves (introduced by Mukai) on a hyperelliptic Jacobian.

Elementary components of Hilbert schemes of points

We generalize the Bialynicki-Birula decomposition to singular schemes and apply it to the Hilbert scheme of points on an affine space. We find an infinite family of small, elementary and generically smooth components of the Hilbert scheme of points of the affine four-space. Our method gives easily verifiable sufficient conditions for proving that a point of the Hilbert scheme is smooth and lies on an elementary component. We also present a necessary condition for smoothability of a finite subscheme given by a homogeneous ideal.

On generically non‐reduced components of Hilbert schemes of smooth curves

AbstractA classical example of Mumford gives a generically non‐reduced component of the Hilbert scheme of smooth curves in such that a general element of the component is contained in a smooth cubic surface in . In this article we use techniques from Hodge theory to give further examples of such (generically non‐reduced) components of Hilbert schemes of smooth curves without any restriction on the degree of the surface containing it. As a byproduct we also obtain generically non‐reduced components of certain Hodge loci.

Irreducible components of Hilbert schemes of rational curves with given normal bundle

We develop a new general method for computing the decomposition type of the normal bundle to a projective rational curve. This method is then used to detect and explain an example of a Hilbert scheme that parametrizes all the rational curves in $\mathbb{P}^s$ with a given decomposition type of the normal bundle and that has exactly two irreducible components. This gives a negative answer to the very old question whether such Hilbert schemes are always irreducible. We also characterize smooth non-degenerate rational curves contained in rational normal scroll surfaces in terms of the splitting type of their restricted tangent bundles, compute their normal bundles and show how to construct these curves as suitable projections of a rational normal curve.

Borel degenerations of arithmetically Cohen–Macaulay curves in ℙ3

We investigate Borel ideals on the Hilbert scheme components of arithmetically Cohen–Macaulay (ACM) codimension two schemes in ℙn. We give a basic necessary criterion for a Borel ideal to be on such a component. Then considering ACM curves in ℙ3 on a quadric we compute in several examples all the Borel ideals on their Hilbert scheme component. Based on this we conjecture which Borel ideals are on such a component, and for a range of Borel ideals we prove that they are on the component.

Detaching embedded points

We show that if $D \subset \mathbb P^N$ is obtained from a codimension two local complete intersection $C$ by adding embedded points of multiplicity $\leq 3$, then $D$ is a flat limit of $C$ and isolated points. As applications, we determine the irreducible components of Hilbert schemes of space curves with high arithmetic genus, show the smoothness of the Hilbert component whose general member is a plane curve union a point in $\mathbb P^3$, and construct a Hilbert component whose general member has an embedded point.

The Hilbert schemes of locally Cohen–Macaulay curves in $${\mathbb{P}^3}$$ may after all be connected

Progress on the problem whether the Hilbert schemes of locally Cohen–Macaulay curves in \({\mathbb{P}^3}\) are connected has been hampered by the lack of an answer to a question raised by Robin Hartshorne in (Commun. Algebra 28:6059–6077, 2000) and more recently in (American Institute of Mathematics, Workshop components of Hilbert schemes, problem list, 2010. http://aimpl.org/hilbertschemes): does there exist a flat irreducible family of curves whose general member is a union of d ≥ 4 disjoint lines on a smooth quadric surface and whose special member is a locally Cohen–Macaulay curve in a double plane? In this paper we give a positive answer to this question: for every d we construct a family with the required properties, whose special fiber is an extremal curve in the sense by Martin-Deschamps and Perrin (Ann. Sci. E.N.S. 4e Série 29:757–785, 1996). From this we conclude that every effective divisor in a smooth quadric surface is in the connected component of its Hilbert scheme that contains extremal curves.

Rational Components of Hilbert Schemes

The Groebner stratum of a monomial ideal \mathfrak{j} is an affine variety that parameterizes the family of all ideals having \mathfrak{j} as initial ideal (with respect to a fixed term ordering). The Groebner strata can be equipped in a natural way with a structure of homogeneous variety and are in a close connection with Hilbert schemes of subschemes in the projective space \mathbf{P}^n . Using properties of the Groebner strata we prove some sufficient conditions for the rationality of components of {\mathcal{H}\text{ilb}_{p(z)}^n} . We show for instance that all the smooth, irreducible components in {\mathcal{H}\text{ilb}_{p(z)}^n} (or in its support) and the Reeves and Stillman component H_{RS} are rational. We also obtain sufficient conditions for isomorphisms between strata corresponding to pairs of ideals defining a same subscheme, that can strongly improve an explicit computation of their equations.

The hilbert scheme component of the intersection of two quadrics

We obtain an explicit description of the irreducible component, of the Hilbert scheme parametrizing complete intersections of two quadrics in IPn for n≥3.