Narrow Results

We prove that for certain projective varieties V ⊂ P^r (e.g. smooth complete intersections with dim(V) ≥ 4, or complete intersections with dim(V) ≥ 7 and codim_V (Sing(V)) ≥ 6), there are only finitely many components of the Hilbert scheme parametrizing irreducible, smooth, projective, codimension two subvarieties of V not of general type.

We study families of ropes of any codimension that are supported on lines. We construct suitable smooth parameter spaces and conclude that all ropes of fixed degree and genus lie in the same component of the corresponding Hilbert scheme. We show that this component is generically smooth if the genus is small enough unless the characteristic of the ground field is two and the curves under consideration have degree two. In this case the component is non-reduced.

We study structures of embedded projective manifolds swept out by cubic varieties. We show if an embedded projective manifold is swept out by high-dimensional smooth cubic hypersurfaces, then it admits an extremal contraction which is a linear projective bundle or a cubic fibration. As an application, we give a characterization of smooth cubic hypersurfaces. We also classify embedded projective manifolds of dimension at most five swept out by copies of the Segre threefold ℙ¹ × ℙ². In the course of the proof, we classify projective manifolds of dimension five swept out by planes.

We study maximal families W of the Hilbert scheme, H(d, g)_sc, of smooth connected space curves whose general curve C lies on a smooth surface S of degree s. We give conditions on C under which W is a generically smooth component of H(d, g)_sc and we determine dim W. If s = 4 and W is an irreducible component of H(d, g)_sc, then the Picard number of S is at most 2 and we explicitly describe, also for s ≥ 5, non-reduced and generically smooth components in the case Pic(S) is generated by the classes of a line and a smooth plane curve of degree s - 1. For curves on smooth cubic surfaces the first author finds new classes of non-reduced components of H(d, g)_sc, thus making progress in proving a conjecture for such families.

Given a smooth curve on a smooth surface, the Hilbert scheme of points on the surface is stratified according to the length of the intersection with the curve. The strata are highly singular. We show that this stratification admits a natural log-resolution, namely the stratified blowup. As a consequence, the induced Poisson structure on the Hilbert scheme of a Poisson surface has unobstructed deformations.

We study the deformations of a smooth curve $C$ on a smooth projective $3$-fold $V$, assuming the presence of a smooth surface $S$ satisfying $C\subset S\subset V$. Generalizing a result of Mukai and Nasu, we give a new sufficient condition for a first order infinitesimal deformation of $C$ in $V$ to be primarily obstructed. In particular, when $V$ is Fano and $S$ is $K3$, we give a sufficient condition for $C$ to be (un)obstructed in $V$, in terms of $(-2)$-curves and elliptic curves on $S$. Applying this result, we prove that the Hilbert scheme ${Hilb}^{\mathrm{\text{sc}}}{V}_4$ of smooth connected curves on a smooth quartic $3$-fold $V_4\subset {\mathbb{P}}^4$ contains infinitely many generically non-reduced irreducible components, which are variations of Mumford’s example for ${Hilb}^{\mathrm{\text{sc}}}{\mathbb{P}}^3$.

In this paper, we make a correction to Theorem 3.3 of the aforementioned paper (Internat. J. Math.28(13) (2017) 1750099). We also provide a counterexample to the theorem and point out an error in the proof.

In this paper, we study K3 double structures on minimal rational surfaces $Y$. The results show there are infinitely many non-split abstract K3 double structures on $Y={𝔽}_e$ parametrized by ${\mathbb{P}}^1$, countably many of which are projective. For $Y={\mathbb{P}}^2$ there exists a unique non-split abstract K3 double structure which is non-projective (see [J.-M. Drézet, Primitive multiple schemes, preprint (2020), arXiv:2004.04921, to appear in Eur. J. Math.]). We show that all projective K3 carpets can be smoothed to a smooth K3 surface. One of the byproducts of the proof shows that unless $Y$ is embedded as a variety of minimal degree, there are infinitely many embedded K3 carpet structures on $Y$. Moreover, we show any embedded projective K3 carpet on ${𝔽}_e$ with $e<3$ arises as a flat limit of embeddings degenerating to 2:1 morphism. The rest do not, but we still prove the smoothing result. We further show that the Hilbert points corresponding to the projective K3 carpets supported on ${𝔽}_e$, embedded by a complete linear series are smooth points if and only if $0\le e\le 2$. In contrast, Hilbert points corresponding to projective (split) K3 carpets supported on ${\mathbb{P}}^2$ and embedded by a complete linear series are always smooth. The results in [P. Bangere, F. J. Gallego and M. González, Deformations of hyperelliptic and generalized hyperelliptic polarized varieties, preprint (2020), arXiv:2005.00342] show that there are no higher dimensional analogues of the results in this paper.

W describe the unobstructed components of the Hilbert Scheme of rational curves of fixed degree $k$ in the moduli space ${\mathrm{\text{U}}}_C(r,L)$ of stable vector bundles of rank $r$ and determinant $L$ on a curve $C$. We show that for every $k$, there are $gcd(r,degL)$ such components. We construct obstructed components of the Hilbert Scheme. We also obtain an upper bound on the degree of rational connectedness of ${\mathrm{\text{U}}}_C(r,L)$ which is linear in the dimension.

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.

We know from a result due to Noether–Lefschetz that a very general surface of degree at least 4 in ${\mathbb{P}}^3$ contains only curves which are complete intersections with other surfaces. The main goal of this paper is to construct an explicit and smooth compactification of a parameter space for surfaces in ${\mathbb{P}}^3$ of degree $d,$ for all sufficiently large $d$, containing one conic and one line. The construction also applies to surfaces in ${\mathbb{P}}^3$ containing one plane curve and one line. As an application, we compute the degree of the locus of surfaces of degree $d,$ containing one conic and one line.

The aim of this paper is to investigate the closed subschemes of moduli spaces corresponding to projective varieties which admit an effective action by a given finite group $G$. To achieve this, we introduce the moduli functor $M_h^G$ of $G$-marked Gorenstein canonical models with Hilbert polynomial $h$, and prove the existence of ${𝔐}_h[G]$, the coarse moduli scheme for $M_h^G$. Then we show that ${𝔐}_h[G]$ has a proper and finite morphism onto ${𝔐}_h$ so that its image ${𝔐}_h(G)$ is a closed subscheme. In the end we obtain the canonical representation type decomposition ${{𝒟}}_h[G]$ of ${𝔐}_h[G]$ and use ${{𝒟}}_h[G]$ to study the structure of ${𝔐}_h[G]$.

We consider the question of irreducibility of the Hilbert scheme of points ℋilb_dℙⁿ and its Gorenstein locus. This locus is known to be reducible for d ≥ 14. For d ≤ 11 the irreducibility of this locus was proved in the series of papers [G. Casnati and R. Notari, On the Gorenstein locus of some punctual Hilbert schemes, J. Pure Appl. Algebra 213(11) (2009) 2055–2074; On the irreducibility and the singularities of Gorenstein locus of the Punctual Hilbert scheme of degree 10, J. Pure Appl. Algebra 215(6) (2011) 1243–1254; Irreducibility of the Gorenstein locus of the Punctual Hilbert Scheme of degree 11, preprint (2012)] and Iarrobino conjectured that the irreducibility holds for d ≤ 13. In this paper, we prove that the subschemes corresponding to the Gorenstein algebras having Hilbert function (1, 5, 5, 1) are smoothable, i.e. lie in the closure of the locus of smooth subschemes. This result completes the proof of irreducibility of the Gorenstein locus of ℋilb₁₂ℙⁿ, see Theorem 2.

We define marked sets and bases over a quasi-stable ideal $𝔧$ in a polynomial ring on a Noetherian $K$-algebra, with $K$ a field of any characteristic. The involved polynomials may be non-homogeneous, but their degree is bounded from above by the maximum among the degrees of the terms in the Pommaret basis of $𝔧$ and a given integer $m$. Due to the combinatorial properties of quasi-stable ideals, these bases behave well with respect to homogenization, similarly to Macaulay bases. We prove that the family of marked bases over a given quasi-stable ideal has an affine scheme structure, is flat and, for large enough $m$, is an open subset of a Hilbert scheme. Our main results lead to algorithms that explicitly construct such a family. We compare our method with similar ones and give some complexity results.

We denote by ${{\mathscr{H}}}_{d,g,r}$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree $d$ and genus $g$ in ${\mathbb{P}}^r$. In this paper, we show that for low genus $g$ outside the Brill–Noether range, the Hilbert scheme ${{\mathscr{H}}}_{g+1,g,4}$ is non-empty whenever $g\ge 9$ and irreducible whose only component generically consists of linearly normal curves unless $g=9$ or $g=12$. This complements the validity of the original assertion of Severi regarding the irreducibility of ${{\mathscr{H}}}_{d,g,r}$ outside the Brill–Nother range for $d=g+1$ and $r=4$.

Let X be a compact connected Riemann surface; the holomorphic cotangent bundle of X will be denoted by K_X. Let denote the moduli space of semistable Higgs Gp(2n, ℂ)-bundles over X of fixed topological type. The complex variety has a natural holomorphic symplectic structure. On the other hand, for any ℓ ≥ 1, the Liouville symplectic from on the total space of K_X defines a holomorphic symplectic structure on the Hilbert scheme Hilb^ℓ(K_X) parametrizing the zero-dimensional subschemes of K_X. We relate the symplectic form on Hilb^ℓ(K_X) with the symplectic form on .