Hilbert Scheme Of Smooth Curves Research Articles

On the Hilbert scheme of smooth curves of degree 15 and genus 14 in $$\mathbb {P}^5$$

On the Hilbert scheme of smooth curves of degree 15 and genus 14 in $$\mathbb {P}^5$$

On the Hilbert scheme of smooth curves in ℙ4 of degree d = g + 1 and genus g with negative Brill–Noether number

We denote by [Formula: see text] 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 [Formula: see text] and genus [Formula: see text] in [Formula: see text]. In this paper, we show that for low genus [Formula: see text] outside the Brill–Noether range, the Hilbert scheme [Formula: see text] is non-empty whenever [Formula: see text] and irreducible whose only component generically consists of linearly normal curves unless [Formula: see text] or [Formula: see text]. This complements the validity of the original assertion of Severi regarding the irreducibility of [Formula: see text] outside the Brill–Nother range for [Formula: see text] and [Formula: see text].

On the Hilbert scheme of linearly normal curves in $$\mathbb {P}^4$$ of degree $$d = g+1$$ and genus g

We denote by $\mathcal{H}_{d,g,r}$ 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 article, we show that any non-empty $\mathcal{H}_{g+1,g,4}$ has only one component whose general element is linear normal unless $g=9$. If $g=9$, we show that $\mathcal{H}_{g+1,g,4}$ is reducible with two components and a general element of each component is linearly normal. This establishes the validity of a certain modified version of an assertion of Severi regarding the irreducibility of $\mathcal{H}_{d,g,r}$ for the case $d=g+1$ and $r=4$.

Irreducibility and components rigid in moduli of the Hilbert scheme of smooth curves

Denote by $$\mathcal {H}_{d,g,r}$$ the Hilbert scheme of smooth curves, that is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree d and genus g in $$\mathbb {P}^r$$ . A component of $$\mathcal {H}_{d,g,r}$$ is rigid in moduli if its image under the natural map $$\pi :\mathcal {H}_{d,g,r} \dashrightarrow \mathcal {M}_{g}$$ is a one point set. In this note, we provide a proof of the fact that $$\mathcal {H}_{d,g,r}$$ has no components rigid in moduli for $$g > 0$$ and $$r=3$$ , from which it follows that the only smooth projective curves embedded in $$\mathbb {P}^3$$ whose only deformations are given by projective transformations are the twisted cubic curves. In case $$r \ge 4$$ , we also prove the non-existence of a component of $$\mathcal {H}_{d,g,r}$$ rigid in moduli in a certain restricted range of d, $$g>0$$ and r. In the course of the proofs, we establish the irreducibility of $$\mathcal {H}_{d,g,3}$$ beyond the range which has been known before.

Irreducibility of the Hilbert scheme of smooth curves in $$\mathbb {P}^4$$ P 4 of degree $$g+2$$ g + 2 and genus g

Irreducibility of the Hilbert scheme of smooth curves in $$\mathbb {P}^4$$ P 4 of degree $$g+2$$ g + 2 and genus g

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.

Reducibility of the Hilbert Scheme of Smooth Curves and Families of Double Covers

Let $\mathcal{I}_{d,g,r}$ be the union of irreducible components of the Hilbert scheme whose general points represent smooth irreducible complex curves of degree $d$ and genus $g$ in $\mathbb{P}^r$. Severi claimed in [15] that $\mathcal{I}_{d,g,r}$ is irreducible if $d \geq g+r$. His statement turned out to be correct for $r = 3$ and $4$, while for $r \geq 6$, counterexamples have been found by using families of $m$-sheeted covers of rational curves with $m \geq 3$. In this work we show the existence of an additional component of $\mathcal{I}_{d,g,r}$ whose general elements are double covers of curves of positive genus. In addition, we find upper bounds for the dimension of the possible components of $\mathcal{I}_{d,g,r}$.

Irreducibility of the Hilbert scheme of smooth curves in $$\mathbb {P}^3$$ P 3 of degree g and genus g

We denote by \(\mathcal {H}_{d,g,r}\) 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 note, we show that any non-empty \(\mathcal {H}_{g,g,3}\) is irreducible without any restriction on the genus g. This extends the result obtained earlier by Iliev (Proc Am Math Soc 134:2823–2832, 2006).

Reducible Hilbert Scheme of Smooth Curves with Positive Brill-Noether Number

In this paper we demonstrate various reducible examples of the scheme $\mathcal {I}{’ _{d,g,r}}$ of smooth curves of degee d and genus g in ${\mathbb {P}^r}$ with positive Brill-Noether number. An example of a reducible $\mathcal {I}{’ _{d,g,r}}$ with positive $\rho (d,g,r)$, namely, the example $\mathcal {I}{’ _{2g - 8,g,g - 8}},$, has been known to some people and seems to have first appeared in the literature in Eisenbud and Harris, Irreducibility of some families of linear series with Brill-Noether number $-1$, Ann. Sci. École Norm. Sup. (4) 22 (1989), 33-53. The purpose of this paper is to add a wider class of examples to the list of such reducible examples by using general k-gonal curves. We also show that $\mathcal {I}{’ _{d,g,r}}$ is irreducible for the range of $d \geq 2g - 7$ and $g - d + r \leq 0$.

Reducible Hilbert scheme of smooth curves with positive Brill-Noether number

In this paper we demonstrate various reducible examples of the scheme I d , g , r ′ \mathcal {I}{’ _{d,g,r}} of smooth curves of degee d and genus g in P r {\mathbb {P}^r} with positive Brill-Noether number. An example of a reducible I d , g , r ′ \mathcal {I}{’ _{d,g,r}} with positive ρ ( d , g , r ) \rho (d,g,r) , namely, the example I 2 g − 8 , g , g − 8 ′ , \mathcal {I}{’ _{2g - 8,g,g - 8}}, , has been known to some people and seems to have first appeared in the literature in Eisenbud and Harris, Irreducibility of some families of linear series with Brill-Noether number − 1 -1 , Ann. Sci. École Norm. Sup. (4) 22 (1989), 33-53. The purpose of this paper is to add a wider class of examples to the list of such reducible examples by using general k-gonal curves. We also show that I d , g , r ′ \mathcal {I}{’ _{d,g,r}} is irreducible for the range of d ≥ 2 g − 7 d \geq 2g - 7 and g − d + r ≤ 0 g - d + r \leq 0 .