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

On the Hilbert scheme of linearly normal curves in Pr of relatively high degree

“…For families of curves in P 3 of lower degree d ≤ g + 2, the most updated result is that any non-empty H d,g,3 is irreducible for every d ≥ g; cf. [ [18].…”

Existence and the reducibility of the Hilbert scheme of linearly normal curves in $\mathbb{P}^r$ of relatively high degrees

“…H g,g,3 = H L g,g,3 is irreducible for g ≥ 8 and is empty for g ≤ 7. Even though the non-emptiness of H g,g,3 for g ≥ 8 was not explicitly mentioned in [20], the existence of a smooth irreducible curve in P 3 of degree d = g for g ≥ 9 is assured by a result due to Gruson-Peskine; note that g ≤ π 1 (g, 3) if g ≥ 9, where π 1 (d, r) is the so-called second Castelnuovo genus bound for curves of degree d in P 3 ; cf. We now shift our attention to curves in P r , r ≥ 5.…”

“…(i) Somewhat stronger results hold for curves in P 3 . For r = 3 and d = g + r − 3 = g, by[20, Theorem 2.1] …”

On the Hilbert scheme of linearly normal curves in $\mathbb{P}^r$ with small index of speciality

“…One obvious advantage considering the Hilbert scheme of linearly normal curves according to its index of speciality α is that the residual series of the very ample hyperplane series corresponding to the general element of a component has the fixed dimension α − 1 regardless of the values of the genus g and the degree d. Therefore one may work more effectively in exploring out several properties of the Hilbert scheme under consideration by looking at the family of curves in a fixed projective space P α−1 induced by the residual series of the hyperplane series. For one thing, in case α = 3, the residual series of hyperplane series are nets and it is possible to derive the basic properties such as irreducibility, existence as well as the number of moduli of the Hilbert scheme of linearly normal curves by considering the corresponding property of the Severi variety of plane curves, which has been proven useful to a certain extent in some previous works on the subject; [20,5,21,22]. In this paper we consider Hilbert scheme of linear normal curves with index of speciality α = 4.…”