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

Abstract

Denote by 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 P r . A component of H d,g,r is rigid in moduli if its image under the natural map π : H d,g,r Mg is a one point set. In this note, we provide a proof of the fact that 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 P 3 whose only deformations are given by projective transformations are the twisted cubic curves. In case r ≥ 4, we also prove the non-existence of a component of 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 H d,g,3 beyond the range which has been known before.In the next two sections, we provide a proof of the fact that H g+1,g,3 is irreducible and H d,g,3 does not have a component rigid in moduli if g > 0. This in turn implies that there are no rigid curves in P 3 except for twisted cubic curves. In the subsequent section we also prove that, for r ≥ 4, H d,g,r does not carry any

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

“…Even though we need to impose certain restrictions on the range of the triples (d, g, r) in which the statements work, it may be worthy of coming up with some generalizations for further study along the line of ideas we followed in this paper. For almost all the statements which will be made in this (iii) If g ≤ r + 2α − 2 -which is equivalent to the condition r 1 3 (2d − g + 1) -we get a contradiction by (26). Therefore we arrive at the following statement which can be viewed as a generalization of Theorem 3.2 (a), (b) and Proposition 4.1 (i), (ii).…”

“…Hence we have an irreducible family of nodal curves lying on a Hirzebruch surface F 1 -parametrized by the irreducible Severi variety Σ M1,11−r -whose non-singular model G M2,0 = Σ ∨ M2,0 . Therefore we obtain another irreducible family of (smooth) curves on F 1 which is embedded into P 10 by |K P 2 1 + C E − O S (1)| = (5; 4) as a linearly normal curve with (d, g) = (2r + 6, r + 11) = (26,21). By Castelnuovo-Severi inequality, a general…”

“…(i) By Castelnuovo theory, it is known that the dimension r of a birationally very ample linear series g r d on a curve of genus g is bounded by (cf. [26,Lemma1.5])…”

Linear series with $ρ< 0$ via thrifty lego-building

“…The disadvantage of this much simpler lego-building is of course the somewhat complicated error bounds. Conjecturally, the bound ρ ≥ − dim M g never holds with equality for g > 0; this is a folklore conjecture sometimes called the rigid curves conjecture, and it has recently been verified for r = 3 and in a restricted range of cases with r ≥ 4 by Keem, Kim, and Lopez [KKL19]. A natural question, in light of this conjecture, is: how close can you get?…”

“…Work on similar problems to those discussed above includes [Far03] on regular components of moduli of stable maps to products of projective spaces, [BBF14] on generalizations to nodal curves, and [KKL19] on the problem of curves rigid in moduli.…”