Let $mathcal{I}_{d,g,R}$ be the union of irreducible components of the Hilbert scheme whose general points parametrize smooth, irreducible, curves of degree $d$, genus $g$, which are non--degenerate in the projective space $mathbb{P}^R$. Under some numerical assumptions on $d$, $g$ and $R$, we construct irreducible components of $mathcal{I}_{d,g,R}$ other than the so-called {em distinguished component}, dominating the moduli space $mathcal{M}_g$ of smooth genus--$g$ curves, which are generically smooth and turn out to be of dimension higher than the expected one. The general point of any such a component corresponds to a curve $X subset mathbb{P}^R$ which is a suitable ramified $m$--cover of an irrational curve $Y subset mathbb{P}^{R-1}$, $m geqslant 2$, lying in a surface cone over $Y$. extends some of the results in previous papers of Y. Choi, H. Iliev, S. Kim (cf. [12,13] in Bibliography).

Flamini, F., Supino, P. (2023). On some components of Hilbert schemes of curves. In F.F. Thomas Dedieu (a cura di), The Art of Doing Algebraic Geometry. Trends in Mathematics (pp. 187-215). Birkhäuser Cham [10.1007/978-3-031-11938-5_8].

On some components of Hilbert schemes of curves

Paola Supino
2023-01-01

Abstract

Let $mathcal{I}_{d,g,R}$ be the union of irreducible components of the Hilbert scheme whose general points parametrize smooth, irreducible, curves of degree $d$, genus $g$, which are non--degenerate in the projective space $mathbb{P}^R$. Under some numerical assumptions on $d$, $g$ and $R$, we construct irreducible components of $mathcal{I}_{d,g,R}$ other than the so-called {em distinguished component}, dominating the moduli space $mathcal{M}_g$ of smooth genus--$g$ curves, which are generically smooth and turn out to be of dimension higher than the expected one. The general point of any such a component corresponds to a curve $X subset mathbb{P}^R$ which is a suitable ramified $m$--cover of an irrational curve $Y subset mathbb{P}^{R-1}$, $m geqslant 2$, lying in a surface cone over $Y$. extends some of the results in previous papers of Y. Choi, H. Iliev, S. Kim (cf. [12,13] in Bibliography).
2023
978-3-031-11937-8
Flamini, F., Supino, P. (2023). On some components of Hilbert schemes of curves. In F.F. Thomas Dedieu (a cura di), The Art of Doing Algebraic Geometry. Trends in Mathematics (pp. 187-215). Birkhäuser Cham [10.1007/978-3-031-11938-5_8].
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11590/378370
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 1
  • ???jsp.display-item.citation.isi??? ND
social impact