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ON SOME PROPERTIES OF THE HILBERT SCHEME OF SMOOTH CURVES

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ON SOME PROPERTIES OF THE HILBERT SCHEME OF SMOOTH CURVES

HRISTO ILIEV

abstract

Let Hd,g,r denotes the Hilbert scheme parametrizing smooth irreducible complex curves of degreedand genusg embedded inPr. Severi claimed in [Sev21] that Hd,g,r is irreducible if d ≥ g +r. As it has turned out in recent years, the conjecture is true for r= 3 and 4, while for r≥6 it is incorrect. I prove that Hg,g,3,Hg+3,g,4 and Hg+2,g,4 are irreducible, provided that g ≥ 13, g ≥5 and g ≥11, respectively. This extends results obtained previously by Ein, [Ein86, Ein87] and by Keem and Kim, [KK92]. The technique can also be applied to extend the known irreducibility range of Hd,g,5 with respect to d.

References

[Ein86] L. Ein. Classification of projective surfaces and projective normality. Ann. scient. Ec.

Norm. Sup., 19(4):469–478, 1986.

[Ein87] L. Ein. The irreducibility of the Hilbert scheme of complex space curves.Proc. of symposia in Pure Math., 46:83–87, 1987.

[KK92] C. Keem and S. Kim. Irreducibility of the Hilbert scheme of complex space curves. J. of Alg., 145:240–640, 1992.

[Sev21] F. Severi.Vorlesungen uber algebraische Geometrie. Teubner, Leipzig, 1921.

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