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

Archiv der Mathematik - 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...     

Normal generation of line bundles of high degrees on smooth algebraic curves

We give necessary and sufficient conditions for non-special line bundles of degree2g — 2 and 2g — 3 being not normally generated. We also provide criteria for special line bundles of degreed > 2g — 6 being normally generated. This work was done under JSPS-KOSEF joint research program 1997. The first named author was partially supported by Grant-in-Aid for Scientific Research, the Ministry of Education, #10440051. The second named author was partially supported by BSRI(1998-015-D00023) and SNU(99-5-l-031). During the period when this paper was prepared for publication, the second named author was enjoying the hospitality of ICTP. The third named author was partially supported by Grant-in-Aid for Scientific Research, the Ministry of Education, #09640043.     

Variety of Special Linear Systems on k-Sheeted Coverings

Irreducibility of W¹ _d (X) for d g - (k - 2) [(h + 3)/2] - h+ 1, where X is a curve of genus g which admits an odd prime degree k map onto a general curve C of genus h > 0 is proved. Also, the existence of a component of W¹ _d(X) with expected dimension on a general k-sheeted covering X over a curve C is shown.     

Irreducibility of the Hilbert scheme of smooth curves in $$\mathbb {P}^4$$ of de gree $$ g+2$$ g+2 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 \(\mathcal {H}_{g+2,g,4}\) is irreducible, generically smooth, and has the expected dimension \(4g+11\) without any restriction on the genus g. Our result augments the irreducibility result obtained earlier by Iliev (Proc Am Math Soc 134:2823–2832, 2006), in which several low genus \(g\le 10\) cases have been left untreated.