Construction of space curves with good properties (Q584337)

We show that for all (d,g) in the degree, genus plane which are between a certain piecewise linear curve \(C_ K\) of order \(g\sim \frac{2\sqrt{2}}{3}d^{3/2}\) and a curve \(C_ 1\) of order \(g\sim \frac{2}{3}d^{3/2}\), there are smooth connected curves with degree d and genus g which are of maximal rank. We also show that for all (d,g) between the curve \(C_ K\) and the d- axis, there are smooth connected curves Y with degree d and genus g such that \(h^ 1I_ Y(n)\cdot h^ 2I_ Y(n)=0\) for all \(n\in {\mathbb{Z}}\). Together with a previous result of Sernesi this implies that for all (d,g) in this area there are components \(H_{d,g}\) of the Hilbert scheme with the expected number of moduli.