Non degenerate projective curves with very degenerate hyperplane section

Abstract

We study the Hilbert scheme of non degenerate locally Cohen-Macaulay projective curves with general hyperplane section spanning a linear space of dimension 2 and minimal Hilbert function. The main result is that those curves are almost always the general element of a generically smooth component $ H_{n,d,g} $ of the corresponding Hilbert scheme. Moreover, we show that the curves with maximal cohomology almost always correspond to smooth points of $ H_{n,d,g}.

Even G-liaison classes of some unions of curves

“…[12]) and with a very degenerate hyperplane section (cf. [12,16]), respectively. Now, we want to construct an example to show that the bound is sharp whatever is the value of k (see also Remark 2.7 below).…”

“…Of course, for j < 0, the only global section of E is = 0, while the global sections of C (see [16,Lemma 6,Theorem 4]…”

“…, x k , t, u]) be a curve of maximal cohomology, of degree d + 2 and genus g as described in [12], Theorem 4.2, and more extensively in [16], associated to the exact sequence…”

“…Here, we consider analogous bounds by assuming that the general hyperplane section of C spans a linear space of dimension n − k, 1 k n − 2, and we give examples of curves with that general hyperplane section, by generalizing the construction in [16].…”

“…Moreover, he constructed curves attaining the bounds. In [16], there was a description of the curves with the worst Hilbert function of the general hyperplane section and of the components of the corresponding Hilbert schemes.…”