Linear normality of general linear sections and some graded Betti numbers of 3-regular projective schemes

In this paper we study graded Betti numbers of any nondegenerate 3-regular algebraic set $X$ in a projective space $\mathbb P^{n}$. More concretely, via Generic initial ideals (Gins) method we mainly consider `tailing' Betti numbers, whose homological index is not less than $\mathrm{codim}(X,\mathbb P^{n})$. For this purpose, we first introduce a key definition `$\mathrm{ND(1)}$ property', which provides a suitable ground where one can generalize the concepts such as `being nondegenerate' or `of minimal degree' from the case of varieties to the case of more general closed subschemes and give a clear interpretation on the tailing Betti numbers. Next, we recall basic notions and facts on Gins theory and we analyze the generation structure of the reverse lexicographic (rlex) Gins of 3-regular $\mathrm{ND(1)}$ subschemes. As a result, we present exact formulae for these tailing Betti numbers, which connect them with linear normality of general linear sections of $X\cap \Lambda$ with a linear subspace $\Lambda$ of dimension at least $\mathrm{codim}(X,\mathbb P^{n})$. Finally, we consider some applications and related examples.