Analysis on some infinite modules, inner projection, and applications

Details Analysis on some infinite modules, inner projection, and applications Analysis on some infinite modules, inner projection, and applications

2008-07-31

arxiv.org/abs/0807.4976v3

Mathematics

Algebraic Geometry

20 pages, one section was excluded and extended to another preprint

Scientific paper

We are interested in the algebraic and geometric structures of inner projections, the partial elimination ideal theory initiated by M. Green and geometric applications. By developing the elimination mapping cone theorem for infinitely generated graded modules and using the induced multiplicative maps, we get interesting relations between syzygies of the original variety and those of inner projection images. As results, first of all, we show that the inner projection from any smooth point of $X$ satisfies at least property $\textbf{N}_{2,p-1}$ for a projective reduced connected scheme $X$ of codimension $e$ satisfying property $\textbf{N}_{2,p},~p\ge1$. Further, we obtain the main theorem on `embedded linear syzygies' which is the natural projection-analogue of `restricting linear syzygies' in the linear section case (\cite{EGHP1}). We also obtain that the arithmetic depths of inner projections are equal to that of $X$ which is cut out by quadrics. This gives useful information on the rigidity of Betti diagrams. These uniform behaviors look unusual in a sense that property $\textbf{N}_{2,p}$ and the arithmetic depths of outer projections depend heavily on moving the center of projection in an ambient space (\cite {BS},\cite {KP},\cite{P}). Moreover, these properties have many interesting corollaries such as `rigidity theorem' on property $\textbf{N}_{2,p}, e-1\le p\le e$ as classifications and the sharp lower bound $e\cdot p -\frac{p(p-1)}{2}$ for the number of quadrics vanishing on $X$ with property $\textbf{N}_{2,p},~p\ge 1$. Examples and further questions are suggested.

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