A study on some infinitely generated graded modules, projections, and applications

Abstract

We are interested in the algebraic and geometric structures of inner projections, partial elimination ideals and geometric applications. By developing the graded mapping cone construction and using the induced multiplicative maps on some infinitely generated graded modules, we obtain some (algebraic and geometric) properties of inner projections. As a result, for a projective reduced scheme $\It{X}$ of codimension e in $\mathbb{P}^{n}$ satisfying property $N_{2,p}$, $p\ge1$, we show that the inner projection from any smooth point of $\It{X}$ satisfies at least property $N_{2,p-1}$. 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 ([13]). This uniform behavior looks unusual in a sense that linear syzygies of outer projections heavily depend on moving the center of projection in an ambient space ([10],[25],[28]). Moreover, this property has many interesting corollaries such as ``rigidity theorem`` on property $N_{2,p}$, $e-1\le p \le e$ and the sharp lower bound $e \cdot p - \frac{p(p-1)}{2}$ of the number of quadrics vanishing on X satisfying property $N_{2,p},~ p \ge 1$. We also investigate the depth and Castelnuovo higher normality of inner projections which give useful information on the Betti diagram and the regularity conjecture due to Eisenbud-Goto and multisecants.