Algebraic invariants of projections of varieties and partial elimination ideals (Q2049407)

This paper is concerned with comparing the depth and the Castelnuovo-Mumford regularity of a non-degenerate closed subscheme \(X\) of projective space with that of its (inner, outer or arbitrary) projections, motivated by the regularity conjecture of Eisenbud and Goto. The authors give explicit bounds for the depth and the regularity of the homogeneous ideal defining \(X\) in terms of the respective invariants of its partial elimination ideals. Recall that M. Green introduced the partial elimination ideals of a homogeneous saturated ideal \(I\) in the polynomial ring \(R=k[x_0,\ldots,x_N]\), denoted by \(K_i(I)\), which form an ascending chain of ideals of the polynomial ring \(S=k[x_1,\ldots,x_N]\); the chain starts with the elimination ideal \(J=K_0(I)\) of \(I\) with respect to the variable \(x_0\) and eventually stabilizes at some index, denoted by \(s\). The authors prove in Theorem 3.1 and Theorem 3.4 that the depths of the closed subschemes defined by \(I\) and its partial elimination ideals are related by the inequalities \[ \mathrm{ depth}_R\frac{R}{I} \geq \min_{1\leq i\leq s-1} \left\{ \mathrm{ depth}_S\frac{S}{J}, \mathrm{ depth}_S\frac{S}{K_i(I)},\;\mathrm{ depth}_S\frac{S}{K_s(I)}+1 \right\} \] \[ \mathrm{ depth}_S\frac{S}{J} \geq \min_{1\leq i\leq s} \left\{ \mathrm{ depth}_R\frac{R}{I},\; \mathrm{ depth}_S\frac{S}{K_i(I)}+1 \right\}, \] where the second inequality holds provided \(J\) is generated in degree strictly greater than \(\mathrm{ reg}_S K_s(I)\). Analogous results concerning the regularities of \(I\) and \(J\) are given in Theorem 4.1 and Theorem 4.6 \[ \mathrm{ reg}_R I \leq \max_{1\leq i\leq s} \left\{ \mathrm{ reg}_SJ, \mathrm{ reg}_S K_i(I)+1 \right\} \] \[ \mathrm{ reg}_S J \leq \max_{1\leq i\leq s-1} \left\{ \mathrm{ reg}_RI,\; \mathrm{ reg}_S K_i(I)+i+1,\mathrm{ reg}_S K_s(I)+s-1 \right\}. \] The authors proceed with improving their results, by obtaining more precise statements in the case of inner projections of irreducible projective varieties (Theorem 5.1). The major improvement is that it suffices to consider the partial elimination ideals \(K_i(I)\) for \(i\) at least equal to the degree \(s'\) of the inner projection map, since \(K_i(I)=K_0(I)\) for \(i<s'\) (Lemma 2.9). An important application of the above is the generalization of the first author's results, that the depth and the regularity of quadratic schemes of codimension at least 2 equal those of its projections from smooth inner points. In particular the authors show in Theorem 6.1 that the same result holds given that the chain of partial elimination ideals stabilizes at \(s=1\), \(S/K_1(I)\) is Cohen-Macaulay, and \(I\) is generated in degrees at least equal to the regularity of \(K_1(I)\). Finally, they strengthen the above even further in the irreducible case (Theorem 6.2), by dropping the assumption that \(s=1\).