Generic projections, the equations defining projective varieties and Castelnuovo regularity (Q1579090)

Let \(X\subset\mathbb{P}^N\) be a reduced, irreducible projective variety of degree \(d\) and codimension \(e\geq 4\) and let \(\text{reg}(X)\) be the Castelnuovo-Mumford regularity of \(X\). If \(X\) is smooth of dimension 5 (resp. 6) it is shown that \(\text{reg}(X) \leq(d-e+1) +10\) (resp. \((d-e+1)+20)\). Moreover, for reduced, connected equidimensional locally Cohen-Macaulay curves or surfaces, upper bounds for \(\text{reg}(X)\) are given in terms of \(d,e\) and the arithmetic genus.