Curvilinear base points, local complete intersection and Koszul syzygies in biprojective spaces

Let be a bigraded ideal in the bigraded polynomial ring . Assume that has codimension 2. Then is a finite set of points. We prove that if is a local complete intersection, then any syzygy of the vanishing at , and in a certain degree range, is in the module of Koszul syzygies. This is an analog of a recent result of Cox and Schenck (2003).

     

Moduli of Polarized Calabi–Yau Pairs

We prove that the irreducible components of the moduli space of polarized Calabi–Yau pairs are projective.     

Ion‐Pair SN2 Substitution: Activation Strain Analyses of Counter‐Ion and Solvent Effects

The ion‐pair S_N2 reactions of model systems M_nF^n?1+CH₃Cl (M⁺=Li⁺, Na⁺, K⁺, and MgCl⁺; n=0, 1) have been quantum chemically explored by using DFT at the OLYP/6‐31++G(d,p) level. The purpose of this study is threefold: 1) to elucidate how the counterion M⁺ modifies ion‐pair S_N2 reactivity relative to the parent reaction F^?+CH₃Cl; 2) to determine how this influences stereochemical competition between the backside and frontside attacks; and 3) to examine the effect of solvation on these ion‐pair S_N2 pathways. Trends in reactivity are analyzed and explained by using the activation strain model (ASM) of chemical reactivity. The ASM has been extended to treat reactivity in solution. These findings contribute to a more rational design of tailor‐made substitution reactions.