Research Projects

This project, titled "Projective Closure of Affine Monomial Curves" is aimed at understanding the geometric properties of the projective closure of affine monomial curves. To be precise, we want to see which geometric properties of affine monomial curves are preserved under the projective closure operation. For example, it is well known that Cohen-Macaulayness is not preserved under this operation. The techniques that will be used to carry out this study are mostly homological and combinatorial. This project proposal is based on a thorough review work done on the basis of many research articles on various properties of numerical semigroups. Also, our PI's own experience of working in this area have provided us with the necessary ideas to work on this problem. The PI has embarked on this project after carrying out initial research work on three very important classes of monomial curves, viz., the Bresinsky curve, the Arslan curve and the Backelin curve (described in detail in Other Technical Details). It has been observed that the commonality of certain behaviour among the affine curves does not ensure any commonality in behaviour of their projective closures. This motivated us to ask the following question: Which geometric properties of affine monomial curves are preserved under the projective closure operation?