arXiv subjects
Publications and source records attributed to Elena Drozd.
Classification of curves in a projective space occupies minds of many mathematicians. First step in doing so is classification of curves on a given surface. This brings us to consideration of the nonsingular Del Pezzo Surface in $P^4_k.$ We describe conditions for a divisor to be very ample and condition for a divisor class to contain an irreducible curve.
Let C be an arithmetically Cohen-Macaulay curve of arithmetic genus 4. We prove that the family of such curves of degree 8 in $P^4_k$ is irreducible.
We prove that on a nonsingular quadric hypersurface Q in $P^4_k$ even CI liaison classes of ACM curves are in bijective correspondence with the stable equivalence classes (up to shift in degree) of maximal Cohen-Macaulay graded modules over the coordinate ring R of Q, which in turn, are in bijective correspondence with stable equivalence classes (up to shift in degree) of ACM sheaves on Q . In the situation of a nonsingular quadric hypersurface Q in $P^4_k$ work of Knorrer shows that there is a unique nonfree indecomposable MCM module over R. We also describe the ACM sheaf corresponding to this MCM module and give its cohomology table and its Hilbert polynomial.
We study Gorenstein liaison of codimension two subschemes of an arithmetically Gorenstein scheme X. Our main result is a criterion for two such subschemes to be in the same Gorenstein liaison class, in terms of the category of ACM sheaves on X. As a consequence we obtain a criterion for X to have the property that every codimension 2 arithmetically Cohen-Macaulay subscheme is in the Gorenstein liaison class of a complete intersection. Using these tools we prove that every arithmetically Gorenstein subscheme of $\mathbb{P}^n$ is in the Gorenstein liaison class of a complete intesection and we are able to characterize the Gorenstein liaison classes of curves on a nonsingular quadric threefold in $\mathbb{P}^4$.