Lower Bounds for Waldschmidt Constants of Generic Lines in {mathbb {P}}^3 and a Chudnovsky-Type Theorem

Abstract

The Waldschmidt constant {{,mathrm{{widehat{alpha }}},}}(I) of a radical ideal I in the coordinate ring of {mathbb {P}}^N measures (asymptotically) the degree of a hypersurface passing through the set defined by I in {mathbb {P}}^N. Nagata’s approach to the 14th Hilbert Problem was based on computing such constant for the set of points in {mathbb {P}}^2. Since then, these constants drew much attention, but still there are no methods to compute them (except for trivial cases). Therefore, the research focuses on looking for accurate bounds for {{,mathrm{{widehat{alpha }}},}}(I). In the paper, we deal with {{,mathrm{{widehat{alpha }}},}}(s), the Waldschmidt constant for s very general lines in {mathbb {P}}^3. We prove that {{,mathrm{{widehat{alpha }}},}}(s) ge lfloor sqrt{2s-1}rfloor holds for all s, whereas the much stronger bound {{,mathrm{{widehat{alpha }}},}}(s) ge lfloor sqrt{2.5 s}rfloor holds for all s but s=4, 7 and 10. We also provide an algorithm which gives even better bounds for {{,mathrm{{widehat{alpha }}},}}(s), very close to the known upper bounds, which are conjecturally equal to {{,mathrm{{widehat{alpha }}},}}(s) for s large enough.