The Waldschmidt constant α(I) of a radical ideal I in the coordinate ring of P N measures (asymptotically) the degree of a hypersurface passing through the set defined by I in P N . Nagata's approach to the 14th Hilbert Problem was based on computing such constant for the set of points in 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 α(I).In the paper we deal with α(s), the Waldschmidt constant for s very general lines in P 3 . We prove that α(s) ≥ ⌊ √ 2s − 1⌋ holds for all s, whereas the much stronger bound α(s) ≥ ⌊ √ 2.5s⌋ holds for all s but s = 4, 7 and 10. We also provide an algorithm which gives even better bounds for α(s), very close to the known upper bounds, which are conjecturally equal to α(s) for s large enough.2010 Mathematics Subject Classification. 14N20 and 13F20 and 13P10 and 14C20.

In the present note we study configurations of codimension 2 flats in projective spaces and classify those with the smallest rate of growth of the initial sequence. Our work extends those of Bocci, Chiantini in P 2 and Janssen in P 3 to projective spaces of arbitrary dimension.

The Waldschmidt constant α(I) of a radical ideal I in the coordinate ring of P N measures (asymptotically) the degree of a hypersurface passing through the set defined by I in P N. Nagata's approach to the 14th Hilbert Problem was based on computing such constant for the set of points in 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 α(I). In the paper, we deal with α(s), the Waldschmidt constant for s very general lines in P 3. We prove that α(s) ≥ √ 2s − 1 holds for all s, whereas the much stronger bound α(s) ≥ √ 2.5s holds for all s but s = 4, 7 and 10. We also provide an algorithm which gives even better bounds for α(s), very close to the known upper bounds, which are conjecturally equal to α(s) for s large enough.

Let Zn = p0 + p1 + ··· + pn be a configuration of points in ℙ2, where all points pi except p0 lie on a line, and let I(Zn) be its corresponding homogeneous ideal in 𝕂 [ℙ2]. The resurgence and the Waldschmidt constant of I(Zn) in [5] have been computed. In this note, we compute these two invariants for the defining ideal of a fat point subscheme Zn,c = cp0 + p1 +··· + pn, i.e. the point p0 is considered with multiplicity c. Our strategy is similar to [5].