Recent developments and open problems in linear series

Symbolic powers of planar point configurations

“…i) Assume to the contrary that there is a divisor C ′ 1 of degree deg(C ′ 1 ) = β ′ 1 < β 1 with vanishing vector m (1) . Then C ′ 1 + C 2 has vanishing vector m (1) + m (2) = m and degree…”

“…Note that it might happen that some entries in n (j) are zero or even that both vectors n (1) and n (2) are zero.…”

“…The unsolved up to date conjectures of Nagata and Segre-Harbourne-Gimigliano-Hirschowitz are prominent examples of such problems. We refer to [2] for precise formulation, motivation and more open problems. These problems can be formulated in the language of symbolic powers.…”

“…These problems can be formulated in the language of symbolic powers. For example, the Nagata conjecture predicts that if Z is a set of r 10 general points in the projective plane P 2 and I is the saturated ideal of Z, then the initial degree α(I (m) ) of the m-th symbolic power of I is bounded from below by m √ r. In the present paper we look at a fixed finite set Z of arbitrary points in the projective plane P 2 with defining saturated ideal I = I(Z) and we consider the increasing sequence α(I) < α(I (2) ) < α(I (3) ) < . .…”

Negative curves on algebraic surfaces

“…In addition, we point out that bounding the genus of a set of curves on a given complex surface of non-negative Kodaira dimension immediately leads to a lower bound on their self-intersections. (The latter result was first proved in [5]. )…”