On the dual Hesse arrangement

In the present note we investigate to which extent the configuration of 9 lines intersecting in triples in 12 points is determined by these incidences. We show that up to a projective automorphism there is exactly one such configuration in characteristic zero and one in characteristic 3. We pin down the geometric difference between these two realizations

Realizability of some Böröoczky arrangements over the rational numbers

In this paper, we study the parameter spaces for Böröoczky arrangements Bn of n lines, where n < 12. We prove that up to n = 12, there exist only one arrangement nonrealizable over the rational numbers, that is B11

Containment problem and combinatorics

In this note, we consider two configurations of twelve lines with nineteen triple points (i.e. points where three lines meet). Both of them have the same arrangemental combinatorial features, which means that in both configurations nine of twelve lines have five triple points and one double point, and the remaining three lines have four triple points and three double points. Taking the ideal of the triple points of these configurations we discover that, quite surprisingly, for one of the configurations the containment I(3)⊂I2 holds, while for the other it does not. Hence, for ideals of points defined by arrangements of lines, the (non)containment of a symbolic power in an ordinary power is not determined alone by arrangemental combinatorial features of the configuration. Moreover, for the configuration with the non-containment I(3)⊈I2, we examine its parameter space, which turns out to be a rational curve, and thus establish the existence of a rational non-containment configuration of points. Such rational examples are very rare