### Abstract or Additional Information

Open problems in algebraic geometry, commutative algebra and combinatorics come together when studying the question "How singular can a plane algebraic curve be?" I will introduce a new concept, Hadean numbers, for exploring this problem and relate it to open problems concerning line arrangements and symbolic powers of ideals. Specifically: What arrangements of lines in the plane have no simple crossings? For which finite sets of points in the projective plane is there a homogeneous polynomial vanishing to order at least three at each point but which is not a sum of pairwise products of homogeneous polynomials vanishing at all of the points? (I.e, which point sets have an ideal whose symbolic cube is not contained in its square?) I will motivate these questions and give examples and some recent results.