Interpolation of Polynomials and Singular Curves: Segre and Veronese Varieties

We study an interpolation problem (objects singular at a prescribed finite set) for curves instead of hypersurfaces. We study singular curves in projective and multiprojective spaces. We construct curves that are singular (or with maximal dimension Zariski tangent space) at each point of a prescribed finite set, while the curves have low degree or low “complexity” (e.g., they are complete intersections of hypersurfaces of low degree). We discuss six open problems on the existence and structure of the base locus of the set of all hypersurfaces of a given degree and singular at a prescribed number of general points. The tools come from algebraic geometry, and some of the results are only existence ones or only asymptotic ones (but with as explicit as possible bounds). Some of the existence results are almost constructive, i.e., in our framework, random parameters should give a solution, or otherwise, take other random parameters.