The computation of the dimension of linear systems of plane curves through a bunch of given multiple points is one of the most classic issues in algebraic geometry. In general, it is still an open problem to understand when the points fail to impose independent conditions. Despite many partial results, a complete solution is not known, even if the fixed points are in general position. The answer in the case of general points in the projective plane is predicted by the famous Segre-Harbourne-Gimigliano-Hirschowitz conjecture. When we consider fixed points in special position, even more interesting situations may occur. Recently, Di Gennaro, Ilardi and Vallès discovered a special configuration Z of nine points with a remarkable property: A general triple point always fails to impose independent conditions on the ideal of Z in degree four. The peculiar structure and properties of this kind of unexpected curves were studied by Cook II, Harbourne, Migliore and Nagel. By using both explicit geometric constructions and more abstract algebraic arguments, we classify low-degree unexpected curves. In particular, we prove that the aforementioned configuration Z is the unique one giving rise to an unexpected quartic.

For any (real) algebraic variety X in a Euclidean space V endowed with a nondegenerate quadratic form q, we introduce a polynomial EDpoly X,u (t 2 ) which, for any u ∈ V , has among its roots the distance from u to X. The degree of EDpoly X,u is the Euclidean Distance degree of X. We prove a duality property when X is a projective variety, namely EDpoly X,u (t 2 ) = EDpoly X ∨ ,u (q(u) − t 2 ) where X ∨ is the dual variety of X. When X is transversal to the isotropic quadric Q, we prove that the ED polynomial of X is monic and the zero locus of its lower term is X ∪ (X ∨ ∩ Q) ∨ .

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A real binary tensor consists of 2 d real entries arranged into hypercube format 2 ×d . For d = 2, a real binary tensor is a 2 × 2 matrix with two singular values. Their product is the determinant. We generalize this formula for any d ≥ 2. Given a partition µ ⊢ d and a µ-symmetric real binary tensor t, we study the distance function from t to the variety X µ,R of µ-symmetric real binary tensors of rank one. The study of the local minima of this function is related to the computation of the singular values of t. Denoting with Xµ the complexification of X µ,R , the Euclidean Distance polynomial EDpoly X ∨ µ ,t (ǫ 2 ) of the dual variety of Xµ at t has among its roots the singular values of t. On one hand, the lowest coefficient of EDpoly X ∨ µ ,t (ǫ 2 ) is the square of the µ-discriminant of t times a product of sum of squares polynomials. On the other hand, we describe the variety of µ-symmetric binary tensors that do not admit the maximum number of singular values, counted with multiplicity. Finally, we compute symbolically all the coefficients of EDpoly X ∨ µ ,t (ǫ 2 ) for tensors of format 2 × 2 × 2.

We study an optimization problem with the feasible set being a real algebraic variety X and whose parametric objective function fu is gradient-solvable with respect to the parametric data u. This class of problems includes Euclidean distance optimization as well as maximum likelihood optimization. For these particular optimization problems, a prominent role is played by the ED and ML correspondence, respectively. To our generalized optimization problem we attach an optimization correspondence and show that it is equidimensional. This leads to the notion of algebraic degree of optimization on X. We apply these results to p-norm optimization, and define the p-norm distance degree of X, which coincides with the ED degree of X for p = 2. Finally, we derive a formula for the p-norm distance degree of X as a weighted sum of the polar classes of X under suitable transversality conditions.