We explore the maximum likelihood degree of a homogeneous polynomial F on a projective variety X, MLDF(X), which generalizes the concept of Gaussian maximum likelihood degree. We show that MLDF(X) is equal to the count of critical points of a rational function on X and give different geometric characterizations of it via topological Euler characteristic, dual varieties, and Chern classes.

Let X C P-d be an m-dimensional variety in d-dimensional complex projective space. Let k be a positive integer such that the combinatorial number ( m + k k ) is smaller than or equal to d . Consider the following interpolak tion problem: does there exist a variety Y C P-d of dimension strictly smaller than ( m + k k) , with X C Y , such that the tangent space to Y at a point p is an element of X is k equal to the k th osculating space to X at p , for almost all points p is an element of X ? In this paper we consider this question in the toric setting. We prove that if X is toric, then there is a unique toric variety Y solving the above interpolation problem. We identify Y in the general case and we explicitly compute some of its invariants when X is a toric curve.

We introduce numerical algebraic geometry methods for computing lower bounds on the reach, local feature size, and weak feature size of the real part of an equidimensional and smooth algebraic variety using the variety's defining polynomials as input. For the weak feature size, we also show that nonquadratic complete intersections generically have finitely many geometric bottlenecks, and we describe how to compute the weak feature size directly rather than a lower bound in this case. In all other cases, we describe additional computations that can be used to determine feature size values rather than lower bounds.

Classical work by Salmon and Bromwich classified singular intersections of two quadric surfaces. The basic idea of these results was already pursued by Cayley in connection with tangent intersections of conics in the plane and used by Schafli for the study of hyperde-terminants. More recently, the problem has been revisited with similar tools in the context of geometric modeling and a generalization to the case of two higher dimensional quadric hyper -surfaces was given by Ottaviani. We propose and study a generalization of this question for systems of Laurent polynomials with support on a fixed point configuration. In the non-defective case, the closure of the locus of coefficients giving a non-degenerate multiple root of the system is defined by a polynomial called the mixed discriminant. We define a related polynomial called the multivariate iterated discriminant, generalizing the classical Schafli method for hyperdeterminants. This iterated discriminant is easier to compute and we prove that it is always divisible by the mixed discriminant. We show that tangent intersections can be computed via iteration if and only if the singular locus of a corresponding dual variety has sufficiently high codimension. We also study when point configurations corresponding to Segre-Veronese varieties and to the lattice points of planar smooth polygons, have their iterated discriminant equal to their mixed discriminant.

In this paper we consider the problem of computing Seshadri constants at a general point on a smooth polarized toric surface. We consider the case when the degree of jet separation is small or the core of the associated polygon is a line segment. Our main result is that under these hypothesis the Seshadri constant at the general point can often be determined in terms of easily computable invariants of the surfaces at hand. When the core of the associated polygon is a point we show that the surface can be constructed via consecutive equivariant blow-ups of either P2 or P1× P1. 

The Losev–Manin moduli space parametrizes pointed chains of projective lines. In this paper we study a possible generalization to families of pointed degenerate toric varieties. Geometric properties of these families, such as flatness and reducedness of the fibers, are explored via a combinatorial characterization. We show that such families are described by a specific type of polytope fibration which generalizes the twisted Cayley sums, originally introduced to characterize elementary extremal contractions of fiber type associated to projective Q-factorial toric varieties with positive dual defect. The case of a one-dimensional simplex can be viewed as an alternative construction of the permutohedra.

In this paper we present an efficient algorithm to produce a provably dense sample of a smooth compact affine variety. The procedure is partly based on computing bottlenecks of the variety. Using geometric information such as the bottlenecks and the local reach we also provide bounds on the density of the sample needed in order to guarantee that the homology of the variety can be recovered from the sample. An implementation of the algorithm is provided together with numerical experiments and a computational comparison to the algorithm by Dufresne et al. [Sampling real algebraic varieties for topological data analysis, arXiv:1802.07716, 2018].

A bottleneck of a smooth algebraic variety X subset of C-n is a pair (x, y) of distinct points x, y is an element of X such that the Euclidean normal spaces at x and y contain the line spanned by x and y. The narrowness of bottlenecks is a fundamental complexity measure in the algebraic geometry of data. In this paper we study the number of bottlenecks of affine and projective varieties, which we call the bottleneck degree. The bottleneck degree is a measure of the complexity of computing all bottlenecks of an algebraic variety, using, for example, numerical homotopy methods. We show that the bottleneck degree is a function of classical invariants such as Chern classes and polar classes. We give the formula explicitly in low dimension and provide an algorithm to compute it in the general case.

Generalizing the famous Bernstein-Kushnirenko theorem, Khovanskii proved in 1978 a combinatorial formula for the arithmetic genus of the compactification of a generic complete intersection associated to a family of lattice polytopes. Recently, an analogous combinatorial formula, called the discrete mixed volume, was introduced by Bihan and shown to be nonnegative. By making a footnote of Khovanskii in his paper explicit, we interpret this invariant as the (motivic) arithmetic genus of the non-compact generic complete intersection associated to the family of lattice polytopes.

The Klein and Wiman configurations are highly symmetric configurations of lines in the projective plane arising from complex reflection groups. One noteworthy property of these configurations is that all the singularities of the configuration have multiplicity at least 3. In this paper we study the surface X obtained by blowing up P-2 in the singular points of one of these line configurations. We study invariant curves on X in detail, with a particular emphasis on curves of negative self-intersection. We use the representation theory of the stabilizers of the singular points to discover several invariant curves of negative self-intersection on X, and use these curves to study Nagata-type questions for linear series on X. The homogeneous ideal I of the collection of points in the configuration is an example of an ideal where the symbolic cube of the ideal is not contained in the square of the ideal; ideals with this property are seemingly quite rare. The resurgence and asymptotic resurgence are invariants which were introduced to measure such failures of containment. We use our knowledge of negative curves on X to compute the resurgence of I exactly. We also compute the asymptotic resurgence and Waldschmidt constant exactly in the case of the Wiman configuration of lines, and provide estimates on both for the Klein configuration.