Segre class computations and practical applications with Martin Helmer

To appear in Mathematics of Computation (arXiv)

Let \(X \subset Y\) be closed (possibly singular) subschemes of a smooth projective toric variety \(T\). We show how to compute the Segre class s(X,Y) as a class in the Chow group of \(T\). Building on this, we give effective methods to compute intersection products in projective varieties, to determine algebraic multiplicity without working in local rings, and to test pairwise containment of subvarieties of \(T\). Our methods may be implemented without using Groebner bases; in particular any algorithm to compute the number of solutions of a zero-dimensional polynomial system may be used.

The geometry of SDP-exactness in quadratic optimization with Diego Cifuentes and Bernd Sturmfels

Submitted (arXiv)

Consider the problem of minimizing a quadratic objective subject to quadratic equations. We study the semialgebraic region of objective functions for which this problem is solved by its semidefinite relaxation. For the Euclidean distance problem, this is a bundle of spectrahedral shadows surrounding the given variety. We characterize the algebraic boundary of this region and we derive a formula for its degree.

Computing images of polynomial maps with Mateusz Michałek and Emre Can Sertöz

Submitted (arXiv)

The image of a polynomial map is a constructible set. While computing its closure is standard in computer algebra systems, a procedure for computing the constructible set itself is not. We provide a new algorithm, based on algebro-geometric techniques, addressing this problem. We also apply these methods to answer a question of W. Hackbusch on the non-closedness of site-independent cyclic matrix product states for infinitely many parameters.

The Euclidean distance degree of smooth complex projective varieties with Paolo Aluffi

To appear in Algebra and Number Theory (arXiv)

We obtain several formulas for the Euclidean distance degree (ED degree) of an arbitrary nonsingular variety in projective space: in terms of Chern and Segre classes, Milnor classes, Chern-Schwartz-MacPherson classes, and an extremely simple formula equating the Euclidean distance degree of \(X\) with the Euler characteristic of an open subset of \(X\).

The Chern-Mather class of the multiview variety with Daniel Lowengrub

Communications in Algebra, 46:6, 2488-2499 (arXiv)

The multiview variety associated to a collection of $N$ cameras records which sequences of image points in $\mathbb{P}^{2N}$ can be obtained by taking pictures of a given world point $x \in \mathbb{P}^3$ with the cameras. In order to reconstruct a scene from its picture under the different cameras it is important to be able to find the critical points of the function which measures the distance between a general point $u \in \mathbb{P}^{2N}$ and the multiview variety. In this paper we calculate a specific degree 3 polynomial that computes the number of critical points as a function of $N$. In order to do this, we construct a resolution of the multiview variety, and use it to compute its Chern-Mather class.

Tritangent planes to space sextics: the algebraic and tropical stories with Yoav Len

In: Combinatorial Algebraic Geometry. Fields Institute Communications, vol 80. Springer (arXiv)

We discuss the classical problem of counting planes tangent to general canonical sextic curves at three points. We determine the number of real tritangents when such a curve is real. We then revisit a curve constructed by Emch with the greatest known number of real tritangents, and conversely construct a curve with very few real tritangents. Using recent results on the relation between algebraic and tropical theta characteristics, we show that the tropicalization of a canonical sextic curve has 15 tritangent planes.

Equations and tropicalization of Enriques surfaces with Barbara Bolognese and Joachim Jelisiejew

In: Combinatorial Algebraic Geometry. Fields Institute Communications, vol 80. Springer (arXiv)

In this article we explicitly compute equations of an Enriques surface via the involution on a K3 surface. We also discuss its tropicalization and compute the tropical homology, thus recovering a special case of the result of Itenberg-Katzarkov-Mikhalkin-Zharkov, and establish a connection between the dimension of the tropical homology groups and the Hodge numbers of the corresponding algebraic Enriques surface.

Computing Segre classes in arbitrary projective varieties

Journal of Symbolic Computation, Volume 82, 2017, 26-37 (arXiv)

We give an algorithm for computing Segre classes of subschemes of arbitrary projective varieties by computing degrees of a sequence of linear projections. Based on the fact that Segre classes of projective varieties commute with intersections by general effective Cartier divisors, we can compile a system of linear equations which determine the coefficients for the Segre class pushed forward to projective space. The algorithm presented here comes after several others which solve the problem in special cases, where the ambient variety is for instance projective space; to our knowledge, this is the first algorithm to be able to compute Segre classes in projective varieties with arbitrary singularities.

Classification of the monomial Cremona transformations of the plane


We classify all monomial planar Cremona maps by multidegree using recent methods developed by Aluffi (arXiv:1308.4152). Following the main result, we prove several more properties of the set of these maps, and also extend the results to the more general "r.c. monomial" maps.

Monomial principalization in the singular setting

Journal of Commutative Algebra 7 (3), 353-362 (arXiv)

We generalize an algorithm by Goward for principalization of monomial ideals in nonsingular varieties to work on any scheme of finite type over a field. The normal crossings condition considered by Goward is weakened to the condition that components of the generating divisors meet as complete intersections. This leads to a substantial generalization of the notion of monomial scheme; we call the resulting schemes "regular crossings (r.c.) monomial." We prove that r.c. monomial subschemes in arbitrarily singular varieties can be principalized by a sequence of blow-ups at codimension 2 r.c. monomial centers.