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

(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

(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

Submitted (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

Submitted (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

To appear in the Journal of Symbolic Computation (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

(arXiv)

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

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.