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Two-dimensional linear spaces of symmetric matrices are classified by Segre symbols. After reviewing known facts from linear algebra and projective geometry, we address new questions motivated by algebraic statistics and optimization. We compute the reciprocal curve and the maximum likelihood degrees, and we study strata of pencils in the Grassmannian.
In this paper we introduce a certain class of families of Hessenberg varieties arising from Springer theory for symmetric spaces. We study the geometry of those Hessenberg varieties and investigate their monodromy representations in detail using the
We examine quadratic surfaces in 3-space that are tangent to nine given figures. These figures can be points, lines, planes or quadrics. The numbers of tangent quadrics were determined by Hermann Schubert in 1879. We study the associated systems of p
In this paper we prove the following results in the plane. They are related to each other, while each of them has its own interest. First we obtain an $epsilon_0$-increment on intersection between pencils of $delta$-tubes, under non-concentration c
In [GH], Griffiths and Harris asked whether a projective complex submanifold of codimension two is determined by the moduli of its second fundamental forms. More precisely, given a nonsingular subvariety $X^n subset {mathbb P}^{n+2}$, the second fund
We study smoothing of pencils of curves on surfaces with normal crossings. As a consequence we show that the canonical divisor of $overline{mathcal{M}}_{g,n}$ is not pseudo-effective in some range, implying that $overline{mathcal{M}}_{12,6},overline{