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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.
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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 geometry of complete intersections of quadrics. We obtain decompositions of these monodromy representations into irreducibles and compute the Fourier transforms of the IC complexes associated to these irreducible representations. The results of the paper refine (part of) the Springer correspondece for the split symmetric pair (SL(N),SO(N)) in [CVX2].
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 polynomial equations, also in the space of complete quadrics, and we solve them using certified numerical methods. Our aim is to show that Schuberts problems are fully real.
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 conditions. In fact we show it is equivalent to the discretized sum-product problem, thus the $epsilon_0$ follows from Bourgains celebrated result. Then we prove a couple of new results on radial projections. We also discussion about the dependence of $epsilon_0$ and make a new conjecture. A tube condition on Frostman measures, after careful refinement, is also given.
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 fundamental form $II_{X,x}$ at a point $x in X$ is a pencil of quadrics on $T_x(X)$, defining a rational map $mu^X$ from $X$ to a suitable moduli space of pencils of quadrics on a complex vector space of dimension $n$. The question raised by Griffiths and Harris was whether the image of $mu^X$ determines $X$. We study this question when $X^n subset {mathbb P}^{n+2}$ is a nonsingular intersection of two quadric hypersurfaces of dimension $n >4$. In this case, the second fundamental form $II_{X,x}$ at a general point $x in X$ is a nonsingular pencil of quadrics. Firstly, we prove that the moduli map $mu^X$ is dominant over the moduli of nonsingular pencils of quadrics. This gives a negative answer to Griffiths-Harriss question. To remedy the situation, we consider a refined version $widetildemu^X$ of the moduli map $mu^X$, which takes into account the infinitesimal information of $mu^X$. Our main result is an affirmative answer in terms of the refined moduli map: we prove that the image of $widetildemu^X$ determines $X$, among nonsingular intersections of two quadrics.
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 $overline{mathcal{M}}_{12,6},overline{mathcal{M}}_{12,7},overline{mathcal{M}}_{13,4}$ and $overline{mathcal{M}}_{14,3}$ are uniruled. We provide upper bounds for the Kodaira dimension of $overline{mathcal{M}}_{12,8}$ and $overline{mathcal{M}}_{16}$. We also show that the moduli of $(4g+5)$-pointed hyperelliptic curves $mathcal{H}_{g,4g+5}$ is uniruled. Together with a recent result of Schwarz, this concludes the Kodaira classification for moduli of pointed hyperelliptic curves.
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