Suppose that X is a compact complex manifold of dimension 8 embedded in the complex projective space. Could it happen that X is tangent to an 8-dimensional projective subspace along a curve? After reviewing this classical problem (and its classical solution), I will discuss logarithmic generalizations. This will quickly take us to dangerous tropical waters with non-archimedean amoebas lurking in the darkness. I will then discuss applications (joint with Bernd Sturmfels) to solving elimination theory problems without the hassle of Groebner bases. The software implementing these magic algorithms was written up by Josephine Yu.

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