78 research outputs found

    Lines on quartic surfaces

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    Rational quintics in the real plane

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    Relative enumerative invariants of real nodal del Pezzo surfaces

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    Relative enumerative invariants of real nodal del Pezzo surfaces

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    Asymptotically maximal families of hypersurfaces in toric varieties

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    A real algebraic variety is maximal (with respect to the Smith-Thom inequality) if the sum of the Betti numbers (with Z2\mathbb{Z}_2 coefficients) of the real part of the variety is equal to the sum of Betti numbers of its complex part. We prove that there exist polytopes that are not Newton polytopes of any maximal hypersurface in the corresponding toric variety. On the other hand we show that for any polytope Δ\Delta there are families of hypersurfaces with the Newton polytopes (λΔ)λ∈N(\lambda\Delta)_{\lambda \in \mathbb{N}} that are asymptotically maximal when λ\lambda tends to infinity. We also show that these results generalize to complete intersections.Comment: 18 pages, 1 figur

    On total reality of meromorphic functions

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    We show that if a meromorphic function of degree at most four on a real algebraic curve of an arbitrary genus has only real critical points then it is conjugate to a real meromorphic function after a suitable projective automorphism of the image.Comment: 13 page

    A Non-Algebraic Patchwork

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    Itenberg and Shustin's pseudoholomorphic curve patchworking is in principle more flexible than Viro's original algebraic one. It was natural to wonder if the former method allows one to construct non-algebraic objects. In this paper we construct the first examples of patchworked real pseudoholomorphic curves in ÎŁn\Sigma_n whose position with respect to the pencil of lines cannot be realised by any homologous real algebraic curve.Comment: 6 pages, 1 figur
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