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        Questions tagged [genus]

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        Biadjacency permanent upper bound in terms of genus of graph?

        Take $M$ to be biadjacency of a planar balanced bipartite on $2n$ vertices with genus $g$. Is it true for every $\epsilon\in(0,1)$ there is a $c_\epsilon>0$ such that $$\log\log(permanent(M))\leq\...
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        1answer
        52 views

        Maximum number of perfect matchings in a graph of genus $g$ balanced $k$-partite graph

        What is the maximum number of perfect matchings a genus $g$ balanced $k$-partite graph (number of vertices for each color in all possible $k$-colorings is within a difference of $1$) can have? I am ...
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        98 views

        Tangent Bundle of reducible genus one curves

        I need to know what can be said in general about the tangent bundle of reducible curves over complex numbers with arithmetic genus one, say $I_N$. As far as I know for any Simpson semistable torsion ...
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        94 views

        Max-min genus of a bipartite graph

        As usual, the genus of a graph with a prescribed circular ordering of the edges at each vertex is defined as the minimum genus of an orientable surface in which the graph can be drawn without edge ...
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        2answers
        155 views

        Number of non-equivalent graph embeddings

        Given a graph $G$, there is a minimal integer $g$ associated with it which captures the minimum genus a surface needs to have so that $G$ embeds in the surface without edge crossings. Is there a way ...
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        1answer
        109 views

        Is bipartite graph genus bound by $O(\mbox{max deg})$?

        We know that planar graphs have $O(1)$ degree. We know balanced (each color has same number of vertices) complete bipartite graphs have genus $O(n^2)$. If maximum and average degree are $O(n^\...
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        212 views

        Convex hull with genus information

        Are there convexity generalizations that admit genus information? For example in genus $1$ is there a way to think of this polyhedron as convex while this polyhedron as non-convex? Any two points can ...
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        1answer
        333 views

        Example of projective variety that do not contain algebraic curves of genus strictly greater to $1$

        Does exist a smooth, complex, projective variety $X$ of dimension $d\geq2$ such that $X$ does not contain smooth, complex, projective curves of wichever genus $g\geq2$? Answer by Bertie: No, it does ...
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        1answer
        103 views

        Non-orientable genus of union of graphs

        It is known that the orientable genus of union of two (disjoint) graphs is the sum of their genus. So, it is natural to ask What can be said about the non-orientable genus of union of two (disjoint) ...
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        1answer
        217 views

        $k$-planar graphs and genus

        Is there a simple function that connects $k$ in $k$-planar graphs and genus of such graphs? If there is no simple function is there any non-trivial upper and lower bound?
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        60 views

        Genus tradeoffs in bipartite graph

        Given $G$ as bipartite graph of genus $g(G)$ with number of vertices of each color being $N$ with $A$ as $N\times N$ biadjacency matrix. Denote $\bar{G}$ to bipartite graph of genus $g(\bar{G})$ of $N\...
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        484 views

        Hyperelliptic Curve [closed]

        Consider the curve given by $z^{2g-2}y^2=\displaystyle\prod_{i=1}^{2g}(x-a_iz)$. This is a hyperelliptic curve and has genus $g-1$. At the same time it is a curve defined by an equation of $d=2g$ and ...
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        1answer
        382 views

        S genus of quadratic forms

        Let $f$ be a non-degenerate quadratic form with integral coefficients. The genus of $f$ is the set of quadratic forms up to integral equivalence which are equivalent to $f$ over the $p$-adic integers $...
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        2answers
        505 views

        Rationality of curve does not depend on base change

        By a curve I mean an integral one-dimensional scheme of finite type over a spectrum of a field. Let $C$ be a curve over an arbitrary field $k$. It's probably a very well known fact, that $C$ is ...
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        1answer
        541 views

        Properties of subvarieties of a simple abelian variety

        Let $A$ be a simple abelian variety over a field $k$. (For simplicity, we assume char $k =0$.) Let $X$ be a smooth projective geometrically connected variety over $k$ of positive dimension. Suppose ...

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