# Questions tagged [genus]

The genus tag has no usage guidance.

20
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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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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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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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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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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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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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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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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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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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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 ...