# Questions tagged [genus]

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2answers
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### Simple proof that the arithmetic genus is non-negative

I take an irreducible and reduced closed curve $C\subseteq \mathbb{P}^n$, defined over an algebraically closed field $k$ and define the arithmetic genus $p_a(C)$ as the integer such that the Hilbert ...
1answer
70 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 ...
0answers
106 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 ...
0answers
127 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 ...
2answers
187 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 ...
1answer
134 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 $O(n^\alpha)$ ...
2answers
241 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 ...
1answer
372 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 ...
1answer
129 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) ...
1answer
238 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?
0answers
65 views

2answers
548 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 ...
1answer
630 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 ...
3answers
2k views

### Minimal genus, adjunction inequality

Let's consider closed simply-connected 4-manifold $M$ and some $a\in H^2(M)$. It is very natural question to estimate minimal $g$ that $a$ can be presented as embedded surface of genus $g$. As I know ...
1answer
260 views

### smooth curves of genus 3 over an algebraic closed field

Is there a way to "easily" compute and describe the Moduli space of smooth curves of genus 3 without stacks and stable curves? In Hartshorne's Algebraic Geometry there is a nice excercise (Chapter ...
3answers
868 views

### Genus and Spinor genus of a lattice

Hi, I'm looking for a motivation for the names genus and spinor genus of a lattice (and spinor norm of an isometry). Is there any relation between the genus of a lattice and the genus of an algebraic ...
0answers
651 views

### Riemann-Roch for ARBITRARY Function Fields

I know that on an algebraic function field in one variable over any base field, there is a good divisor theory for it and a Riemann-Roch Theorem; in particular, there is a 'good' notion of 'genus'. (...
7answers
3k views

### How do you see the genus of a curve, just looking at its function field?

Yuhao asked in the 20-questions seminar: The genus of a curve is a birational invariant; the function field of a curve determines it up to birational equivelance. How do you see the genus directly ...