Questions tagged [genus]

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6
votes
2answers
275 views

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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
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 ...
4
votes
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 ...
3
votes
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 $O(n^\alpha)$ ...
5
votes
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 ...
0
votes
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 ...
3
votes
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) ...
5
votes
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?
1
vote
0answers
65 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\...
3
votes
0answers
527 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 ...
3
votes
1answer
429 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 $...
0
votes
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 ...
4
votes
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 ...
5
votes
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 ...
3
votes
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 ...
2
votes
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 ...
2
votes
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'. (...
26
votes
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 ...