Questions tagged [genus]

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Integral solutions to f(x, y, z) = n where f is a cubic form

I'm looking to see if there is an integral solution to $f(x,y,z)=n$ where f is a cubic form. Especially interesting is the diagonal case: $$ ax^3+by^3+cz^3=n $$ for fixed integers $a,b,c,n$. If there ...
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2 votes
1 answer
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Where can I find a picture of the complete 9-map on a triple torus that corresponds to Heffter’s table?

What I’m looking for is the analogue of Figure 5 in the paper by Saul Stahl, The Othe Map Coloring Theorem, Mathematics Magazine 1985, which is a complete 8-map $M_8$ on the double torus $S_2$ that ...
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1 vote
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Is there an effective genus theory for indefinite quadratic forms?

For positive definite quadratic forms, there is a way to check if two forms are isomorphic by arithmetic equivalence over $GL_n(Z)$ by computing configurations of vector up to some norm and then ...
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1 vote
0 answers
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Degree and genus of projected curve

Let $C\subset\mathbb{P}^n$ be a normal curve over an algebraically closed field of characteristic $0$. Assume that $C$ is not contained in any hyperplane. We may assume that $P=[0:\cdots:0:1]$ is on $...
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  • 305
6 votes
2 answers
417 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 ...
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1 vote
1 answer
80 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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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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1 vote
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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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4 votes
2 answers
209 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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  • 13k
3 votes
1 answer
143 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 $O(n^\alpha)$ ...
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5 votes
2 answers
282 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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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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3 votes
1 answer
141 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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5 votes
1 answer
247 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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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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3 votes
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554 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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3 votes
1 answer
498 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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2 answers
572 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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4 votes
1 answer
694 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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5 votes
3 answers
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 ...
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3 votes
1 answer
262 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 ...
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3 votes
3 answers
1k 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 ...
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2 votes
0 answers
687 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'. (...
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26 votes
7 answers
4k 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 ...
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