# Questions tagged [genus]

The tag has no usage guidance.

31 questions
Filter by
Sorted by
Tagged with
279 views

### If a graph embedded on a surface is divided by a curve into a right and left that do not intersect can it be embedded on a surface of smaller genus?

Suppose we have a graph $G$ embedded on a (smooth, orientable etc) surface $Q$. Suppose there is a cycle $C$ of $G$ such that $C$ does not separate our surface $Q$ into two connected regions and ...
• 161
311 views

### Existence of space curves of given genus and degree

In Hartshorne's Algebraic Geometry Chapter IV, Section 6, he summarizes known results on the existence of smooth space curves of degree $d$ and genus $g$ for $g\le 12$ and $d \le 10$. He shows the ...
• 489
85 views

I am trying to understand the genus of a lattice from Conway and Sloane textbook. They said two quadratic forms $Q_1$ and $Q_2$ lie in the same genus if they are equivalent over $\mathbb{R}$ and over ...
• 1
43 views

### How to calculate the genus of a 3 dimensional curve (f(kx,ky,kz)) using the Newton polyhedra?

Given a plane affine curve $\sum_{i,j}a_{i,j}k_x^ik_y^j = 0$, its genus can be calculated as the number of integral points of the interior of the convex hull of $\{(i,j) \mid a_{i,j} \neq 0\}$. How to ...
2k views

### Is there an algorithm for the genus of a knot?

A Seifert surface of a knot is a surface whose boundary is the knot. The genus of a knot is the minimal genus among all the Seifert surfaces of the knot. My question is, is any algorithm known to ...
• 4,489
94 views

### Primitive representation of integers by some form on the genus of a quadratic form

Some time ago, I asked a question about equidistribution on a paper of Duke and Schulze-Pillot that was usefully answered. However, on the answer there was a statement that was unimportant for me back ...
• 303
1 vote
137 views

### Derivation for genus-degree formula from algebraic functions field theory

This is a copy of my question from math.stackexchange: https://math.stackexchange.com/questions/4517289/derivation-for-genus-degree-formula-from-algebraic-functions-field-theory. I didn't get any ...
119 views

### 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 ...
• 8,904
172 views

### 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 ...
1 vote
78 views

### 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 ...
1 vote
175 views

• 13.5k
586 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 ...
532 views

• 4,741
265 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 ...
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 ...
• 171
733 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'. (...
• 468