# A Lie algebra method for rational parametrization of Severi-Brauer surfaces

@article{Graaf2005ALA, title={A Lie algebra method for rational parametrization of Severi-Brauer surfaces}, author={Willem A. de Graaf and Michael C. Harrison and Jana P{\'i}lnikov{\'a} and Josef Schicho}, journal={Journal of Algebra}, year={2005}, volume={303}, pages={514-529} }

## 33 Citations

### Rational parametrisation for degree 6 Del Pezzo surfaces using lie algebras

- MathematicsISSAC '06
- 2006

We give an algorithm for deciding whether a given Del Pezzo surface of degree 6 is rational over the ground field, and for computing a proper parametrisation in the affirmative case. This problem is…

### Explicit n-descent on elliptic curves III. Algorithms

- MathematicsMath. Comput.
- 2015

This is the third in a series of papers in which we study the n-Selmer group of an elliptic curve, with the aim of representing its elements as genus one normal curves of degree n. The methods we…

### Radical parametrization of algebraic curves and surfaces

- Mathematics
- 2013

Parametrization of algebraic curves and surfaces is a fundamental topic in CAGD (intersections; osets and conchoids; etc.) There are many results on rational parametrization, in particular in the…

### Morphisms to Brauer–Severi Varieties, with Applications to Del Pezzo Surfaces

- Mathematics
- 2017

We classify morphisms from proper varieties to Brauer–Severi varieties, which generalizes the classical correspondence between morphisms to projective space and globally generated invertible sheaves.…

### Computational birational geometry of minimal rational surfaces

- Mathematics
- 2009

The classification of minimal rational surfaces and the birational links between them by Iskovskikh, Manin and others is a well-known subject in the theory of algebraic surfaces. We explain…

### Parametrizing algebraic varieties using Lie algebras

- Mathematics
- 2006

In the thesis we present a new method for parametrizing algebraic varieties over the field of characteristic zero. The problem of parametrizing is reduced to a problem of finding an isomorphism of…

### Effective radical parametrization of trigonal curves

- MathematicsArXiv
- 2011

This article presents a method to decide whether a given algebraic curve is trigonal, and in the affirmative case to compute a map from C to the projective line whose fibers cut out the linear system.

### Point counting on curves using a gonality preserving lift

- Mathematics, Computer Science
- 2016

This work sketches how this can be done efficiently for curves of gonality at most four, with an in-depth treatment of curves of genus at most five over finite fields of odd characteristic, including an implementation in Magma.

### Parametrizing algebraic varieties using

- Mathematics
- 2006

In the thesis we present a new method for parametrizing algebraic varieties over the field of characteristic zero. The problem of parametrizing is reduced to a problem of finding an isomorphism of…

## References

SHOWING 1-10 OF 33 REFERENCES

### Sheaf algorithms using the exterior algebra

- Mathematics
- 2002

In this chapter we explain constructive methods for computing the cohomology of a sheaf on a projective variety. We also give a construction for the Beilinson monad, a tool for studying the sheaf…

### Affine varieties and lie algebras of vector fields

- Mathematics
- 1993

In this article, we associate to affine algebraic or local analytic varieties their tangent algebra. This is the Lie algebra of all vector fields on the ambient space which are tangent to the…

### Finding maximal orders in semisimple algebras over Q

- Mathematics, Computer Sciencecomputational complexity
- 2005

A polynomial time ff-algorithm is presented to solve the algorithmic problem of constructing a maximal order in a semisimple algebra over an algebraic number field.

### Explicit descent for Jacobians of cyclic coevers of the projective line.

- Mathematics
- 1997

We develop a general method for bounding Mordell-Weil ranks of Jacobians of arbitrary curves of the form y p = f(x). As an example, we compute the Mordell-Weil ranks over Q and Q( p 3) for a…

### On solving relative norm equations in algebraic number fields

- MathematicsMath. Comput.
- 1997

A theorem is proved which enables us to determine whether a given relative norm equation of the form |N F/e (η)| = |θ| has any solutions η ∈ M at all and, if so, to compute a complete set of nonassociate solutions.

### Lattice basis reduction for indefinite forms and an application

- Mathematics, Computer ScienceDiscret. Math.
- 1996

### Solvability of norm equations over cyclic number fields of prime degree

- MathematicsMath. Comput.
- 1996

An algorithm is described which takes as input α and the minimal polynomial of α over Q, and determines if α is a norm of an element of L, and it is shown that, if the authors ignore the time needed to obtain a complete factorization of α and a completefactorization of the discriminant of α, then the algorithm runs in timePolynomial in the size of the input.

### Proper parametrization of surfaces with a rational pencil

- MathematicsISSAC
- 2000

We give an algorithm for the following problem: given a surface defined over Q, and a rational pencil on it, compute a proper parametrization with coefficients in Q.

### Algorithmic properties of maximal orders in simple algebras over Q

- Mathematicscomputational complexity
- 2005

It is obtained that the problem of deciding if the index of a given central simple algebra over an algebraic number field isd, whered is a given natural number, belongs to the complexity class N P ∩co N P.