## Description

MultiGradedRationalMap provides functions for computing the degree of a multi-graded rational map.

In the paper Degree and birationality of multi-graded rational maps, a new algebra called the saturated special fiber ring was introduced. This algebra is related to several features in the study of rational maps.

Some functions of this package are capable of working in the multi-graded setting. Let $R$ be the multi-homogeneous polynomial ring $R=k[x_{1,0},x_{1,1},...,x_{1,r_1}, x_{2,0},x_{2,1},...,x_{2,r_2}, ......, x_{m,0},x_{m,1},...,x_{m,r_m}]$ where the multidegree of a varible $x_{i,j}$ is $\{0,...,1,...,0\}$. Let $\mathbf{m}$ be the multi-homogeneous irrelevant ideal $\mathbf{m}=(x_{1,0},x_{1,1},...,x_{1,r_1})\cap (x_{2,0},x_{2,1},...,x_{2,r_2}) \cap ... \cap (x_{m,0},x_{m,1},...,x_{m,r_m})$ of $R$. Let $I$ be a multi-homogeneous ideal in $R$, which is generated by multi-homogeneous polynomials of the same multi-degree. The saturated special fiber ring of $I$ is defined by the algebra $$\oplus_{n=0}^\infty [(I^n)^{sat}]_{n*d}.$$

The main idea of this package is to exploit this algebra to compute the degree and test the birationality of rational maps.

We also implement the Jacobian dual criterion in the multi-graded setting.

Overlap with other packages: - The package Cremona performs several computations related to rational and birational maps between irreducible projective varieties. Among other things, it can compute the degree of a rational map, test birationality and find the inverse of a birational map. There is a deterministic implementation and a fast probabilistic implementation.

- The package RationalMaps computes several things related to rational maps between projective varieties. Among other things, it can detect birationality and compute the inverse of a rational map. It contains an implementation of the remarkable Jacobian dual criterion.

- The package Parametrization mostly deals with rational parametrizations of rational curves defined over ℚ. It includes a function to compute the inverse of a rational map.

- The present implementation of this package can only handle rational maps where the source is a multiprojective space. On the other hand, the packages Cremona, RationalMaps and Parametrization can handle more general varieties.

Acknowledgements: The author is grateful to the organizers of the Macaulay2 workshop in Leipzig. The author is grateful to Laurent Busé for his support on the preparation of this package.

## Version

This documentation describes version 0.1 of MultiGradedRationalMap.

## Source code

The source code from which this documentation is derived is in the file MultiGradedRationalMap.m2.

## Exports

• Functions and commands
• Methods
• "degreeOfMap(Ideal)" -- see degreeOfMap -- computes the degree of a rational map
• "degreeOfMapIter(Ideal,ZZ)" -- see degreeOfMapIter -- computes the degree of a rational map
• "gensSatSpecialFib(Ideal)" -- see gensSatSpecialFib -- computes generators of the saturated special fiber ring
• "gensSatSpecialFib(Ideal,ZZ)" -- see gensSatSpecialFib -- computes generators of the saturated special fiber ring
• "Hm1Rees0(Ideal)" -- see Hm1Rees0 -- computes the module [Hm^1(Rees(I))]_0
• "isBiratMap(Ideal)" -- see isBiratMap -- tests the birationality of a rational with the Jacobian dual criterion
• "jacobianDualRank(Ideal)" -- see jacobianDualRank -- computes the full Jacobian dual rank
• "partialJDRs(Ideal)" -- see partialJDRs -- computes the partial Jacobian dual ranks
• "satSpecialFiber(Ideal)" -- see satSpecialFiberIdeal -- computes the defining equations of the saturated special fiber ring
• "satSpecialFiberIdeal(Ideal,ZZ)" -- see satSpecialFiberIdeal -- computes the defining equations of the saturated special fiber ring
• "upperBoundDegreeSingleGraded(Ideal)" -- see upperBoundDegreeSingleGraded -- computes an upper bound for the degree of a rational map
• Symbols

## For the programmer

The object MultiGradedRationalMap is .