# segre -- Segre embedding

## Synopsis

• Usage:
segre phi
segre R
• Inputs:
• phi, , with source a closed subvariety $X\subseteq\mathbb{P}^{n_1}\times\mathbb{P}^{n_2}\times\cdots\times\mathbb{P}^{n_k}$ of a product of projective spaces
• R, , or a polynomial ring, the coordinate ring of the subvariety $X\subseteq\mathbb{P}^{n_1}\times\mathbb{P}^{n_2}\times\cdots\times\mathbb{P}^{n_k}$
• Outputs:
• , the restriction to $X$ of the Segre embedding of $\mathbb{P}^{n_1}\times\mathbb{P}^{n_2}\times\cdots\times\mathbb{P}^{n_k}$, where the linear span of the image is identified with a projective space

## Description

More properly, this method accepts and returns objects of the class MultihomogeneousRationalMap.

 i1 : phi = first graph quadroQuadricCremonaTransformation(3,1) o1 = -- rational map -- source: subvariety of Proj(QQ[x , x , x , x ]) x Proj(QQ[y , y , y , y ]) defined by 0 1 2 3 0 1 2 3 { x y - x y , 3 2 2 3 x y + x y , 3 1 1 3 x y + x y , 2 1 1 2 x y - x y + x y + x y 0 0 1 1 2 2 3 3 } target: Proj(QQ[x, y, z, t]) defining forms: { x , 0 x , 1 x , 2 x 3 } o1 : MultihomogeneousRationalMap (birational map from threefold in PP^3 x PP^3 to PP^3) i2 : segre phi o2 = -- rational map -- source: subvariety of Proj(QQ[x , x , x , x ]) x Proj(QQ[y , y , y , y ]) defined by 0 1 2 3 0 1 2 3 { x y - x y , 3 2 2 3 x y + x y , 3 1 1 3 x y + x y , 2 1 1 2 x y - x y + x y + x y 0 0 1 1 2 2 3 3 } target: Proj(QQ[t , t , t , t , t , t , t , t , t , t , t , t ]) 0 1 2 3 4 5 6 7 8 9 10 11 defining forms: { x y , 0 1 x y , 0 2 x y , 0 3 x y , 1 0 x y , 1 1 x y , 2 0 -x y , 1 2 x y , 2 2 x y , 3 0 -x y , 1 3 x y , 2 3 x y 3 3 } o2 : MultihomogeneousRationalMap (rational map from threefold in PP^3 x PP^3 to PP^11)

## Ways to use segre :

• "segre(PolynomialRing)"
• "segre(QuotientRing)"
• "segre(RationalMap)"

## For the programmer

The object segre is .