# toGrass(EmbeddedProjectiveVariety) -- embedding of an ordinary Gushel-Mukai fourfold or a del Pezzo variety into GG(1,4)

## Synopsis

• Function: toGrass
• Usage:
toGrass X
• Inputs:
• X, , an ordinary Gushel-Mukai fourfold, or a del Pezzo variety of dimension at least 4 (e.g., a sixfold projectively equivalent to $\mathbb{G}(1,4)\subset\mathbb{P}^9$)
• Outputs:
• , an embedding of $X$ into the Grassmannian $\mathbb{G}(1,4)\subset\mathbb{P}^9$, Plücker embedded

## Description

 i1 : x = gens ring PP_(ZZ/33331)^8; i2 : X = projectiveVariety ideal(x_4*x_6-x_3*x_7+x_1*x_8, x_4*x_5-x_2*x_7+x_0*x_8, x_3*x_5-x_2*x_6+x_0*x_8+x_1*x_8-x_5*x_8, x_1*x_5-x_0*x_6+x_0*x_7+x_1*x_7-x_5*x_7, x_1*x_2-x_0*x_3+x_0*x_4+x_1*x_4-x_2*x_7+x_0*x_8); o2 : ProjectiveVariety, 5-dimensional subvariety of PP^8 i3 : time toGrass X -- used 11.5312 seconds o3 = multi-rational map consisting of one single rational map source variety: 5-dimensional subvariety of PP^8 cut out by 5 hypersurfaces of degree 2 target variety: GG(1,4) ⊂ PP^9 o3 : MultirationalMap (rational map from X to GG(1,4)) i4 : show oo o4 = -- multi-rational map -- ZZ source: subvariety of Proj(-----[x0 , x0 , x0 , x0 , x0 , x0 , x0 , x0 , x0 ]) defined by 33331 0 1 2 3 4 5 6 7 8 { x0 x0 - x0 x0 + x0 x0 , 4 6 3 7 1 8 x0 x0 - x0 x0 + x0 x0 , 4 5 2 7 0 8 x0 x0 - x0 x0 + x0 x0 + x0 x0 - x0 x0 , 3 5 2 6 0 8 1 8 5 8 x0 x0 - x0 x0 + x0 x0 + x0 x0 - x0 x0 , 1 5 0 6 0 7 1 7 5 7 x0 x0 - x0 x0 + x0 x0 + x0 x0 - x0 x0 + x0 x0 1 2 0 3 0 4 1 4 2 7 0 8 } ZZ target: subvariety of Proj(-----[x , x , x , x , x , x , x , x , x , x ]) defined by 33331 0,1 0,2 1,2 0,3 1,3 2,3 0,4 1,4 2,4 3,4 { x x - x x + x x , 2,3 1,4 1,3 2,4 1,2 3,4 x x - x x + x x , 2,3 0,4 0,3 2,4 0,2 3,4 x x - x x + x x , 1,3 0,4 0,3 1,4 0,1 3,4 x x - x x + x x , 1,2 0,4 0,2 1,4 0,1 2,4 x x - x x + x x 1,2 0,3 0,2 1,3 0,1 2,3 } -- rational map 1/1 -- map 1/1, one of its representatives: { 5418x0 - 821x0 + 5588x0 - 3585x0 - 1758x0 - 15576x0 + 9147x0 - 14993x0 - 4736x0 , 0 1 2 3 4 5 6 7 8 11632x0 - 4732x0 - 10523x0 - 11526x0 - 1991x0 - 1831x0 - 9701x0 + 12320x0 - 2015x0 , 0 1 2 3 4 5 6 7 8 16371x0 - 7244x0 + 4935x0 + 15111x0 + 3749x0 - 12977x0 + 15511x0 + 7287x0 + 6751x0 , 0 1 2 3 4 5 6 7 8 - 13960x0 - 3219x0 + 8239x0 - 10597x0 + 7747x0 + 273x0 - 6285x0 + 2934x0 - 4471x0 , 0 1 2 3 4 5 6 7 8 - 12638x0 - 12017x0 - 2651x0 + 7012x0 - 9505x0 + 3559x0 - 2170x0 - 59x0 - 265x0 , 0 1 2 3 4 5 6 7 8 - 4096x0 + 10456x0 - 2284x0 + 11208x0 + 5756x0 - 6263x0 + 599x0 + 7817x0 - 6486x0 , 0 1 2 3 4 5 6 7 8 5896x0 + 11711x0 - 9239x0 + 9726x0 + 9682x0 + 2295x0 - 6875x0 - 16024x0 - 7246x0 , 0 1 2 3 4 5 6 7 8 - 8687x0 + 14564x0 + 3651x0 - 6141x0 - 7924x0 + 3227x0 - 5479x0 + 13427x0 + 11982x0 , 0 1 2 3 4 5 6 7 8 89x0 + 11710x0 + 1284x0 - 12079x0 + 11673x0 - 2256x0 + 12732x0 - 7459x0 - 5231x0 , 0 1 2 3 4 5 6 7 8 - 25x0 + 12923x0 + 1000x0 + 871x0 + 15902x0 - 3782x0 - 7479x0 - 5250x0 + 11717x0 0 1 2 3 4 5 6 7 8 }