# Fano(ZZ,EmbeddedProjectiveVariety) -- Fano scheme of a projective variety

## Synopsis

• Function: Fano
• Usage:
Fano(k,X)
Fano(k,X,AffineChartGrass=>...)
Fano X
Fano(X,AffineChartGrass=>...)
• Inputs:
• k, an integer, optional with default value equal to $\mathrm{dim}(X)$
• Outputs:
• , the subvariety of the Grassmannian $\mathbb{G}(k,\mathrm{ambient}(X))$ that parametrizes the $k$-planes lying on $X$

## Description

This function is based internally on the function plucker, provided by the package Resultants. In particular, note that by default the computation is done on a randomly chosen affine chart on the Grassmannian. To change this behavior, you can use the AffineChartGrass option.

 i1 : K = ZZ/33331; i2 : L = linearSpan {point PP_K^4,point PP_K^4}; -- a line in P^4 o2 : ProjectiveVariety, curve in PP^4 i3 : p := Fano L o3 = point of coordinates [12627, 14941, 11188, -4847, 15437, 16492, 13501, 8759, 6817, 1] o3 : ProjectiveVariety, a point in PP^9 (subvariety of codimension 6 in GG(1,4) ⊂ PP^9) i4 : Fano p o4 = curve in PP^4 cut out by 3 hypersurfaces of degree 1 o4 : ProjectiveVariety, curve in PP^4 i5 : assert(Fano p == L)

If the input is a subvariety $Y\subset\mathbb{G}(k,\mathbb{P}^n)$, then the output is the variety $W\subset\mathbb{P}^n$ swept out by the linear spaces corresponding to points of $Y$. As an example, we now compute a surface scroll $W\subset\mathbb{P}^4$ over an elliptic curve $Y\subset\mathbb{G}(1,\mathbb{P}^4)$.

 i6 : G = GG_K(1,4); o6 : ProjectiveVariety, GG(1,4) i7 : Y := (G * random({{1},{1},{1},{1},{1}},0_G)) % G -- an elliptic curve o7 = curve in PP^9 cut out by 10 hypersurfaces of degrees 1^5 2^5 o7 : ProjectiveVariety, curve in PP^9 (subvariety of codimension 5 in G) i8 : W = Fano Y; -- surface swept out by the lines of Y o8 : ProjectiveVariety, surface in PP^4

We can recover the subvariety $Y\subset\mathbb{G}(k,\mathbb{P}^n)$ by computing the Fano variety of $k$-planes contained in $W$.

 i9 : Fano(1,W) -- variety of lines contained in W o9 = curve in PP^9 cut out by 10 hypersurfaces of degrees 1^5 2^5 o9 : ProjectiveVariety, curve in PP^9 (subvariety of codimension 5 in G) i10 : assert(oo == Y)