FanoScheme is a package in MAGMA for computation with Fano schemes of embedded projective varieties. Let \(X\subset \mathbb{P}^n\) be an embedded projective variety. Then the Fano scheme \(\mathbf{F}_k (X)\) of \(k\)-planes in \(X\) is the fine moduli space that parametrizes those \(k\)-planes contained in \(X\). The scheme \(\mathbf{F}_k (X)\) is a subscheme of the Grassmannian \(\mathbb{G}(k,n)\).
Moreover, a Grassmannian \(\mathbb{G}(k,n)\) is the same as the Fano scheme \(\mathbf{F}_k(\mathbb{P}^n)\).
Returns the Fano scheme \(\mathbf{F}_k(X)\) as a subscheme of a Grassmannian \(\mathbb{G}(k, r)\) embedded in the projective space grassAmbient. The dimension of grassAmbient must be equal to \(\binom{r+1}{k+1}\) where \(r\) is the dimension of the ambient projective space of \(X\), otherwise an error occurs. The returned Fano scheme is a subscheme of grassAmbient.
Returns the Fano scheme \(\mathbf{F}_k(X)\) as a subscheme of a Grassmannian \(\mathbb{G}(k, r)\) embedded in a projective space of dimension \(\binom{r+1}{k+1}\). It creates a projective space ambientSpace of dimension \(\binom{r+1}{k+1}\) and then calls FanoScheme(X, k, grassAmbient).
Example 1 : The famous Cayley-Salmon theorem asserts that a smooth cubic surface in \(\mathbb{P}^3\) contains exactly 27 lines. We will use
FanoSchemeto demonstrate the theorem.> KK:=Rationals(); > KK; Rational Field > P<x,y,z,w>:=ProjectiveSpace(KK,3); > P; Projective Space of dimension 3 over Rational Field Variables: x, y, z, w > grassAmbient<p_0,p_1,p_2,p_3,p_4,p_5>:=ProjectiveSpace(KK,5); > grassAmbient; Projective Space of dimension 5 over Rational Field Variables: p_0, p_1, p_2, p_3, p_4, p_5 > X:=Scheme(P, x^3+y^3+z^3+w^3); > X; Scheme over Rational Field defined by x^3 + y^3 + z^3 + w^3 > Y:=FanoScheme(X,1,grassAmbient); > Dimension(Y); 0 > Degree(Y); 27Example 2 : The smooth quadric \(X\subset \mathbb{P}^3\) defined by \(xy-zw=0\) has two disjoint family of lines, namely its two sets of rulings. Let's examine the Fano scheme \(\mathbf{F}_1(X)\). We will see that the Fano scheme \(\mathbf{F}_1(X)\) has two irreducible components. They are curves of degree 2. Upon inspecting the equations for each compnent, we see that they are two disjoint conics in the Grassmannian \(\mathbb{G}(1,3)\).
KK; Rational Field > P<x,y,z,w>:=ProjectiveSpace(KK,3); > P; Projective Space of dimension 3 over Rational Field Variables: x, y, z, w > grassAmbient<p_0,p_1,p_2,p_3,p_4,p_5>:=ProjectiveSpace(KK,5); > grassAmbient; Projective Space of dimension 5 over Rational Field Variables: p_0, p_1, p_2, p_3, p_4, p_5 X:=Scheme(P, x*y-z*w); X; Scheme over Rational Field defined by x*y - z*w > Y:=FanoScheme(X,1,grassAmbient); for component in IrreducibleComponents(Y) do component; printf "Dimension of component = %o\n", Dimension(component); printf "Degree of component = %o\n", Degree(component); print "-----"; end for; Scheme over Rational Field defined by p_2*p_3 + p_5^2, p_0 - p_5, p_1, p_4 Dimension of component = 1 Degree of component = 2 ----- Scheme over Rational Field defined by p_1*p_4 + p_5^2, p_0 + p_5, p_2, p_3 Dimension of component = 1 Degree of component = 2 -----
Returns the Grassmannian \(\mathbb{G}(k, r)\) of \(k\)-planes in an \(n\)-projective space \(P\). It works by calling FanoScheme(P, k, grassAmbient). The returned Grassmannian is a subscheme of the ambient projective space grassAmbient which must have dimension \(\binom{n+1}{k+1}-1\), otherwise an error occurs.
Returns the Grassmannian \(\mathbb{G}(k,P)\) of \(k\)-planes in the \(n\)-projective space \(P\) by calling FanoScheme(P, k). The returned Grassmannian is a subscheme of an ambient projective space of dimension \(\binom{n+1}{k+1}-1\).
Returns the Grassmannian \(\mathbb{G}(k,P)\) of \(k\)-planes in the \(n\)-projective space \(P\) by calling FanoScheme(P, k, grassAmbient). The returned Grassmannian is a subscheme of the ambient projective space grassAmbient which must have dimension \(\binom{n+1}{k+1}-1\), otherwise an error occurs.
Example 3 : We create the Grassmannian \(\mathbb{G}(1,3)\) and display its Plücker relation.
> KK:=Rationals(); > KK; Rational Field > grassAmbient<p_0,p_1,p_2,p_3,p_4,p_5>:=ProjectiveSpace(KK,5); > grassAmbient; Projective Space of dimension 5 over Rational Field Variables: p_0, p_1, p_2, p_3, p_4, p_5 > G:=Grassmannian(1,3,grassAmbient); > G; Scheme over Rational Field defined by p_2*p_3 - p_1*p_4 + p_0*p_5