# Singularities of lifting of type [300b] -- The lifting of [300b] defined by bilaison acquires singularities in dimension 3

We describe a way of lifting case [300b] by performing a bilaison construction. Let $X$ be a degree 7 residual component of an intersection of three quadrics containing a linear space of codimension 3.

 i1 : loadPackage "RandomIdeals" o1 = RandomIdeals o1 : Package i2 : kk = ZZ/101; i3 : R = kk[x_0..x_7]; i4 : T = random(R^1,R^{-1,-1,-1}); 1 3 o4 : Matrix R <--- R i5 : I = ideal(T); o5 : Ideal of R i6 : J = randomElementsFromIdeal({2,2,2},I); o6 : Ideal of R i7 : X=J:I; o7 : Ideal of R

Consider $SS$ the intersection of the two components of the complete intersection

 i8 : SS=X+I; o8 : Ideal of R i9 : SingSS= radical ideal singularLocus saturate SS; o9 : Ideal of R i10 : degree SingSS o10 = 6 i11 : dim SingSS o11 = 1

Perform a bilaison uo by degree 2 of $SS$ in $X$ and get $BT$ an AG variety of degree 17 and type [300b]

 i12 : JJ = randomElementsFromIdeal({3},SS); o12 : Ideal of R i13 : IDD=X+JJ; o13 : Ideal of R i14 : PP=IDD:SS; o14 : Ideal of R i15 : BB= randomElementsFromIdeal({5},PP); o15 : Ideal of R i16 : BU=BB+X; o16 : Ideal of R i17 : BT=BU:PP; o17 : Ideal of R i18 : degree BT o18 = 17

We look for singularities of $BT$. Since the computation of singular locus is too long, we just check the rank of the jacobian matrix in one of the components of $SingSS$ and get that $BT$ is singular in that locus.

 i19 : minors(4,(map(R/(decompose SingSS)_0,R)) (jacobian BT)) o19 = ideal () R o19 : Ideal of --------------------------------------------------------------------------- (x + 4x , x + 24x , x + 10x , x + 5x , x + 44x , x - 37x , x + 38x ) 6 7 5 7 4 7 3 7 2 7 1 7 0 7

## See also

• [QQ] -- Quaternary Quartic Forms and Gorenstein rings (Kapustka, Kapustka, Ranestad, Schenck, Stillman, Yuan, 2021)