Check smoothness of GM threefold

[Chris Dare]

Hi All —

Im running into a bit of an issue in checking whether a particular variety is indeed a smooth Gushel-Mukai threefold (of the 1st type). The general construction (as taken from A. Iliev, The Fano Surface of the Gushel-Threefold ) is to take V = C^5 (or any other field of characteristic 0), let G = Gr(2, V), let X be a (complete) intersection of G with a P^7 and a quadric Q; if the resulting variety is smooth, it is a Fano threefold of picard number 1 and degree 10 and called a Gushel-Mukai threefold of the 1st type.

My code for trying to construct one is as follows:

i1 : ringP20 = QQ[x_0..x_20]

o1 = ringP20

o1 : PolynomialRing

i2 : G=Grassmannian(2,5,ringP20)

o2 = ideal (...

o2 : Ideal of ringP20

i3 : I=G+ideal(x_8..x_20)

o3 = ideal (...

o3 : Ideal of ringP20

i4 : J=I+ideal(x_0*x_7 + x_1*x_6 + x_2*x_5 + x_3*x_4)

o4 = ideal (...

o4 : Ideal of ringP20

i5 : X=Proj(ringP20/J)

o5 = X

o5 : ProjectiveVariety

i6 : dim X

o6 = 3

i7 : degree X

o7 = 10

At first glance everything looks good, so what I want to do is make sure this particular variety is smooth in order to verify that this system of equations gives a GM threefold of the first kind. But when I run

i15 : Xs = singularLocus X

the computer just keeps running... for hours. I let it run all afternoon and wasn't able to get Macaulay2 to store the singular locus in an object to check to make sure the singularLocus is empty (i.e. check dim = 0, deg = 0).

Any advice is appreciated.

— Best,

Chris

[H XW]

Chris,

If singularLocus uses the Jacobian criterion, it is rather time
consuming in computing the minors of a matrix with polynomial entries.
In my experience, 20 variables are too many for a direct
computation, unless the ideal is very simple.
A more efficient way is to manually find an affine covering, says
{U_i}, such that, by some manipulations on the defining equations, we
can decompose U_i=V_i\tims Y_i, where Y_i is known to be smooth, and
V_i might be singular, but with a smaller embedded dimension or
simpler defining equations. Then use M2 to compute singularLocus V_i.

Best,
Xiaowen

Chris Dare <cd...@ucsb.edu> 于2022年12月22日周四 07:09写道：

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