# Noether-Lefschetz examples -- examples from Section 6.2 in [QQ]

We give examples of specific quartics interesting in Noether-Lefschetz loci for K3 surfaces, and where they fit in the Betti classification.

 i1 : kk = ZZ/101; i2 : R = kk[x_0..x_3];

The first example illustrates Corollary 6.18.

 i3 : Q618 = (x_0^2+x_1^2+x_2^2+x_3^2)^2+x_0^4+x_1^4+x_2^4+x_3^4 4 2 2 4 2 2 2 2 4 2 2 2 2 2 2 4 o3 = 2x + 2x x + 2x + 2x x + 2x x + 2x + 2x x + 2x x + 2x x + 2x 0 0 1 1 0 2 1 2 2 0 3 1 3 2 3 3 o3 : R i4 : minimalBetti inverseSystem Q618 0 1 2 3 4 o4 = total: 1 16 30 16 1 0: 1 . . . . 1: . . . . . 2: . 16 30 16 . 3: . . . . . 4: . . . . 1 o4 : BettiTally i5 : quarticType Q618 o5 = [000]

We illustrate Remark 6.19, considering a double quadric:

 i6 : Q619 = (x_0^2+x_1^2+x_2^2+x_3^2)^2 4 2 2 4 2 2 2 2 4 2 2 2 2 2 2 4 o6 = x + 2x x + x + 2x x + 2x x + x + 2x x + 2x x + 2x x + x 0 0 1 1 0 2 1 2 2 0 3 1 3 2 3 3 o6 : R i7 : minimalBetti inverseSystem Q619 0 1 2 3 4 o7 = total: 1 16 30 16 1 0: 1 . . . . 1: . . . . . 2: . 16 30 16 . 3: . . . . . 4: . . . . 1 o7 : BettiTally

Next, we illustrate Remark 6.20. The first example is that of the Vinberg most singular K3 surface. This is of type [331].

 i8 : Q620V = x_0^4-x_1*x_2*x_3*(x_1+x_2+x_3) 4 2 2 2 o8 = x - x x x - x x x - x x x 0 1 2 3 1 2 3 1 2 3 o8 : R i9 : minimalBetti inverseSystem Q620V 0 1 2 3 4 o9 = total: 1 11 20 11 1 0: 1 . . . . 1: . 3 3 1 . 2: . 7 14 7 . 3: . 1 3 3 . 4: . . . . 1 o9 : BettiTally i10 : quarticType Q620V o10 = [331]

The second example illustrating Remark 6.20 is that a general element of the Dwork pencil has type [000].

 i11 : Q620D = x_0^4+x_1^4+x_2^4+x_3^4-8*x_0*x_1*x_2*x_3 4 4 4 4 o11 = x + x + x - 8x x x x + x 0 1 2 0 1 2 3 3 o11 : R i12 : minimalBetti inverseSystem Q620D 0 1 2 3 4 o12 = total: 1 16 30 16 1 0: 1 . . . . 1: . . . . . 2: . 16 30 16 . 3: . . . . . 4: . . . . 1 o12 : BettiTally i13 : quarticType Q620D o13 = [000]

The third example illustrating Remark 6.20 is that the K3 quartics $S_{t}\subset \mathbb{P}^{5}$ given by $$x_{1}^{4}+\dots+x_{5}^{4}-t(x_{1}^{2}+\dots+x_{5}^{2})^{2}=x_{1}+\dots+x_{5}=0$$ for general $t$ are of type [000]. However, $S_{0}$ is of type [550].

 i14 : x5=x_0+x_1+x_2+x_3 o14 = x + x + x + x 0 1 2 3 o14 : R i15 : Q = x_0^4+x_1^4+x_2^4+x_3^4+x5^4-random(kk)*(x_0^2+x_1^2+x_2^2+x_3^2+x5^2)^2; i16 : minimalBetti inverseSystem Q 0 1 2 3 4 o16 = total: 1 16 30 16 1 0: 1 . . . . 1: . . . . . 2: . 16 30 16 . 3: . . . . . 4: . . . . 1 o16 : BettiTally i17 : Q = x_0^4+x_1^4+x_2^4+x_3^4+x5^4; i18 : minimalBetti inverseSystem Q 0 1 2 3 4 o18 = total: 1 6 10 6 1 0: 1 . . . . 1: . 5 5 . . 2: . 1 . 1 . 3: . . 5 5 . 4: . . . . 1 o18 : BettiTally i19 : quarticType Q o19 = [550]