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;
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i2 : R = kk[x_0..x_3];
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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
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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
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i5 : quarticType Q618
o5 = [000]
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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
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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
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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
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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
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i10 : quarticType Q620V
o10 = [331]
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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
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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
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i13 : quarticType Q620D
o13 = [000]
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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
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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;
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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
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i17 : Q = x_0^4+x_1^4+x_2^4+x_3^4+x5^4;
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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
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i19 : quarticType Q
o19 = [550]
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