We construct a singular Calabi-Yau of dimension 3 with 2 nodes, of codimension 4, of degree 20. This example cannot be smoothed. However, there is the example of the determinantal ($3 \times 3$ minors of a $4 \times 4$ matrix of linear forms, which is a smooth Calabi-Yau 3-fold of degree 20.
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The original example code
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Now we analyze the singularities of this Calabi-Yau. First, note that $X$ is singular at the point$(1, 0, \ldots, 0)$.
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Now we find other singular points. We would like to just compute the singular locus as the $4 \time 4$ minors of the $8 \times 16$ Jacobian matrix, which is $127,400$ determinants, but this is pretty large.
We find that there is exactly one more singular point, and that it is of multiplicity 2.
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This implies that the singular locus is a finite set of points, and is at most 2 points, and we know one of the points, and the other pointis off of the hyperplane $x_7 = 0$.
However, we must check that this point is singular on $X$, as our ideal is only a subset of the ideal of the singular locus.
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We now know that the singular locus consists of these 2 points, as the degree of $J_1$ is 2, so there can be no other components of dimension 0.