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Emerton
Emerton
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The union of two planes in $\mathbb A^4$ which meet at a point is not Cohen--Macaulay, and so in particular not a local complete intersection.

More generally, any smooth subvariety of a smooth variety is a local complete intersection, so any non-Cohen--Macaulay subvariety whose components are smooth gives an example of a union of local complete intersections which is not itself a local complete intersection.

[Added: Your question has the caveat "with an appropriate scheme structure"; is there any other structure besides the reduced scheme structure on the union which you had in mind?]

The union of two planes in $\mathbb A^4$ which meet at a point is not Cohen--Macaulay, and so in particular not a local complete intersection.

More generally, any smooth subvariety of a smooth variety is a complete intersection, so any non-Cohen--Macaulay subvariety whose components are smooth gives an example of a union of local complete intersections which is not itself a local complete intersection.

[Added: Your question has the caveat "with an appropriate scheme structure"; is there any other structure besides the reduced scheme structure on the union which you had in mind?]

The union of two planes in $\mathbb A^4$ which meet at a point is not Cohen--Macaulay, and so in particular not a local complete intersection.

More generally, any smooth subvariety of a smooth variety is a local complete intersection, so any non-Cohen--Macaulay subvariety whose components are smooth gives an example of a union of local complete intersections which is not itself a local complete intersection.

[Added: Your question has the caveat "with an appropriate scheme structure"; is there any other structure besides the reduced scheme structure on the union which you had in mind?]

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Emerton
Emerton
  • 57.6k
  • 6
  • 209
  • 259

The union of two planes in $\mathbb A^4$ which meet at a point is not Cohen--Macaulay, and so in particular not a local complete intersection.

More generally, any smooth subvariety of a smooth variety is a complete intersection, so any non-Cohen--Macaulay subvariety whose components are smooth gives an example of a union of local complete intersections which is not itself a local complete intersection.

[Added: Your question has the caveat "with an appropriate scheme structure"; is there any other structure besides the reduced scheme structure on the union which you had in mind?]