Let $$ L_{1}=V(X_{0},X_{1})\subseteq\mathbb{P}^{3}, $$ $$ L_{2}=V(X_{2},X_{3})\subseteq\mathbb{P}^{3}. $$ I want to prove that $$ I(L_{1}\cup L_{2})=(X_{0}X_{2},X_{0}X_{3},X_{1}X_{2},X_{1}X_{3}). $$ I kown that $$ I(L_{1} \cup L_{2})=I(L_{1})\cap I(L_{2})=(X_{0},X_{1})\cap(X_{2},X_{3}), $$ but I am not able to prove $$ (X_{0},X_{1})\cap(X_{2},X_{3})=(X_{0}X_{2},X_{0}X_{3},X_{1}X_{2},X_{1}X_{3}). $$ It is clear that $$ (X_{0},X_{1})\cap(X_{2},X_{3})\supseteq (X_{0}X_{2},X_{0}X_{3},X_{1}X_{2},X_{1}X_{3}), $$ but the other content is harder. How could we do it?
2026-04-23 06:40:17.1776926417
Ideal of the union of two skew lines in $\mathbb{P}^{3}$
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The generators of the intersection of two monomial ideals can be found by taking the least common multiples of generators of the two ideals; see here.