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Let ${\\cal L}\\s S(\\g)$ be the (graded) ideal defining $\\hbox{\\rm Sing}\\,\\g$ and let $2r$ be the dimension of a $G$-orbit of a regular element in $\\g$. Then ${\\cal L}^k = 0$ for any $k<r$. On the other hand, there exists a remarkable $G$-module $M\\s {\\cal L}^r$ which already defines $\\hbox{\\rm Sing}\\,\\g$. 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Let $S(\\g)$ be the $G$-module of polynomial functions on $\\g$ and let $\\hbox{\\rm Sing}\\,\\g$ be the closed algebraic cone of singular elements in $\\g$. Let ${\\cal L}\\s S(\\g)$ be the (graded) ideal defining $\\hbox{\\rm Sing}\\,\\g$ and let $2r$ be the dimension of a $G$-orbit of a regular element in $\\g$. Then ${\\cal L}^k = 0$ for any $k<r$. On the other hand, there exists a remarkable $G$-module $M\\s {\\cal L}^r$ which already defines $\\hbox{\\rm Sing}\\,\\g$. 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