▁of ▁all ▁cub ic ▁curves , ▁the ▁general ▁family ▁to ▁which ▁these ▁examples ▁belong . ▁It ▁was ▁noticed ▁in ▁the ▁form ulation ▁of ▁Bé z out ' s ▁theorem ▁that ▁such ▁singular ▁points ▁must ▁be ▁counted ▁with ▁multip licity ▁( 2 ▁for ▁a ▁double ▁point , ▁ 3 ▁for ▁a ▁c us p ), ▁in ▁account ing ▁for ▁inter sections ▁of ▁curves . ▁ ▁It ▁was ▁then ▁a ▁short ▁step ▁to ▁define ▁the ▁general ▁notion ▁of ▁a ▁singular ▁point ▁of ▁an ▁algebraic ▁variety ; ▁that ▁is , ▁to ▁allow ▁higher ▁dimensions . ▁ ▁The ▁general ▁position ▁of ▁singular ities ▁in ▁algebraic ▁geometry ▁Such ▁singular ities ▁in ▁algebraic ▁geometry ▁are ▁the ▁easiest ▁in ▁principle ▁to ▁study , ▁since ▁they ▁are ▁defined ▁by ▁polynomial ▁equations ▁and ▁therefore ▁in ▁terms ▁of ▁a ▁coordinate ▁system . ▁One ▁can ▁say ▁that ▁the ▁extr insic ▁meaning ▁of ▁a ▁singular ▁point ▁isn ' t ▁in ▁question ; ▁it ▁is ▁just ▁that ▁in ▁intr insic ▁terms ▁the ▁coordinates ▁in ▁the ▁ambient ▁space ▁don ' t ▁straightforward ly ▁translate ▁the ▁geometry ▁of ▁the ▁algebraic ▁variety ▁at ▁the ▁point . ▁Int ensive ▁studies ▁of ▁such ▁singular ities ▁led ▁in ▁the ▁end ▁to ▁He isu ke ▁Hir on aka ' s ▁fundamental ▁theorem ▁on ▁resolution ▁of ▁singular ities ▁( in ▁bir ational ▁geometry ▁in ▁characteristic ▁ 0 ). ▁This ▁means ▁that ▁the ▁simple ▁process ▁of ▁" l ifting " ▁a ▁piece ▁of ▁string ▁off ▁itself , ▁by ▁the ▁" ob vious " ▁use ▁of ▁the ▁cross - over ▁at ▁a ▁double ▁point , ▁is