= Smooth-locus group of a singular Weierstrass cubic
{title2=$C_{\mathrm{sm}}(K)$}
For a singular projective Weierstrass cubic, its <smooth locus of a variety> contains the point at infinity and carries the <chord-and-tangent group law>. The singular point is excluded. Over an <algebraic closure>, a node gives the <multiplicative algebraic group> and a cusp gives the <additive group>. A node whose branches are not defined over the base <field> gives a nonsplit one-dimensional <algebraic torus>. This group is distinct from an <elliptic curve>, whose projective model is smooth.
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