[Math] An irreducible curve of degree 3 has one singular point

algebraic-geometry

Good morning, i got stuck with these exercises.

  1. Let $X$ be an hypersurface of degree 3 and suppose that $X$ has two singular points $P$ and $Q$. Let $L_{PQ}$ the line containing $P$ and $Q$. Show that $L_{PQ}\subset X$.

  2. Let $F(x,y,z)$ be an homogeneous polynomial, $k$ algebraic closed and let $C=Z(F)\subset\mathbb P_k^2$ be an irreducible curve. Prove that is $deg(F) =3$ then $X$ has at most one singular point.

Best Answer

For 2. Assume that there are 2 distinct singular points $p$ and $q$. Then the multiplicity of $p$ and $q$ are each greater than $2$. Let $L$ be a line joining $p$ and $q$. Then by Bezout's Theorem, $3=(deg(F))(deg(L))\geq (\text{multiplicty of } L\cap C\text{ at }p)+(\text{multiplicty of } L\cap C\text{ at }q)\geq2+2=4$. This is a contradiction.

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