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\large
\textbf{18.721 Introduction to Algebraic Geometry\hspace{228pt}Fall~2021}\\\vspace{4pt}
\textbf{Problem Set \#1\hspace{220pt}Due: 09/17/2021}
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\subsection*{Description}

Solutions are to be prepared in typeset form (typically via latex) and submitted electronically as a pdf file via \href{https://www.gradescope.com/courses/281407}{Gradescope} by midnight on the due date.
Late problem sets will not be accepted -- your lowest score is dropped, so you can afford to skip one problem set without penalty.

Collaboration is permitted/encouraged, but you must write up your own solutions and explicitly identify your collaborators, as well as any resources you consulted that are not listed above. If there are none, please indicate this by writing \textbf{Sources consulted:\ none} at the top of your submission.

\medskip

\noindent
\begin{enumerate}
	\item Let $f(x,y,z)$ be an homogeneous irreducible polynomial of degree $>1$. Prove that the locus $f = 0$ in $\P^2$ contains three points that do not lie on a line.

	\item
		\begin{enumerate}
			\item Consider the curve $X\colon y^2 = x^3 \subset \A^2$.
				Construct a surjective map $\varphi: \A^1 \to X \subset \A^2$, i.e., find a parametrization of $X$.
			\item Consider the induced map on the algebras
				\begin{align*}
					\varphi^* \colon& \C[x,y]/(y^2 - x^3) \to \C[t]\\
													& g(x,y) \mapsto g(\varphi(t)),
				\end{align*}
				Show that $\varphi^*$ is not an isomorphism.
			\item In class, we saw that for $Y\colon y^2 = x^3 + x^2$ the map $\varphi(t) \colonequals (t^2 - 1, t^3 - t)$ is a parametrization for $Y$.
				Show that in this setting,
				\begin{align*}
					\varphi^* \colon& \C[x,y]/(y^2 - x^3 - x^2) \to \C[t]\\
													& g(x,y) \mapsto g(t^2 - 1, t^3 - t),
				\end{align*}
				is also not an isomorphism.
		\end{enumerate}


	\item Show that a cubic curve in $\P^2$, i.e., defined by a degree 3 homogeneous irreducible polynomial, has at most $1$ singular point.



	\item Let $C$ be a smooth cubic curve in $\P^2$ defined by $f$, and let $(0,1,0)$ be a flex point of $C$ whose tangent is $z = 0$.
		\begin{enumerate}
			\item Show that the coefficients of $x^2 y,$ $x y^2,$ and $y^3$ in $f$ are zero.
			\item Show that with a suitable choice of coordinates, one can reduce the defining polynomial to the form $f = y^2 z + x^3 + a x z^2 + b z^3$, where $x^3 + a x + b$ is a polynomial with distinct roots.
			\item Show that after a change of variables, one can take either $a = 1$ or $b = 1$.
		\end{enumerate}

	\item Let $f(x)$ be a polynomial in $\C[x_1, \dots, x_n]$.
		Show that $\{x: f(x) =0 \} = \A^n$ if and only if $f$ is the zero polynomial.

	\item Let $p$ be a prime number. Recall that $\F_p \colonequals \Z/p \Z$ is a field.
		\begin{enumerate}
			\item Consider the polynomial $g(x,y) = x^2 y + y^2 x \in \F_2[x,y]$. Show that $g(x,y) = 0$ for every $(x,y) \in \F^2 _2$.
			\item Find a nonzero polynomial in $\F_2[x,y, z]$ involving all three variables that vanishes at every point of $\F_2^3$.
			\item
				Let $p$ be any prime number.
				Find a nonzero polynomial in $\F_p[x]$ that vanishes at all points of $\F_p$.
				(Hint: Fermat's little theorem )
		\end{enumerate}
\end{enumerate}

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