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\begin{document}
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	\large
	\textbf{18.721 Introduction to Algebraic Geometry\hspace{228pt}Fall~2021}\\\vspace{4pt}
	\textbf{Problem Set \#2\hspace{220pt}Due: 09/24/2021}
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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 $f, g \in k[x]$, where $\deg f = \ell$ and $\deg g = m$. Show the following properties regarding resultants:
		\begin{enumerate}
			\item $\Res(f, g, x) = (-1)^{\ell m} \Res(g, f, x)$.
			\item For $\lambda, \mu \neq 0$, we have $\Res(\lambda f, \mu g, x) = \lambda^m \mu^{\ell} \Res(f, g, x)$.
			\item There are polynomials $A, B \in k[x]$ such that $A f + B g = \Res(f, g, x)$.
		\end{enumerate}

	\item Solve the following system of equations in $\P^3(x,y,z,t)$
		\begin{equation*}
			\begin{aligned}
				x^{2} + y t + z t  &= t^2,\\
				y^{2} + x t + z t  &= t^2,\\
				z^{2} + x t + y t  &= t^2.
			\end{aligned}
		\end{equation*}
		You may use a computer algebra system to carry intermediate calculations.
	\item Let $C$ be a cubic curve with a node.
		\begin{enumerate}
			\item Determine the degree of $C^*$.
			\item Determine the number of flexes, bitangents, nodes, and cusps of $C$ and of $C^*$.
		\end{enumerate}

	\item Let $C$ be a singular plane curve of degree $d$ where all of its singularities are nodes or cusps.
		Let $\overline{C}$ be the desingularization of $C$.
		Compute the genus of $\overline{C}$ in the following settings:
		\begin{enumerate}
			\item $C$ has a unique singular point, a cusp.
			\item $C$ has a unique singular point, a node.
			\item $C$ has a $k+n$ singular points, $k$ cusps and $n$ nodes.
		\end{enumerate}

\end{enumerate}


\end{document}
