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\def\AA{\mathbb{A}}
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\title{Algebraic Curves --- Exercises}
\begin{document}
\maketitle
\section{Sheet 1 --- 12 May 2020}
\begin{exercise}[1.8 of~\cite{fulton}]
Let $k$ be a field.
Show that the algebraic subsets of~$\AA^1(k)$ are just the finite subsets,
together with $\AA^1(k)$ itself.
\end{exercise}
\begin{exercise}[1.9 of~\cite{fulton}]
If $k$ is a finite field, show that every subset of~$\AA^1(k)$ is algebraic.
\end{exercise}
\begin{exercise}[1.14 of~\cite{fulton}]
Let $F$ be a nonconstant polynomial in $k[X_1, \dots, X_n]$,
$k$ algebraically closed.
Show that $\AA^n(k) \setminus V(F)$ is infinite if $n \ge 1$,
and $V(F)$ is infinite if $n \ge 2$.
Conclude that the complement of any proper algebraic set is infinite.
(\emph{Hint}: see problem~1.4 of~\cite{fulton}.)
\end{exercise}
\begin{exercise}[1.15 of~\cite{fulton}]
Let $V \subset \AA^n(k)$, $W \subset \AA^m(k)$ be algebraic sets.
Show that
\[
V \times W =
\{ (a_1, \dots, a_n, b_1, \dots, b_m) \mid
(a_1, \dots, a_n) \in V,
(b_1, \dots, b_m) \in W \}
\]
is an algebraic set in $\AA^{n+m}(k)$.
It is called the \emph{product} of~$V$ and~$W$.
\end{exercise}
\begin{exercise}[1.16 of~\cite{fulton}]
Let $V, W$ be algebraic sets in $\AA^n(k)$.
Show that $V = W$ if and only if $I(V) = I(W)$.
\end{exercise}
\begin{exercise}[1.17 of~\cite{fulton}]
\begin{enumerate}
\item Let $V$ be an algebraic set in~$\AA^n(k)$,
$P \in \AA^n(k)$ a point not in~$V$.
Show that there is a polynomial $F \in k[X_1, \dots, X_n]$
such that $F(Q) = 0$ for all $Q \in V$,
but $F(P) = 1$.
(\emph{Hint}: $I(V) \ne I(V \cup \{P\})$.)
\item Let $P_1, \dots, P_r$ be distinct points in~$\AA^n(k)$,
not in an algebraic set~$V$.
Show that there are polynomials $F_1, \dots, F_r \in I(V)$
such that $F_i(P_j) = 0$ if $i \ne j$,
and $F_i(P_i) = 1$.
(\emph{Hint}: Apply the preceding part to the union of~$V$
and all but one point.)
\item With $P_1, \dots, P_r$ and~$V$ as in the preceding point,
and $a_{ij} \in k$ for $1 \le i,j \le r$,
show that there are $G_i \in I(V)$
with $G_i(P_j) = a_{ij}$ for all $i$ and~$j$.
(\emph{Hint}: Consider $\sum_j a_{ij} F_j$.)
\end{enumerate}
\end{exercise}
\begin{exercise}[1.22 of~\cite{fulton}]
Let $I$ be an ideal in a ring~$R$,
$\pi \colon R \to R/I$ the natural homomorphism.
\begin{enumerate}
\item Show that for every ideal~$J'$ of~$R/I$,
$\pi^{-1}(J') = J$ is an ideal of~$R$ containing~$I$,
and for every ideal~$J$ of~$R$ containing~$I$,
$\pi(J) = J'$ is an ideal of~$R/I$.
This sets up a natural one-to-one correspondence
between $\{\text{ideals of~$R/I$}\}$
and $\{\text{ideals of~$R$ that contain~$I$}\}$.
\item Show that $J'$ is a radical ideal if and only if~$J$ is radical.
Similarly for prime and maximal ideals.
\item Show that $J'$ is a finitely generated if $J$ is.
Conclude that $R/I$ is Noetherian if~$R$ is Noetherian.
Any ring of the form $k[X_1, \dots, X_n]/I$ is Noetherian.
\end{enumerate}
\end{exercise}
\section{Sheet 2 --- 19 May 2020}
Let $k$ be an algebraically closed field.
\begin{exercise}[2.4 of~\cite{fulton}]
Let $V \subset \AA^n$ be a nonempty variety.
Show that the following are equivalent:
\begin{enumerate*}[label=(\roman*)]
\item $V$ is a point;
\item $\Gamma(V) = k$;
\item $\dim_k \Gamma(V) < \infty$.
\end{enumerate*}
\end{exercise}
\begin{exercise}[2.14 of~\cite{fulton}, important]
A set $V \subset \AA^n(k)$ is called
a \emph{linear subvariety} of~$\AA^n(k)$ if
$V = V(F_1, \dots, F_r)$
for some polynomials~$F_i$ of degree~$1$.
\begin{enumerate*}[label=(\alph*)]
\item Show that if $T$ is an affine change of coordinates on~$\AA^n$,
then $V^T$ is also a linear subvariety of~$\AA^n(k)$.
\item If $V \ne \varnothing$,
show that there is an affine change of coordinates~$T$ of~$\AA^n$
such that $V^T = V(X_{m+1}, \dots, X_n)$.
(\emph{Hint}: use induction on~$r$.)
So $V$ is a variety.
\item Show that the $m$ that appears in part~(b)
is independent of the choice of~$T$.
It is called the \emph{dimension} of~$V$.
Then $V$ is isomorphic (as a variety) to~$\AA^m(k)$.
(\emph{Hint}: Suppose there were an affine change of coordinates~$T$
such that $V(X_{m+1}, \dots, X_n)^T = V(X_{s+1}, \dots, X_n)$, $m < s$;
show that $T_{m+1}, \dots, T_n$ would be dependent.)
\end{enumerate*}
\end{exercise}
\begin{exercise}[2.15 of~\cite{fulton}, important]
Let $P = (a_1, \dots, a_n)$, $Q = (b_1, \dots, b_n)$
be distinct points of~$\AA^n$.
The \emph{line} through $P$~and~$Q$ is defined to be
$\{ a_1 + t(b_1 - a_1), \dots a_n + t(b_n - a_n) \mid t \in k \}$.
\begin{enumerate*}[label=(\alph*)]
\item Show that if~$L$ is the line through $P$ and~$Q$,
and $T$ is an affine change of coordinates,
then $T(L)$ is the line through $T(P)$ and~$T(Q)$.
\item Show that a line is a linear subvariety of dimension~$1$,
and that a linear subvariety of dimension~$1$
is the line through any two of its points.
\item Show that, in $\AA^2$,
a line is the same thing as a hyperplane.
\item Let $P, P' \in \AA^2$,
$L_1, L_2$ two distinct lines through~$P$,
$L_1', L_2'$ distinct lines through~$P'$.
Show that there is an affine change of coordinates~$T$ of~$\AA^2$
such that $T(P) = P'$ and $T(L_i) = L_i'$, $i = 1,2$.
\end{enumerate*}
\end{exercise}
\begin{exercise}[2.44 of~\cite{fulton}, important]
Let $V$ be a variety in~$\AA^n$,
$I = I(V) \subset k[X_1, \dots, X_n]$.
$P \in V$,
and let $J$ be an ideal of $k[X_1, \dots, X_n]$ that contains~$I$.
Let $J'$ be the image of~$J$ in $\Gamma(V)$.
Show that there is a natural homomorphism~$\phi$
from $\mathcal{O}_P(\AA^n)/J\mathcal{O}_P(\AA^n)$
to $\mathcal{O}_P(V)/J'\mathcal{O}_P(V)$,
and that $\phi$ is an isomorphism.
In particular,
$\mathcal{O}_P(\AA^n)/I\mathcal{O}_P(\AA^n)$
is isomorphic to $\mathcal{O}_P(V)$.
\end{exercise}
\section{Sheet 3 --- 26 May 2020}
\def\ord{\mathrm{ord}}
\begin{exercise}[2.29 of~\cite{fulton}]
Let $R$ be a DVR with quotient field~$K$,
$\ord$ the order function on~$K$.
\begin{enumerate*}[label=(\alph*)]
\item If $\ord(a) < \ord(b)$,
show that $\ord(a + b) = \ord(a)$.
\item If $a_1, \dots, a_n \in K$,
and for some $i$,
$\ord(a_i) < \ord(a_j)$ (all $j \ne i$),
then $a_1 + \dots + a_n \ne 0$.
\end{enumerate*}
\end{exercise}
\begin{exercise}[3.6 of~\cite{fulton}]
Irreducible curves with given tangent lines~$L_i$ of multiplicity~$r_i$
may be constructed as follows:
if $\sum r_i = m$, let $F = \prod L_i^{r_i} + F_{m+1}$,
where $F_{m+1}$ is chosen to make $F$ irreducible
(see Problem~2.34 of~\cite{fulton}).
\end{exercise}
\begin{exercise}[3.11 of~\cite{fulton}]
Let $V \subset \AA^n$ be an affine variety, $P \in V$.
The \emph{tangent space} $T_P(V)$ is defined to be
$\{(v_1, \dots, v_n) \in \AA^n \mid
\text{for all $G \in I(V)$, $\sum G_{X_i}(P)v_i = 0$}\}$.
If $V = V(F)$ is a hypersurface, $F$ irreducible,
show that $T_P(V) = \{(v_1, \dots, v_n) \mid \sum F_{X_i}(P)v_i = 0\}$.
How does the dimension of~$T_P(V)$ relate to the multiplicity of~$F$ at~$P$?
\end{exercise}
\begin{exercise}[3.12 of~\cite{fulton}]
A simple point~$P$ on a curve~$F$ is called a \emph{flex} if
$\ord_P^F(L) \ge 3$,
where $L$ is the tangent to~$F$ at~$P$.
The flex is called \emph{ordinary} if $\ord_P^F(L) = 3$,
a \emph{higher} flex otherwise.
\begin{enumerate*}[label=(\alph*)]
\item Let $F = Y - X^n$. For which $n$ does $F$ have a flex at $P = (0,0)$,
and what kind of flex?
\item Suppose $P = (0,0)$, $L = Y$ is the tangent line,
$F = Y + aX^2 + \dots$.
Show that $P$ is a flex on~$F$ if and only if $a = 0$.
Give a simple criterion for calculating $\ord_P^F(Y)$,
and therefore determining if~$P$ is a higher flex.
\end{enumerate*}
\end{exercise}
\begin{exercise}[3.16 of~\cite{fulton}]
Let $F \in k[X_1, \dots, X_r]$ define a hypersurface in~$\AA^r$.
Write $F = F_m + F_{m-1} + \dots$,
and let $m = \nu_P(F)$ where $P = (0,0)$.
Suppose that $F$ is irreducible, and let $\mathcal{O} = \mathcal{O}_P(V(F))$,
$\mathfrak{m}$ its maximal ideal.
Show that $\chi(n) = \dim_k(\mathcal{O}/\mathfrak{m}^n)$
is a polynomial of degree $r - 1$
for sufficiently large~$n$,
and that the leading coefficient of~$\chi$ is
$\nu_P(F)/(r - 1)!$.
Can you find a definition for the multiplicity of a local ring
that makes sense in all the cases you know?
\end{exercise}
\begin{exercise}
Let $K$ be a field with a valuation as in Definition~5.3.
Prove that
\begin{enumerate}
\item $R = \{z \in K \mid v(z) >= 0\}$ is a ring.
\item For all $z \in K^*$ we have $z \in R$ or $z^{-1} \in R$.
\item For all $z, w \in R$ we have $v(z) \le v(w)$ iff $w \in (z)$.
\item The group of units of $R$ is $R^* = \{z \in R \mid v(z) = 0\}$
\item $R$ is a local ring with maximal ideal
$\mathfrak{m} = \{z \in R \mid v(z) > 0\}$
and quotient field $K$.
\item $R$ is a PID.
\end{enumerate}
\end{exercise}
\section{Sheet 4 --- 02 June 2020}
\begin{exercise}[3.14 of~\cite{fulton}]
Let $V = V(X^2 - Y^3, Y^2 - Z^3) \subset \AA^3$,
$P = (0,0,0)$, $\mathfrak m = \mathfrak m_P(V)$.
Find $\dim_k(\mathfrak m/ \mathfrak m^2)$.
(See problem~1.40 of~\cite{fulton}.)
\end{exercise}
\begin{exercise}[3.17 of~\cite{fulton}]
Find the intersection numbers of various pairs of curves
from the examples of Section~1 of \cite{fulton},
at the point $P = (0,0)$.
\end{exercise}
\begin{exercise}[3.23 of~\cite{fulton}]
A point $P$ on a curve~$F$ is called a \emph{hypercusp}
if $\nu_P(F) > 1$,
$F$ has only one tangent line~$L$ at $P$,
and $I(P, L \cap F) = \nu_P(f) + 1$.
Generalize the results of problem~3.22 of~\cite{fulton} to this case.
\end{exercise}
\begin{exercise}[3.24 of~\cite{fulton}]
The object of this problem
is to find a property of the local ring $\mathcal O_P(F)$
that determines whether or not $P$ is an ordinary multiple point on~$F$.
Let $F$ be an irreducible plane curve,
$P = (0,0)$, $\nu = \nu_P(F) > 1$.
Let $\mathfrak m = \mathfrak m_P(F)$.
For $G \in k[X,Y]$,
denote its residue in $\Gamma(F)$ by~$g$;
and for $g \in \mathfrak m$,
denote its residue in $\mathfrak m/\mathfrak m^2$ by~$\bar g$.
\begin{enumerate*}[label=(\alph*)]
\item Show that the map from
$\{\text{forms of degree~$1$ in $k[X,Y]$}\}$ to $\mathfrak m/\mathfrak m^2$
taking $aX + bY$ to $\overline{ax + by}$ is an isomorphism of vector spaces
(see problem~3.13 of~\cite{fulton}).
\item Suppose $P$ is an ordinary multiple point,
with tangents $L_1, \dots, L_\nu$.
Show that $I(P, F \cap L_i) > \nu$
and $\bar l_i \ne \lambda \bar l_j$ for all $i \ne j$, all $\lambda \in k$.
\item Suppose there are $G_1, \dots, G_\nu \in k[X,Y]$
such that $I(P, F \cap G_i) > \nu$
and $\bar g_i \ne \lambda \bar g_j$ for all $i \ne j$,
and all $\lambda \in k$.
Show that $P$ is an ordinary multiple point on~$F$.
(\emph{Hint}: Write $G_i = L_i + \text{higher terms}$,
$\bar l_i = \bar g_i \ne 0$,
and $L_i$ is the tangent to~$G_i$,
so $L_i$ is tangent to~$F$ by property~(5) of intersection numbers.
Thus $F$ has $\nu$ tangents at~$P$.)
\item Show that $P$ is an ordinary multiple point on~$F$
if and only if there are $g_1, \dots g_\nu \in \mathfrak m$
such that $\bar g_i \ne \lambda \bar g_j$ for all $i \ne j$, $\lambda \in k$,
and $\dim \mathcal O_P(F) / (g_i) > \nu$.
\end{enumerate*}
\end{exercise}
\begin{exercise}[important]
Let $F$ be an irreducible cubic curve.
Show that $F$ has at most one multiple point.
Show that such a multiple point must be either a node or a cusp.
\end{exercise}
\begin{exercise}
Let $F$ and $G$ be irreducible plane curves, and let $P$ be a point.
Check that the intersection product $I(P,F \cap G)$
as given in Definition~6.2 of Lecture~7
satisfies properties (3), (4), and~(7) given at the end of Lecture~7.
\end{exercise}
\section{Sheet 5 --- 09 June 2020}
\begin{exercise}[important, 4.28 of~\cite{fulton}]
For simplicity of notation,
in this problem we let
$X_0, \dots, X_n$ be coordinates for $\PP^n$,
$Y_0, \dots, Y_m$ be coordinates for $\PP^m$,
$T_{00}, T_{01}, \dots, T_{0m}, T_{10}, \dots, T_{nm}$
coordinates for~$\PP^N$,
where $N = (n+1)(m+1) - 1 = n + m + nm$.
Define $S \colon \PP^n \times \PP^m \to \PP^N$
by the formula:
\[
S([x_0 : \dots : x_n], [y_0 : \dots : y_m]) =
[x_0y_0 : x_0y_1 : \dots : x_ny_m.
\]
$S$ is called the \emph{Segre embedding}
of $\PP^n \times \PP^m$ in $\PP^{n+m+nm}$.
\begin{enumerate*}[label=(\alph*)]
\item Show that $S$ is a well-defined, one-to-one mapping.
\item Show that if $W$ is an algebraic subset of~$\PP^N$,
then $S^{-1}(W)$ is an algebraic subset of $\PP^n \times \PP^m$.
\item Let
$V = V(\{T_{ij}T_{kl} - T_{il}T_{kj} \mid
i,k = 0, \dots, n; j,l = 0, \dots, m\}) \subset \PP^N$.
Show that $S(\PP^n \times \PP^m) = V$.
In fact, $S(U_i \times U_j) = V \cap U_{ij}$,
where $U_{ij} = \{[t] \mid t_{ij} \ne 0\}$.
\item Show that $V$ is a variety.
\end{enumerate*}
\end{exercise}
\begin{exercise}[important]
Show that the Segre embedding $S(\PP^1 \times \PP^1) \subset \PP^3$
is the quadric surface.
On the quadric surface, there are two families of lines.
Show that two lines intersect if they come from different families,
and are parallel if they are from the same family.
\end{exercise}
\begin{exercise}[important, after 4.26 of~\cite{fulton}]
\begin{enumerate*}[label=(\alph*)]
\item Define maps
$\phi_{i,j} \colon \AA^{n+m} \to U_i \times U_j \subset \PP^n \times \PP^m$.
Using $\phi_{n+1,m+1}$,
define the ``biprojective closure'' of an algebraic set in~$\AA^{n+m}$.
Choose two items of Proposition~3 of~\S4.3 in~\cite{fulton},
and prove their analogues in the current setting.
\item Generalize part~(a) to maps
$\phi \colon \AA^{n_1} \times \AA^{n_r} \times \AA^m \to
\PP^{n_1} \times \PP^{n_r} \times \AA^m$.
Show that this sets up a correspondence between
$\{\text{nonempty affine varieties in $\AA^{n_1 + \dots + m}$}\}$
and
$\{\text{varieties in $\PP^{n_1} \times \dots \times \AA^m$
that intersect $U_{n_1+1} \times \dots \times \AA^m$}\}$.
Show that this correspondence preserves function fields and local rings.
\end{enumerate*}
\end{exercise}
\begin{exercise}[4.27 of~\cite{fulton}]
Show that the pole set of a rational function on a variety
in any multispace is an algebraic subset.
\end{exercise}
\begin{exercise}[4.19 of~\cite{fulton}]
If $I = (F)$ is the ideal of an affine hypersurface, show that $I^* = (F^*)$.
\end{exercise}
\begin{exercise}[4.20 of~\cite{fulton}]
Let $V = V(Y- X^2, Z - X^3) \subset \AA^3$.
Prove:
\begin{enumerate}[label=(\alph*)]
\item $I(V) = (Y - X^2, Z - X^3)$.
\item $ZW - XY \in I(V)^* \subset k[X,Y,Z,W]$,
but $ZW - XY \notin ((Y - X^2)^*, (Z - X^3)^*)$.
\end{enumerate}
So if $I(V) = (F_1, \dots, F_r)$,
it does not follow that $I(V)^* = (F_1^*, \dots, F_r^*)$.
\end{exercise}
\begin{exercise}[4.11 of~\cite{fulton}]
A set $V \subset \PP^n(k)$ is called a
\emph{linear subvariety} of~$\PP^n(k)$
if $V = V(H_1, \dots, H_r)$,
where each $H_i$ is a form of degree~$1$.
\begin{enumerate*}[label=(\alph*)]
\item Show that if $T$ is a projective change of coordinates,
then $V^T = T^{-1}(V)$ is also a linear subvariety.
\item Show that there is a projective change of coordinates~$T$
of~$\PP^n$ such that
$V^T = V(X_{m+2}, \dots, X_{n+1}$),
so $V$ is a variety.
\item Show that the $m$ that appears in part~(b) is independent
of the choice of~$T$.
It is called the \emph{dimension} of~$V$ ($m = -1$ if $V = \varnothing$).
\end{enumerate*}
\end{exercise}
\begin{exercise}[5.2 of~\cite{fulton}, see also Lecture~9, part of Example~7.14]
Show that the following curves are irreducible;
find their multiple points,
and the multiplicities and tangents at the multiple points.
\begin{enumerate}[label=(\alph*)]
\item $XY^4 + YZ^4 + XZ^4$.
\item $X^2Y^3 + X^2Z^3 + Y^2Z^3$.
\item $Y^2Z - X(X-Z)(X - \lambda Z)$, $\lambda \in k$.
\item $X^n + Y^n + Z^n$, $n > 0$.
\end{enumerate}
\end{exercise}
\section{Sheet 6 --- 16 June 2020}
\begin{exercise}[5.5 of~\cite{fulton}]
Let $P = [0 : 1 : 0]$,
$F$ a curve of degree~$n$, $F = \sum F_i(X,Z)Y^{n-i}$,
$F_i$ a form of degree~$i$.
Show that $\nu_P(F)$ is the smallest~$\nu$ such that $F_\nu \ne 0$,
and the factors of~$F_\nu$ determine the tangents to~$F$ at~$P$.
\end{exercise}
\begin{exercise}[5.18 of~\cite{fulton}]
Show that there is only one conic passing through the five points
$[0 : 0 : 1]$, $[0 : 1 : 0]$, $[1 : 0 : 0]$, $[1 : 1 : 1]$, and $[1 : 2 : 3]$;
show that it is nonsingular.
\end{exercise}
\begin{exercise}[5.23 of~\cite{fulton}, important, slightly changed]
A problem about flexes (see Problem~3.12 of~\cite{fulton}):
Let $F$ be a projective plane curve of degree~$n$,
and assume $F$ contains no lines.
Let $F_i = F_{X_i}$ and $F_{ij} = F_{X_iX_j}$,
forms of degree $n - 1$ and $n - 2$ respectively.
Form a $3 \times 3$ matrix
with the entry in the $(i, j)$th place being~$F_{ij}$.
Let $h$ be the determinant of this matrix,
a form of degree~$3(n - 2)$.
This $H$ is called the \emph{Hessian} of~$F$.
\begin{enumerate}
\item
Show that $H$ vanishes identically on an irreducible plane curve
iff the curve is a line.
\end{enumerate}
\noindent
The following theorem shows the relationship between flexes and the Hessian.
\medskip\noindent
\textbf{Theorem.} ($\text{char}(k) = 0$)
\begin{enumerate}[label=(\roman*)]
\item
$P \in H \cap F$ if and only if $P$ is either a flex
or a multiple point of~$F$.
\item
$I(P, H \cap F) = 1$ if and only if $P$ is an ordinary flex.
\end{enumerate}
Outline of the proof.
\begin{enumerate}[start=2]
\item
Let $T$ be a projective change of coordinates.
Then the Hessian of~$F^T = (\det(T))^2(H^T)$.
So we can assume $P = [0 : 0 : 1]$;
write $f(X, Y) = F(X, Y, 1)$
and $h(X, Y) = H(X, Y, 1)$.
\item
$(n - 1)F_j = \sum_i X_i F_{ij}$. (Use Euler's Theorem.)
\item
$I(P, f \cap h) = I(P, f \cap g)$
where $g = f_y^2 f_{xx} + f_x^2 f_{yy} - 2 f_x f_y f_{xy}$.
(\emph{Hint}: Perform row and column operations on the matrix for~$h$.
Add $x$ times the first row plus $y$ times the second row to the third row,
then apply the preceding part.
Do the same with the columns.
Then calculate the determinant.)
\item
If $P$ is a multiple point on~$F$, then $I(P, f \cap g) > 1$.
\item
Suppose $P$ is a simple point,
$Y = 0$ is the tangent line to $F$ at~$P$,
so $f = y + ax^2 + bxy + cy^2 + ex^2y + \dots$.
Then $P$ is a flex if and only if $a = 0$,
and $P$ is an ordinary flex if and only if $a = 0$ and $d \ne 0$.
A short calculation shows that
$g = 2a + 6dx + (8ac - 2b^2 + 2e)y + \text{higher terms}$,
which concludes the proof.
\end{enumerate}
\medskip\noindent
\textbf{Corollary.}
\begin{enumerate}[label=(\textit{\roman*})]
\item
A nonsingular curve of degree $ > 2$ always has a flex.
\item
A nonsingular cubic has nine flexes, all ordinary.
\end{enumerate}
\end{exercise}
\begin{exercise}[5.26 of~\cite{fulton}]
($\text{char}(k) = 0$)
Let $F$ be an irreducible curve of degree $n$ in $\PP^2$.
Suppose $P \in \PP^2$,
with $\nu_P(F) = r \ge 0$.
Then for all but a finite number of lines~$L$ through~$P$,
$L$ intersects $F$ in $n - r$ distinct points other than~$P$.
We outline a proof:
\begin{enumerate}
\item
We may assume $P = [0 : 1 : 0]$.
If $L_\lambda = \{[\lambda : t : 1] \mid t \in k\} \cup \{P\}$,
we need only consider the~$L_\lambda$.
Then $F = A_r(X,Z)Y^{n-r} + \dots + A_n(X, Z)$, $A_r \ne 0$.
(See Problems~4.24, 5.5 of~\cite{fulton}.)
\item
Let $G_\lambda(t) = F(\lambda, t, 1)$.
It is enough to show that for all but a finite number of~$\lambda$,
$G_\lambda$ has $n - r$ distinct points.
\item
Show that $G_\lambda$ has $n - r$ distinct roots if $A_r(\lambda,1) \ne 0$,
and $F \cap F_Y \cap L_\lambda = \{P\}$
(see Problem~1.53 of~\cite{fulton}).
\end{enumerate}
\end{exercise}
\section{Sheet 7 --- 23 June 2020}
\begin{exercise}[5.33 of~\cite{fulton}]
Let $C$ be an irreducible cubic,
$L$ a line such that $L \bullet C = P_1 + P_2 + P_3$,
$P_i$ distinct.
Let $L_i$ be the tangent line to~$C$ at~$P_i$:
$L_i \bullet C = 2P_i + Q_i$ for some $Q_i$.
Show that $Q_1$, $Q_2$, $Q_3$ lie on a line.
($L^2$ is a conic!)
\end{exercise}
\begin{exercise}[5.37 of~\cite{fulton}]
Suppose $\mathcal O$ is a flex on~$C$.
\begin{enumerate*}[label=(\alph*)]
\item Show that the flexes form a subgroup of~$C$;
as an abelian group,
this subgroup is isomorphic to $\mathbb Z/(3) \times \mathbb Z/(3)$.
\item Show that the flexes
are exactly the elements of order three in the group.
(I.e.,
exactly those elements~$P$ such that $P \oplus P \oplus P = \mathcal O$.)
\item Show that a point~$P$ is of order two in the group
if and only if the tangent to~$C$ at $P$ passes through~$\mathcal O$.
\item Let $C = Y^2Z - X(X - Z)(X - \lambda Z)$, $\lambda \ne 0,1$,
$\mathcal O = [0 : 1 : 0]$.
Find the points of order two.
\item Show that the points of order two on a nonsingular cubic form a group
isomorphic to $\mathbb Z/(2) \times \mathbb Z/(2)$.
\item Let $C$ be a nonsingular cubic, $P \in C$.
How many lines through $P$ are tangent to $C$ at some point $Q \ne P$?
(The answer depends on whether $P$ is a flex.)
\end{enumerate*}
\end{exercise}
\begin{exercise}[5.41 of~\cite{fulton}]
Let $C$ be a nonsingular cubic,
$\mathcal O$ a flex on~$C$.
Let $P_1, \dots, P_{3m} \in C$.
Show that $P_1 \oplus \dots \oplus P_{3m} = \mathcal O$
if and only if there is a curve~$F$ of degree~$m$
such that $F \bullet C = \sum_{i=1}^{3m} P_i$.
(\emph{Hint}: Use induction on~$m$.
Let $L \bullet C = P_1 + P_2 + Q$,
$L' \bullet C = P_3 + P_4 + R$,
$L'' \bullet C = Q + R + S$,
and apply induction to $S, P_5, \dots, P_{3m}$;
use Noether's Theorem.)
\end{exercise}
\begin{exercise}[5.43 of~\cite{fulton}]
For which points $P$ on a nonsingular cubic~$C$
does there exist a nonsingular conic that intersects~$C$ only at~$P$.
\end{exercise}
\begin{exercise}[6.14 of~\cite{fulton}]
Let $X$, $Y$ be varieties,
$F \colon X \to Y$ a mapping.
Let $X = \bigcup_\alpha U_\alpha$,
$Y = \bigcup_\alpha V_\alpha$,
with $U_\alpha$, $V_\alpha$ open subvarieties,
and suppose $f(U_\alpha) \subset V_\alpha$ for all $\alpha$.
\begin{enumerate*}[label=(\alph*)]
\item Show that $f$ is a morphism if and only if each restriction
$f_\alpha \colon U_\alpha \to V_\alpha$ of~$f$ is a morphism.
\item If each $U_\alpha$, $V_\alpha$ is affine,
$f$ is a morphism
if and only if each $\tilde f(\Gamma(V_\alpha)) \subset \Gamma(U_\alpha)$.
\end{enumerate*}
\end{exercise}
\begin{exercise}[6.16 of~\cite{fulton}]
Let $f \colon X \to Y$ be a morphism of varieties,
$X' \subset X$, $Y' \subset Y$ subvarieties
(open or closed).
Assume $f(X') \subset Y'$.
Then the restriction of~$f$ to~$X'$
is a morphism from $X'$ to~$Y'$.
(Use Problems~6.14 and~2.9 of~\cite{fulton}.)
\end{exercise}
\section{Sheet 8 --- 30 June 2020}
\begin{exercise}
Show that the twisted cubic (projective or affine, whichever you prefer)
has dimension $1$ and is therefore a curve (i.e. variety of dimension $1$).
\end{exercise}
\begin{exercise}[6.40 of~\cite{fulton}]
If there is a dominating rational map from $X$ to~$Y$,
then $\dim(Y) \le \dim(X)$.
\end{exercise}
\begin{exercise}[6.41 of~\cite{fulton}]
Every $n$-dimensional variety is birationally equivalent
to a hypersurface in~$\AA^{n+1}$ (or~$\PP^{n+1}$).
\end{exercise}
\begin{exercise}[6.43 of~\cite{fulton}]
Let $C$ be a projective curve, $P \in C$.
Then there is a birational morphism $f \colon C \to C'$,
$C'$ a projective plane curve,
such that $f^{-1}(f(P)) = \{P\}$.
We outline a proof:
\begin{enumerate}[label=(\alph*)]
\item
We can assume:
$C \subset \PP^{n+1}$.
Let $T, X_1, \dots, X_n, Z$ be coordinates for $\PP^{n+1}$;
Then $C \cap V(T)$ is finite;
$C \cap V(T,Z) = \varnothing$;
$P = [0: \dots : 0 : 1]$;
and $k(C)$ is algebraic over~$k(u)$,
where $u = \bar T / \bar Z \in k(C)$.
\item
For each $\lambda = (\lambda_1, \dots, \lambda_n) \in k^n$,
let $\phi_\lambda \colon C \to \PP^2$
be defined by the formula
$\phi([t : x_1 : \dots : x_n : z]) = [t : \sum \lambda_i x_i : z]$.
Then $\phi_\lambda$ is a well-defined morphism,
and $\phi_\lambda(P) = [0 : 0 : 1]$.
Let $C'$ be the closure of $\phi_\lambda(C)$.
\item
The variable $\lambda$ can be chosen so $\phi_\lambda$
is a birational morphism from~$C$ to~$C'$,
and $\phi_\lambda^{-1}([0 : 0 : 1]) = \{P\}$.
(Use Problem~6.32 of~\cite{fulton}
and the fact that $C \cap V(T)$ is finite.)
\end{enumerate}
\end{exercise}
\section{Sheet 9 --- 07 July 2020}
Assume $k$ is algebraically closed of characteristic~$0$.
\begin{exercise}[6.43 of~\cite{fulton}, important]
Let $C$ be a projective curve, $P \in C$.
Then there is a birational morphism $f \colon C \to C'$,
$C'$ a projective plane curve,
such that $f^{-1}(f(P)) = \{P\}$.
We outline a proof:
\begin{enumerate}[label=(\alph*)]
\item
We can assume:
$C \subset \PP^{n+1}$.
Let $T, X_1, \dots, X_n, Z$ be coordinates for $\PP^{n+1}$;
Then $C \cap V(T)$ is finite;
$C \cap V(T,Z) = \varnothing$;
$P = [0: \dots : 0 : 1]$;
and $k(C)$ is algebraic over~$k(u)$,
where $u = \bar T / \bar Z \in k(C)$.
\item
For each $\lambda = (\lambda_1, \dots, \lambda_n) \in k^n$,
let $\phi_\lambda \colon C \to \PP^2$
be defined by the formula
$\phi([t : x_1 : \dots : x_n : z]) = [t : \sum \lambda_i x_i : z]$.
Then $\phi_\lambda$ is a well-defined morphism,
and $\phi_\lambda(P) = [0 : 0 : 1]$.
Let $C'$ be the closure of $\phi_\lambda(C)$.
\item
The variable $\lambda$ can be chosen so $\phi_\lambda$
is a birational morphism from~$C$ to~$C'$,
and $\phi_\lambda^{-1}([0 : 0 : 1]) = \{P\}$.
(Use Problem~6.32 of~\cite{fulton}
and the fact that $C \cap V(T)$ is finite.)
\end{enumerate}
\end{exercise}
\begin{exercise}[6.46 of~\cite{fulton}, important]
Let $k(\PP^1) = k(T)$, $T = X/Y$ (see Problem~4.8 of~\cite{fulton}).
For any variety~$V$, and $f \in k(V)$, $f \notin k$,
the subfield $k(f)$ generated by $f$ is naturally isomorphic to~$K(T)$.
Thus a nonconstant $f \in k(V)$
corresponds to a homomorphism from $k(T)$ to~$k(V)$,
and hence to a dominating rational map from~$V$ to~$\PP^1$.
The corresponding map is usualy denoted also by~$f$.
If this rational map is a morphism,
show that the pole set of~$f$ is $f^{-1}([1:0])$.
\end{exercise}
\begin{exercise}[7.2 of~\cite{fulton}]
\begin{enumerate*}[label=(\alph*)]
\item
For each of the curves~$F$ in~\S3.1 of~\cite{fulton}, find~$F'$;
show that $F'$ is nonsingular in the first five examples,
but not in the sixth.
\item
Let $F = Y^2 - X^5$. What is $F'$? What is $(F')'$?
What must be done to resolve the singularity of the curve $Y^2 = X^{2n+1}$?
\end{enumerate*}
\end{exercise}
\begin{exercise}[7.6 of~\cite{fulton}]
If $P$ is an ordinary cusp on~$C$, show that $f^{-1}(P) = \{P_1\}$,
where $P_1$ is a simple point on~$C'$.
\end{exercise}
\section{Sheet 10 --- 14 July 2020}
\begin{exercise}
Show that a nonconstant morphism between two curves has finite fibre.
\end{exercise}
\begin{exercise}
Show that principal divisors form a subgroup of $\Div C$.
\end{exercise}
\begin{exercise}
Make sure you understand the sequence of equalities
in the last step of the proof of Lemma~12.9,~ii), Lecture~20
(specifically the third ``='').
\end{exercise}
\begin{exercise}
Show that if $C$ is in good position, then so is $C'$
(see notation from Lecture~19 and hints therein).
\end{exercise}
\begin{exercise}[7.12 of~\cite{fulton}]
Find a quadratic transformation of $Y^2Z^2 - X^4 - Y^4$
with only ordinary multiple points.
Do the same with $Y^4 + Z^4 - 2X^2(Y - Z)^2$.
\end{exercise}
\section{Sheet 11 --- 21 July 2020}
\begin{exercise}
\begin{enumerate}
\item
Show that if a rational map of curves has degree one,
then it is a birational map.
\item
If $f_0, \dots, f_r$ are functions in~$k(C)$ for some smooth curve~$C$
that are regular and do not vanish simultaneously on points of~$C$,
then $[f_0 : \dots : f_r]$ is a morphism from $C$ to~$\PP^r$.
\end{enumerate}
\end{exercise}
\begin{exercise}
Prove Lemma~12.16 from Lecture~22.
\end{exercise}
\begin{exercise}
Prove parts i) and iii) of Proposition~12.18 from Lecture~22.
\end{exercise}
\begin{exercise}
If $P$ is a base point of a divisor $D$ on a curve, then $L(D) = L(D-P)$.
\end{exercise}
\begin{thebibliography}{1}
\bibitem{fulton}
Fulton, William.
Algebraic curves. An introduction to algebraic geometry.
Notes written with the collaboration of Richard Weiss.
Reprint of 1969 original.
Advanced Book Classics.
{\it Addison-Wesley Publishing Company,
Advanced Book Program,
Redwood City, CA}\/,
1989.
{xxii}+226 pp.
ISBN: 0-201-51010-3
\end{thebibliography}
\end{document}