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\begin{center}
\bf{Changes and Corrections for Andy Stubbs}\\
\bf{Thesis Title: Hyperelliptic Curves}
\end{center}
\noindent
p3: replace $\Theta$ by $\infty$ in three occurrences.\\
p3, last line of paragraph 1: I suggest ``explicitly'' rather than
technically.\\
p4, line 2: ... but { \bf this} is only useful ...\\
p4, para 5, line 7: Euclidean.\\
p5, line 1: ... can {\bf be} more ...\\
p5, para 3, line 2: Euclidean.\\
p6, line -1: should remark here that the weights are the orders of
the poles at $\infty$.\\
p7, para 3, line -2: $-$ in the formula; line -1: a $+$ sign.\\
p7, line -1: " ... non-singular curve, {\bf the normalisation of $\C$}, ..."\\
p8, last para,
line 1: the explanation here is a little thin. The functions must
certainly be regular away from $\OO$ to blow it down but choosing ones
with poles at "infinity" then combines the rational mapping with the
covering of the Kummer. I would guess the weight you refer to is the valuation
associated with $\Theta$ (or $ \Theta^+ + \Theta^- $) pulled back to $S^2 \C$.
This connection seems to be missing in genus 3 as well.\\
line -3: "...Jacobian give{\bf n} by"\\
line -1: $F(x_1, x_2)$ not $F(x, u)$.\\
p9, (2.9): ditto.\\
p9, para 2, line 1: $.., \kappa_3$ {\bf of}.\\
p11, Definition 2.1: As discussed in the viva this needs amplification particularly
of the usage; (g-1)st power. This will then sort out the second half, I hope. It
needs to be clear that in all cases $\Theta$, or whatever, has codimension 1 ie
is a divisor.\\
p11, Conjecture 2.4: a more grammatical wording is "...equations of degree at most
four".\\
p11 line -3: "equation{\bf s}".\\
p11 line -1: " in $L({\bf 2} \Theta)$.\\
p12 line 3: "bundle $L$" (font).\\
p12 para 1, line 8: You should give a reference (at the least) where it is
established that $\phi_{\Theta}$ is a principal polarization and hence
${\rm deg}(\phi_{\Theta})=1$.\\
p12: It would be better to work entirely in terms of divisors and
avoid line bundles. $\Theta^n$ could then be replaced by $n\Theta$ here as
elsewhere.\\
p12 (2.16) ${\rm dim} L (n \Theta )$.\\
p12 last para, line 1: "...of $L (4 \Theta )$, ..."\\
p12 line -4: " ... the basis for the Jacobian is ..." This must be re-expressed.
You are saying that the basis is indexed by 2-torsion and characterise the even and
odd members. Make it clear, also, how $a.b$ is defined. It would also be better
to write $({\Bbb Z}/2{\Bbb Z})^g$\\
p13 line 1: " ... set; if $a$ ... " (font).\\
p13 line 3: $ 2^{2g-1}$.\\
p13 para 3, line 1: objects\\
p13 para 3, line 2: $z_i$. (\LaTeX problem)\\
p13 section 2.2: "Addition on the {\bf Jacobian} of a curve of genus $g$"\\
As discussed in the viva this is not in its logically correct place.
I suggest moving it to a suitable point in Chapter 6, along with section 2.3.
(which should be entitled "Overview of Addition Algorithm")
However, 2.4 could be left in its present position.\\
p13 sect 2.2 line 3: explain/define $I_{P_i} ( {\cal C}, {\cal D} )$.\\
p13 sect 2.2 para 2, line 2: $0 \le n < m_i$.\\
p20 Lemma 2.20: \LaTeX error in derivative in the first line.\\
p20 Lemma 2.21: $S^g {\cal C}$.\\
p20 last line: V{\bf a}ndermonde.\\
p21 line 2: \LaTeX error in $\frac{dx_i}{y_i}$.\\
p23 para 2, line 5: $a_0$ not $a_1.$\\
p27 Conjecture 3.10: this needs rephrasing. What is an "elementary basis"?
"modulo the function field of the curve" ?\\
p30 line -9: $\kappa_3$ (twice)\\
p30 line -7: Should begin: $ \kappa_3 = b_0^2 - ....$\\
p30 line -5: $o_n ( \kappa_3 )$.\\
p30 line -2: " ... {\bf more} general"\\
p31 line -4: " ... {\bf five} independent..."\\
p32 para 3, line 1: "computation{\bf al} aid"\\
p32 para 3, line 2: " ... the{\bf re} exists a basis ... "\\
p32 line -3: "weight of {\bf the} ... "\\
p33 line 1: "symmetrising"\\
p33 Theorem 3.13: ${\mathbb P}^8$\\
p34 para 2, line 7: delete "are".\\
p34 lines -6,-5: "However, as -1 acts trivially on the Kummer variety and $c_0, ..., c_3$
furnish an embedding of this, it may be ...."\\
Chapters 4 and 5, general: where code is interspersed with commentary could you
make the code more "arresting" by, say, having it in bold type?\\
p43 4.2, line 1: " ... to each"; delete "with".\\
p43 4.2, line 3: "a weight $d$ with $y$".\\
p44 line -13: the function ${ \tt get\underline{\;}orbit}$ has lost its parameter declaration.\\
p46 4.3 Distillation: as discussed in the viva an example or two of output to
illustrate the algorithm operating on actual monomials should be given.\\
p52 para 2, line 1: system{\bf at}ic.\\
p52 para 2, line 3: " $\kappa_0, ... , \kappa_3$.\\
p52 line -3: "... pole only at $(x,u)=(x_1,-y_1)$ but not at $(x,u)= ...$,"\\
p53 para 2, line 2: " coefficient {\bf is} ...".\\
p54 line 3: \LaTeX error "$phi_{i_m}$".\\
p63 line 3: delete "merely" and second "even".\\
p63 para 2, line 1: equation $eq[1]=0$ ?\\
p63 proof of (5.4), line 1: "Theorem 5.2"?\\
p63 lines -7, -6: " ... basis for the {\bf span} of the three ... "\\
p64 5.3 Relations in $L(4 \Theta )$ for Genus 3. (?)\\
p70 line 1: "our comp-".\\
p72 line -6: delete one "given".\\
\vspace{1in}
\hfill{John Merriman}
\end{document}