I have written a paragraph of text and one line is protruding out into the right margin. I am completely baffled, can anyone help?
My text is thus
"… cohomology of the base manifold. These invariants describe the obstruction to the presence of global sections, which we may interpret as a measure of the twisting of a bundle.
Today there are three main expositions of characteristic classes: Chern-Weil theory; the approach of Grothendieck; and the study of universal bundles. Each exposition has its merits and its drawbacks, and we shall not have time to present all three. We will focus initially on the Chern-Weil approach as presented in Madsen \& Tornehave \cite{Madsen} in which characteristic classes are derived by taking invariant polynomials in the curvature form, a construction well studied in differential geometry. This approach is preferred as it leads to simple proofs of the important naturality and Whitney-sum properties. We will then discuss briefly the approach of Grothendieck as presented in Bott \& Tu \cite{Bott} which deserves attention simply for its elegance. Here a class is constructed for a rank 2, oriented vector bundle, living in the cohomology of the base manifold, which is seen to pull back its information from the fibre above, counting the number of obstructions to the presence of a global section. Then, using the Leray-Hirsch theorem and some clever algebra as well as a result from cohomology, characteristic classes are derived from this for general complex vector bundles. This approach allows us to present a clear computation.
Neither of these first two expositions really address the important question of {\textit{why}} such characteristic classes…"
And here's how it is coming out!
Any ideas?
Thanks,
Adam.
Best Answer
There are several possibilitys to solve your problem.
Because you gave no MWE I created a simple one (I left out your cites, say you will see ? instead) and used package
geometry
to get the layout you showed in your picture. (Remark: It would be easier next time you provide such a MWE by your own, please!)I added package
showframe
to show the typing area.Then I added command
\hyphenation{Gro-then-dieck}
to show that only this command will give you the result you want.But you do not need
\hyphenation
if you use packagemicrotype
. Only withmicrotype
you get in this example a proper justifing left and right without a hyphenation of Grothendieck. (Play with the margins ingeometry
and you will get situations that an hyphenation occurs.)I suggest to call package
microtype
in all your documents.MWE (with activated
microtype
; comment it and uncomment the line forhyphenation
to test the other version; comment both to have your situation):and the result:
and with both uncommented lines you get an hyphenation in Grothendieck: