By William Fulton
Third Preface, 2008
This textual content has been out of print for a number of years, with the writer protecting copyrights.
Since I proceed to listen to from younger algebraic geometers who used this as
their first textual content, i'm joyful now to make this variation on hand for free of charge to anyone
interested. i'm such a lot thankful to Kwankyu Lee for creating a cautious LaTeX version,
which was once the foundation of this variation; thank you additionally to Eugene Eisenstein for aid with
As in 1989, i've got controlled to withstand making sweeping adjustments. I thank all who
have despatched corrections to past types, in particular Grzegorz Bobi´nski for the most
recent and thorough record. it's inevitable that this conversion has brought some
new mistakes, and that i and destiny readers might be thankful for those who will ship any error you
find to me at firstname.lastname@example.org.
Second Preface, 1989
When this ebook first seemed, there have been few texts to be had to a amateur in modern
algebraic geometry. on the grounds that then many introductory treatises have seemed, including
excellent texts through Shafarevich,Mumford,Hartshorne, Griffiths-Harris, Kunz,
Clemens, Iitaka, Brieskorn-Knörrer, and Arbarello-Cornalba-Griffiths-Harris.
The earlier 20 years have additionally obvious a great deal of development in our understanding
of the themes lined during this textual content: linear sequence on curves, intersection thought, and
the Riemann-Roch challenge. it's been tempting to rewrite the e-book to mirror this
progress, however it doesn't appear attainable to take action with out leaving behind its elementary
character and destroying its unique function: to introduce scholars with a bit algebra
background to a couple of the information of algebraic geometry and to aid them gain
some appreciation either for algebraic geometry and for origins and purposes of
many of the notions of commutative algebra. If operating in the course of the ebook and its
exercises is helping organize a reader for any of the texts pointed out above, that would be an
First Preface, 1969
Although algebraic geometry is a hugely built and thriving box of mathematics,
it is notoriously tricky for the newbie to make his manner into the subject.
There are a number of texts on an undergraduate point that provide a superb remedy of
the classical concept of airplane curves, yet those don't organize the scholar adequately
for smooth algebraic geometry. however, such a lot books with a latest approach
demand substantial history in algebra and topology, frequently the equivalent
of a 12 months or extra of graduate learn. the purpose of those notes is to boost the
theory of algebraic curves from the point of view of recent algebraic geometry, but
without over the top prerequisites.
We have assumed that the reader knows a few uncomplicated houses of rings,
ideals, and polynomials, equivalent to is frequently coated in a one-semester path in modern
algebra; extra commutative algebra is constructed in later sections. Chapter
1 starts off with a precis of the evidence we'd like from algebra. the remainder of the chapter
is focused on simple homes of affine algebraic units; we've given Zariski’s
proof of the $64000 Nullstellensatz.
The coordinate ring, functionality box, and native earrings of an affine sort are studied
in bankruptcy 2. As in any sleek therapy of algebraic geometry, they play a fundamental
role in our guidance. the final examine of affine and projective varieties
is persisted in Chapters four and six, yet in basic terms so far as precious for our examine of curves.
Chapter three considers affine aircraft curves. The classical definition of the multiplicity
of some extent on a curve is proven to rely purely at the neighborhood ring of the curve at the
point. The intersection variety of airplane curves at some degree is characterised by way of its
properties, and a definition by way of a definite residue classification ring of a neighborhood ring is
shown to have those houses. Bézout’s Theorem and Max Noether’s Fundamental
Theorem are the topic of bankruptcy five. (Anyone acquainted with the cohomology of
projective types will realize that this cohomology is implicit in our proofs.)
In bankruptcy 7 the nonsingular version of a curve is developed via blowing
up issues, and the correspondence among algebraic functionality fields on one
variable and nonsingular projective curves is proven. within the concluding chapter
the algebraic method of Chevalley is mixed with the geometric reasoning of
Brill and Noether to turn out the Riemann-Roch Theorem.
These notes are from a direction taught to Juniors at Brandeis college in 1967–
68. The path used to be repeated (assuming the entire algebra) to a gaggle of graduate students
during the in depth week on the finish of the Spring semester. we've retained
an crucial characteristic of those classes through together with a number of hundred difficulties. The results
of the starred difficulties are used freely within the textual content, whereas the others diversity from
exercises to functions and extensions of the theory.
From bankruptcy three on, okay denotes a hard and fast algebraically closed box. every time convenient
(including with no remark the various difficulties) we've got assumed ok to
be of attribute 0. The minor changes essential to expand the idea to
arbitrary attribute are mentioned in an appendix.
Thanks are because of Richard Weiss, a scholar within the path, for sharing the task
of writing the notes. He corrected many blunders and superior the readability of the text.
Professor PaulMonsky supplied a number of important feedback as I taught the course.
“Je n’ai jamais été assez loin pour bien sentir l’application de l’algèbre à l. a. géométrie.
Je n’ai mois aspect cette manière d’opérer sans voir ce qu’on fait, et il me sembloit que
résoudre un probleme de géométrie par les équations, c’étoit jouer un air en tournant
une manivelle. los angeles prime fois que je trouvai par le calcul que le carré d’un
binôme étoit composé du carré de chacune de ses events, et du double produit de
l’une par l’autre, malgré los angeles justesse de ma multiplication, je n’en voulus rien croire
jusqu’à ce que j’eusse fai l. a. determine. Ce n’étoit pas que je n’eusse un grand goût pour
l’algèbre en n’y considérant que los angeles quantité abstraite; mais appliquée a l’étendue, je
voulois voir l’opération sur les lignes; autrement je n’y comprenois plus rien.”
Les Confessions de J.-J. Rousseau
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Extra info for Algebraic Curves: An Introduction to Algebraic Geometry
Calculate χ(n) = dimk (O /mn ). (b) Let O = O P (Ar (k)). Show that χ(n) is a polynomial of degree r in n, with leading coefficient 1/r ! 36). 16. Let F ∈ k[X 1 , . . , X n ] define a hypersurface in Ar . Write F = F m + F m+1 + · · · , and let m = m P (F ) where P = (0, 0). Suppose F is irreducible, and let O = O P (V (F )), m its maximal ideal. Show that χ(n) = dimk (O /mn ) is a polynomial of degree r − 1 for sufficiently large n, and that the leading coefficient of χ is m 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?
Let V be a vector space, W a subspace, T : V → V a one-to-one linear map such that T (W ) ⊂ W , and assume V /W and W /T (W ) are finite-dimensional. (a) Show that T induces an isomorphism of V /W with T (V )/T (W ). (b) Construct an isomorphism between T (V )/(W ∩ T (V )) and (W + T (V ))/W , and an isomorphism between W /(W ∩ T (V )) and (W + T (V ))/T (V ). 49(c) to show that dimV /(W + T (V )) = dim(W ∩ T (V ))/T (W ). (d) Conclude finally that dimV /T (V ) = dimW /T (W ). 11 Free Modules Let R be a ring, X any set.
Write F = F m +F m+1 +· · ·+F n , where F i is a form in k[X , Y ] of degree i , F m = 0. We define m to be the multiplicity of F at P = (0, 0), write m = m P (F ). Note that P ∈ F if and only if m P (F ) > 0. Using the rules for derivatives, it is easy to check that P is a simple point on F if and only if m P (F ) = 1, and in this case F 1 is exactly the tangent line to F at P . If m = 2, P is called a double point; if m = 3, a triple point, etc. 1. 6). The L i are called the tangent lines to F at P = (0, 0); r i is the multiplicity of the tangent.
Algebraic Curves: An Introduction to Algebraic Geometry by William Fulton