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 completely satisfied 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 root of this version; thank you additionally to Eugene Eisenstein for support with
As in 1989, i've got controlled to withstand making sweeping alterations. I thank all who
have despatched corrections to previous models, particularly Grzegorz Bobi´nski for the most
recent and thorough record. it really is inevitable that this conversion has brought some
new blunders, and that i and destiny readers could be thankful if you happen to will ship any error you
find to me at firstname.lastname@example.org.
Second Preface, 1989
When this booklet first seemed, there have been few texts to be had to a amateur in modern
algebraic geometry. considering the fact that then many introductory treatises have seemed, including
excellent texts via Shafarevich,Mumford,Hartshorne, Griffiths-Harris, Kunz,
Clemens, Iitaka, Brieskorn-Knörrer, and Arbarello-Cornalba-Griffiths-Harris.
The prior 20 years have additionally noticeable a great deal of progress in our understanding
of the themes coated during this textual content: linear sequence on curves, intersection idea, 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 objective: to introduce scholars with a bit algebra
background to some of the guidelines of algebraic geometry and to aid them gain
some appreciation either for algebraic geometry and for origins and functions of
many of the notions of commutative algebra. If operating in the course of the e-book and its
exercises is helping organize a reader for any of the texts pointed out above, that might 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 means into the subject.
There are a number of texts on an undergraduate point that provide a superb therapy of
the classical thought of aircraft curves, yet those don't arrange the coed adequately
for glossy algebraic geometry. however, such a lot books with a contemporary approach
demand substantial heritage in algebra and topology, frequently the equivalent
of a yr or extra of graduate research. the purpose of those notes is to improve 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 is aware a few easy homes of rings,
ideals, and polynomials, akin to is usually lined in a one-semester direction in modern
algebra; extra commutative algebra is built in later sections. Chapter
1 starts off with a precis of the proof we'd like from algebra. the remainder of the chapter
is interested in easy homes of affine algebraic units; we have now 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 remedy of algebraic geometry, they play a fundamental
role in our education. the overall learn of affine and projective varieties
is persisted in Chapters four and six, yet basically so far as worthy for our examine of curves.
Chapter three considers affine airplane curves. The classical definition of the multiplicity
of some extent on a curve is proven to rely in simple terms at the neighborhood ring of the curve at the
point. The intersection variety of airplane curves at some degree is characterised by means of its
properties, and a definition when it comes to a undeniable residue classification ring of a neighborhood ring is
shown to have those homes. Bézout’s Theorem and Max Noether’s Fundamental
Theorem are the topic of bankruptcy five. (Anyone acquainted with the cohomology of
projective forms will realize that this cohomology is implicit in our proofs.)
In bankruptcy 7 the nonsingular version of a curve is developed through 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 process 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 collage in 1967–
68. The path was once repeated (assuming all of the algebra) to a gaggle of graduate students
during the extensive week on the finish of the Spring semester. we have now retained
an crucial characteristic of those classes by means of together with numerous hundred difficulties. The results
of the starred difficulties are used freely within the textual content, whereas the others diversity from
exercises to purposes and extensions of the theory.
From bankruptcy three on, okay denotes a hard and fast algebraically closed box. each time convenient
(including with no remark a few of the difficulties) now we have assumed okay to
be of attribute 0. The minor changes essential to expand the speculation to
arbitrary attribute are mentioned in an appendix.
Thanks are because of Richard Weiss, a scholar within the direction, for sharing the task
of writing the notes. He corrected many mistakes and more suitable the readability of the text.
Professor PaulMonsky supplied a number of precious 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 element 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. l. a. superior 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 l. a. 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 resources for Algebraic Curves: An Introduction to Algebraic Geometry
A) If F and G have no common tangents at P , then I t ⊂ (F,G)O for t ≥ m + n − 1. (b) ψ is one-to-one if and only if F and G have distinct tangents at P . Proof of (a). Let L 1 , . . , L m be the tangents to F at P , M 1 , . . , M n the tangents to G. Let L i = L m if i > m, M j = M n if j > n, and let A i j = L 1 · · · L i M 1 · · · M j for all i , j ≥ 0 (A 00 = 1). 35(c)). To prove (a), it therefore suffices to show that A i j ∈ (F,G)O for all i + j ≥ m +n −1. But i + j ≥ m + n − 1 implies that either i ≥ m or j ≥ n.
R n are rings,the cartesian product R 1 × · · · × R n is made into a ring as follows: (a 1 , . . , a n ) + (b 1 , . . , b n ) = (a 1 + b 1 , . . , a n + b n ), and (a 1 , . . , a n )(b 1 , . . , b n ) = (a 1 b 1 , . . , a n b n ). This ring is called the direct product of R 1 , . . , R n , and is written n n i =1 R i . The natural projection maps πi : j =1 R j → R i taking (a 1 , . . , a n ) to a i are ring homomorphisms. The direct product is characterized by the following property: given any ring R, and ring homomorphisms ϕi : R → R i , i = 1, .
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. The line L i is a simple (resp. ) tangent if r i = 1 (resp. ). If F has m distinct (simple) tangents at P , we say that P is an ordinary multiple point of F .
Algebraic Curves: An Introduction to Algebraic Geometry by William Fulton