Algebraic Curves: An Introduction to Algebraic Geometry by William Fulton

By William Fulton

Third Preface, 2008

This textual content has been out of print for a number of years, with the writer conserving copyrights.
Since I proceed to listen to from younger algebraic geometers who used this as
their first textual content, i'm happy now to make this variation on hand at no cost to anyone
interested. i'm so much thankful to Kwankyu Lee for creating a cautious LaTeX version,
which used to be the root of this variation; thank you additionally to Eugene Eisenstein for aid with
the graphics.

As in 1989, i've got controlled to withstand making sweeping alterations. I thank all who
have despatched corrections to previous types, in particular Grzegorz Bobi´nski for the most
recent and thorough record. it truly is inevitable that this conversion has brought some
new blunders, and that i and destiny readers should be thankful if you happen to will ship any error you
find to me at

Second Preface, 1989

When this publication first seemed, there have been few texts on hand to a beginner in modern
algebraic geometry. given 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 previous 20 years have additionally noticeable a great deal of development in our understanding
of the subjects coated during this textual content: linear sequence on curves, intersection conception, and
the Riemann-Roch challenge. it's been tempting to rewrite the booklet to mirror this
progress, however it doesn't appear attainable to take action with out forsaking its elementary
character and destroying its unique objective: to introduce scholars with a bit algebra
background to some of the information 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 ebook and its
exercises is helping arrange a reader for any of the texts pointed out above, that may be an
added benefit.

First Preface, 1969

Although algebraic geometry is a hugely constructed and thriving box of mathematics,
it is notoriously tough for the newbie to make his means into the subject.
There are numerous texts on an undergraduate point that provide a superb therapy of
the classical thought of airplane curves, yet those don't arrange the coed adequately
for smooth algebraic geometry. however, so much books with a latest approach
demand massive heritage in algebra and topology, usually the equivalent
of a yr or extra of graduate learn. the purpose of those notes is to strengthen the
theory of algebraic curves from the perspective of recent algebraic geometry, but
without over the top prerequisites.

We have assumed that the reader is aware a few uncomplicated houses of rings,
ideals, and polynomials, akin 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 want from algebra. the remainder of the chapter
is desirous about easy homes of affine algebraic units; we've given Zariski’s
proof of the $64000 Nullstellensatz.

The coordinate ring, functionality box, and native jewelry of an affine sort are studied
in bankruptcy 2. As in any glossy therapy of algebraic geometry, they play a fundamental
role in our instruction. the overall research of affine and projective varieties
is persisted in Chapters four and six, yet merely so far as valuable for our research 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 aircraft curves at some degree is characterised via its
properties, and a definition when it comes to a definite residue classification ring of an area ring is
shown to have those homes. Bézout’s Theorem and Max Noether’s Fundamental
Theorem are the topic of bankruptcy five. (Anyone accustomed to the cohomology of
projective kinds 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 validated. 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 path taught to Juniors at Brandeis collage in 1967–
68. The direction was once repeated (assuming all of the algebra) to a bunch of graduate students
during the extensive week on the finish of the Spring semester. now we have 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 set algebraically closed box. each time convenient
(including with out 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 as a result of Richard Weiss, a pupil within the path, for sharing the task
of writing the notes. He corrected many blunders and more suitable the readability of the text.
Professor PaulMonsky supplied numerous worthwhile 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 most appropriate 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 l. a. justesse de ma multiplication, je n’en voulus rien croire
jusqu’à ce que j’eusse fai los angeles 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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X n+1 ) = 0 for every choice of homogeneous coordinates (x 1 , . . , x n+1 ) for P ; we then write F (P ) = 0. If F is a form, and F vanishes at one representative of P , then it vanishes at every representative. 2). For any set S of polynomials in k[X 1 , . . , X n+1 ], we let V (S) = {P ∈ Pn | P is a zero of each F ∈ S}. If I is the ideal generated by S, V (I ) = V (S). If I = (F (1) , . . , F (r ) ), where F (i ) = F j(i ) , F j(i ) a form of degree j , then V (I ) = V ({F j(i ) }), so V (S) = V ({F j(i ) }) is the set of zeros of a finite number of forms.

Show that k[[X ]] is a DVR with uniformizing parameter X . Its quotient field is denoted k((X )). 32. 30. Any z ∈ R then determines a power series λi X i , if λ0 , λ1 , . . 30(b). (a) Show that the map z → λi X i is a one-to-one ring homomorphism of R into k[[X ]]. We often write z = λi t i , and call this the power series expansion of z in terms of t . (b) Show that the homomorphism extends to a homomorphism of K into k((X )), and that the order function on k((X )) restricts to that on K . 24, t = X .

0)}; (iii) Rad(I ) = I a (Va (I )) ⊃ (X 1 , . . , X n+1 ) (by the affine Nullstellensatz); and (iv) (X 1 , . . 41). (2) I p (Vp (I )) = I a (C (Vp (I ))) = I a (Va (I )) = Rad(I ). The usual corollaries of the Nullstellensatz go through, except that we must always make an exception with the ideal (X 1 , . . , X n+1 ). In particular, there is a oneto-one correspondence between projective hypersurfaces V = V (F ) and the (nonconstant) forms F that define V provided F has no multiple factors (F is determined up to multiplication by a nonzero λ ∈ k).

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