By William Fulton

ISBN-10: 0201510103

ISBN-13: 9780201510102

ISBN-10: 0805330828

ISBN-13: 9780805330823

Preface

Third Preface, 2008

This textual content has been out of print for a number of years, with the writer retaining copyrights.

Since I proceed to listen to from younger algebraic geometers who used this as

their first textual content, i'm comfortable 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 was once the foundation 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 prior types, specially Grzegorz Bobi´nski for the most

recent and thorough record. it's inevitable that this conversion has brought some

new blunders, and that i and destiny readers might be thankful in the event you will ship any blunders you

find to me at wfulton@umich.edu.

Second Preface, 1989

When this ebook first seemed, there have been few texts on hand to a beginner in modern

algebraic geometry. on the grounds that then many introductory treatises have seemed, including

excellent texts by means of 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 progress in our understanding

of the themes coated during this textual content: linear sequence on curves, intersection thought, and

the Riemann-Roch challenge. it's been tempting to rewrite the publication to mirror this

progress, however it doesn't look 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 tips 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 booklet and its

exercises is helping organize a reader for any of the texts pointed out above, that may be an

added benefit.

PREFACE

First Preface, 1969

Although algebraic geometry is a hugely built and thriving box of mathematics,

it is notoriously tough for the newbie to make his means into the subject.

There are a number of texts on an undergraduate point that provide a good remedy of

the classical idea of airplane curves, yet those don't organize the scholar adequately

for sleek algebraic geometry. nevertheless, so much books with a latest approach

demand huge heritage in algebra and topology, frequently the equivalent

of a 12 months or extra of graduate examine. the purpose of those notes is to improve the

theory of algebraic curves from the perspective of contemporary 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 frequently lined in a one-semester direction in modern

algebra; extra commutative algebra is constructed in later sections. Chapter

1 starts off with a precis of the proof we want from algebra. the remainder of the chapter

is excited about uncomplicated homes of affine algebraic units; we've got given Zariski’s

proof of the real Nullstellensatz.

The coordinate ring, functionality box, and native jewelry of an affine style are studied

in bankruptcy 2. As in any sleek therapy of algebraic geometry, they play a fundamental

role in our education. the overall research of affine and projective varieties

is endured in Chapters four and six, yet basically so far as worthy for our learn of curves.

Chapter three considers affine airplane curves. The classical definition of the multiplicity

of some extent on a curve is proven to count simply 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 by way of a definite 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 accustomed to 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 built via blowing

up issues, and the correspondence among algebraic functionality fields on one

variable and nonsingular projective curves is tested. within the concluding chapter

the algebraic strategy of Chevalley is mixed with the geometric reasoning of

Brill and Noether to end up the Riemann-Roch Theorem.

These notes are from a path taught to Juniors at Brandeis collage in 1967–

68. The path was once repeated (assuming the entire algebra) to a gaggle of graduate students

during the extensive week on the finish of the Spring semester. we've retained

an crucial function of those classes by means of 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 purposes and extensions of the theory.

From bankruptcy three on, ok denotes a set algebraically closed box. every time convenient

(including with out remark the various difficulties) we now have 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 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 desirable the readability of the text.

Professor PaulMonsky supplied numerous valuable 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. l. a. most suitable 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 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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**Extra info for Algebraic Curves: An Introduction to Algebraic Geometry**

**Example text**

If α = ±1, they intersect in two points. When α = ±1, they no not intersect, but the curve is asymptotic to the line. We want to enlarge the plane in such a way that two such curves intersect “at infinity”. One way to achieve this is to identify each point (x, y) ∈ A2 with the point (x, y, 1) ∈ A3 . Every point (x, y, 1) determines a line in A3 that passes through (0, 0, 0) and (x, y, 1). Every line through (0, 0, 0) except those lying in the plane z = 0 corresponds to exactly one such point. The lines through (0, 0, 0) in the plane z = 0 can be thought of as corresponding to the “points at infinity”.

9. 42), and I (P, F ∩G) ≥ dimk (O /(H )). 44), and O P (H ) ⊃ Γ(H ), with Γ(H ) infinite-dimensional by Corollary 4 to the Nullstellensatz. This proves (1). To prove (6), it is enough to show that I (P, F ∩ G H ) = I (P, F ∩ G) + I (P, F ∩ H ) for any F,G, H . We may assume F and G H have no common components, since the result is clear otherwise. 42), and define a k-linear map ψ : O /(F, H ) → O /(F,G H ) by letting ψ(z) = G z, z ∈ O (the bar denotes residues). 10, it is enough to show that the sequence ψ ϕ 0 −→ O /(F, H ) −→ O /(F,G H ) −→ O /(F,G) −→ 0 is exact.

1) The following four conditions are equivalent: (i) Vp (I ) = ; (ii) Va (I ) ⊂ {(0, . . , 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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