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 protecting copyrights.

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

their first textual content, i'm blissful now to make this variation to be had 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

the graphics.

As in 1989, i've got controlled to withstand making sweeping alterations. I thank all who

have despatched corrections to previous models, specifically Grzegorz Bobi´nski for the most

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

new error, and that i and destiny readers may be thankful if you happen to will ship any error you

find to me at wfulton@umich.edu.

Second Preface, 1989

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

algebraic geometry. because then many introductory treatises have seemed, including

excellent texts by way of Shafarevich,Mumford,Hartshorne, Griffiths-Harris, Kunz,

Clemens, Iitaka, Brieskorn-Knörrer, and Arbarello-Cornalba-Griffiths-Harris.

The earlier twenty years have additionally visible a great deal of progress in our understanding

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

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

progress, however it doesn't appear attainable to take action with out forsaking its elementary

character and destroying its unique goal: 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 purposes of

many of the notions of commutative algebra. If operating throughout the publication and its

exercises is helping arrange a reader for any of the texts pointed out above, that might 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 supply an outstanding remedy of

the classical idea of aircraft curves, yet those don't arrange the coed adequately

for glossy algebraic geometry. nevertheless, such a lot books with a latest approach

demand huge history in algebra and topology, usually the equivalent

of a 12 months or extra of graduate learn. the purpose of those notes is to advance the

theory of algebraic curves from the perspective of contemporary algebraic geometry, but

without over the top prerequisites.

We have assumed that the reader understands a few uncomplicated houses of rings,

ideals, and polynomials, corresponding to is frequently lined in a one-semester direction in modern

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

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

is excited by simple houses of affine algebraic units; we've 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 glossy remedy of algebraic geometry, they play a fundamental

role in our training. the final learn of affine and projective varieties

is persisted in Chapters four and six, yet simply so far as important 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 basically at the neighborhood ring of the curve at the

point. The intersection variety of airplane curves at some degree is characterised through its

properties, and a definition when it comes to a undeniable residue category 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 conversant in 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 verified. 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 direction used to be repeated (assuming the entire 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 through 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 functions and extensions of the theory.

From bankruptcy three on, okay denotes a hard and fast algebraically closed box. at any time when convenient

(including with no remark a number of the difficulties) we've got assumed okay to

be of attribute 0. The minor alterations 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 mistakes and enhanced the readability of the text.

Professor PaulMonsky supplied numerous beneficial feedback as I taught the course.

“Je n’ai jamais été assez loin pour bien sentir l’application de l’algèbre à los angeles 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. premier 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 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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**Additional info for Algebraic Curves: An Introduction to Algebraic Geometry**

**Example text**

It is not hard to see that the only ideals in R are the principal ideals (t n ), n a nonnegative integer, so R is a PID. A ring R satisfying the conditions of Proposition 4 is called a discrete valuation ring, written DVR. 5. DISCRETE VALUATION RINGS 23 any other uniformizing parameter is of the form ut , u a unit in R. Let K be the quotient field of R. 2). The exponent n is called the order of z, and is written n = ord(z); we define ord(0) = ∞. Note that R = {z ∈ K | ord(z) ≥ 0}, and m = {z ∈ K | ord(z) > 0} is the maximal ideal in R.

F r generate I as an ideal in O P (V ). For if f ∈ I ⊂ O P (V ), there is a b ∈ Γ(V ) with b(P ) = 0 and b f ∈ Γ(V ); then b f ∈ Γ(V ) ∩ I , so b f = a i f i , a i ∈ Γ(V ); therefore f = (a i /b) f i , as desired. 17. Let V = V (Y 2 − X 2 (X + 1)) ⊂ A2 , and X , Y the residues of X , Y in Γ(V ); let z = Y /X ∈ k(V ). Find the pole sets of z and of z 2 . 22 CHAPTER 2. 18. Let O P (V ) be the local ring of a variety V at a point P . Show that there is a natural one-to-one correspondence between the prime ideals in O P (V ) and the subvarieties of V that pass through P .

2) R has a unique maximal ideal that contains every proper ideal of R. Proof. Let m = {non-units of R}. Clearly every proper ideal of R is contained in m; the lemma is an immediate consequence of this. A ring satisfying the conditions of the lemma is called a local ring; the units are those elements not belonging to the maximal ideal. We have seen that O P (V ) is a local ring, and mP (V ) is its unique maximal ideal. These local rings play a prominent role in the modern study of algebraic varieties.

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