By William Fulton

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 happy now to make this variation to be had for gratis 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 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, particularly Grzegorz Bobi´nski for the most

recent and thorough checklist. it truly is inevitable that this conversion has brought some

new mistakes, and that i and destiny readers should be thankful in case you will ship any error you

find to me at wfulton@umich.edu.

Second Preface, 1989

When this e-book first seemed, there have been few texts to be had to a amateur in modern

algebraic geometry. when you consider that 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 prior 20 years have additionally visible a great deal of progress in our understanding

of the subjects lined 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 look attainable to take action with no 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 assist them gain

some appreciation either for algebraic geometry and for origins and purposes of

many of the notions of commutative algebra. If operating during the e-book 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 constructed and thriving box of mathematics,

it is notoriously tricky for the newbie to make his method into the subject.

There are numerous texts on an undergraduate point that supply an outstanding therapy of

the classical idea 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 historical past in algebra and topology, usually the equivalent

of a yr 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 is aware a few easy homes of rings,

ideals, and polynomials, comparable to is usually coated in a one-semester direction in modern

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

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

is enthusiastic 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 type are studied

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

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

is persevered in Chapters four and six, yet simply so far as priceless for our learn of curves.

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

of some degree on a curve is proven to rely simply at the neighborhood ring of the curve at the

point. The intersection variety of airplane curves at some extent is characterised via its

properties, and a definition when it comes to a undeniable residue classification ring of an area 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 path taught to Juniors at Brandeis collage in 1967–

68. The path used to be repeated (assuming all of the algebra) to a gaggle of graduate students

during the extensive week on the finish of the Spring semester. now we have retained

an crucial function 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 variety from

exercises to functions and extensions of the theory.

From bankruptcy three on, ok denotes a hard and fast algebraically closed box. every time convenient

(including with no remark some of the difficulties) now we 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 direction, 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 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. optimum 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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Note that R = {z ∈ K | ord(z) ≥ 0}, and m = {z ∈ K | ord(z) > 0} is the maximal ideal in R. ∗ Show that the order function on K is independent of the choice of uniformizing parameter. ∗ Let V = A1 , Γ(V ) = k[X ], K = k(V ) = k(X ). (a) For each a ∈ k = V , show that O a (V ) is a DVR with uniformizing parameter t = X − a. (b) Show that O ∞ = {F /G ∈ k(X ) | deg(G) ≥ deg(F )} is also a DVR, with uniformizing parameter t = 1/X . 25. Let p ∈ Z be a prime number. Show that {r ∈ Q | r = a/b, a, b ∈ Z, p doesn’t divide b} is a DVR with quotient field Q.

X n ], denote the residue of G in Γ(V ) by G. Let f ∈ k(V ). Let J f = {G ∈ k[X 1 , . . , X n ] | G f ∈ Γ(V )}. Note that J f is an ideal in k[X 1 , . . , X n ] containing I (V ), and the points of V (J f ) are exactly those points where f is not defined. This proves (1). , 1 · f = f ∈ Γ(V ), which proves (2). Suppose f ∈ O P (V ). ) The ideal mP (V ) = { f ∈ O P (V ) | f (P ) = 0} is called the maximal ideal of V at P . It is the kernel of the evaluation homomorphism f → f (P ) of O P (V ) onto k, so O P (V )/mP (V ) is isomorphic to k.

3) If T is an affine change of coordinates on A2 , and T (Q) = P , then I (P, F ∩G) = I (Q, F T ∩G T ). (4) I (P, F ∩G) = I (P,G ∩ F ). Two curves F and G are said to intersect transversally at P if P is a simple point both on F and on G, and if the tangent line to F at P is different from the tangent line to G at P . We want the intersection number to be one exactly when F and G meet transversally at P . More generally, we require (5) I (P, F ∩G) ≥ m P (F )m P (G), with equality occurring if and only if F and G have not tangent lines in common at P .