By William Fulton

Preface

Third Preface, 2008

This textual content has been out of print for a number of years, with the writer keeping 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 was once the foundation of this version; 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 past models, 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 may be thankful in the event you will ship any mistakes you

find to me at wfulton@umich.edu.

Second Preface, 1989

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

algebraic geometry. due to the fact then many introductory treatises have seemed, including

excellent texts through Shafarevich,Mumford,Hartshorne, Griffiths-Harris, Kunz,

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

The prior twenty 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 ebook to mirror this

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

character and destroying its unique goal: to introduce scholars with a bit algebra

background to a couple of the tips 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 ebook 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 means into the subject.

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

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

for smooth algebraic geometry. however, such a lot books with a latest approach

demand enormous history in algebra and topology, frequently the equivalent

of a yr or extra of graduate examine. 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 easy homes of rings,

ideals, and polynomials, akin 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 proof we want from algebra. the remainder of the chapter

is excited by uncomplicated houses of affine algebraic units; we have now given Zariski’s

proof of the real Nullstellensatz.

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

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

role in our practise. the overall examine of affine and projective varieties

is endured in Chapters four and six, yet purely so far as beneficial for our examine of curves.

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

of some extent on a curve is proven to count purely 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 by way of a undeniable residue category ring of a neighborhood ring is

shown to have those houses. Bézout’s Theorem and Max Noether’s Fundamental

Theorem are the topic of bankruptcy five. (Anyone accustomed to 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 built through 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 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 college in 1967–

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

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

an crucial characteristic of those classes via together with a number of hundred difficulties. The results

of the starred difficulties are used freely within the textual content, whereas the others variety from

exercises to purposes and extensions of the theory.

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

(including with out remark some 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 pupil within the path, for sharing the task

of writing the notes. He corrected many error and enhanced 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 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. most well known 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 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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**Example text**

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 . Find the power series expansion of z = (1 − X )−1 and of (1 − X )(1 + X 2 )−1 in terms of t .

It suffices to show that F m ∈ I , for then F − F m ∈ I , and an inductive argument finishes the proof. Write F = A (α) F (α) . Comparing terms of the same degree, we conclude that F m = A (α) F (α) , so F m ∈ I . m−d α n An algebraic set V ⊂ P is irreducible if it is not the union of two smaller algebraic sets. 4 below, shows that V is irreducible if and only if I (V ) is prime. An irreducible algebraic set in Pn is called a projective variety. Any projective algebraic set can be written uniquely as a union of projective varieties, its irreducible components.

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. ∗ Let R be a DVR with quotient field K ; let m be the maximal ideal of R. (a) Show that if z ∈ K , z ∈ R, then z −1 ∈ m. (b) Suppose R ⊂ S ⊂ K , and S is also a DVR. Suppose the maximal ideal of S contains m. Show that S = R. 27. 24 are the only DVR’s with quotient field k(X ) that contain k.