By William Fulton

Preface

Third Preface, 2008

This textual content has been out of print for numerous 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 pleased now to make this variation on hand for free of charge 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 support 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, in particular Grzegorz Bobi´nski for the most

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

new blunders, and that i and destiny readers might be thankful in case you will ship any blunders 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 through Shafarevich,Mumford,Hartshorne, Griffiths-Harris, Kunz,

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

The prior twenty years have additionally obvious 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 booklet to mirror this

progress, however it doesn't look attainable to take action with no forsaking 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 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 arrange 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 method into the subject.

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

the classical conception of aircraft curves, yet those don't arrange the scholar adequately

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

demand significant historical past in algebra and topology, usually the equivalent

of a 12 months or extra of graduate examine. the purpose of those notes is to increase 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 simple houses of rings,

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

is desirous about easy 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 kind are studied

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

role in our instruction. the overall learn of affine and projective varieties

is endured in Chapters four and six, yet purely so far as precious 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 rely purely at the neighborhood ring of the curve at the

point. The intersection variety of aircraft curves at some extent is characterised by means of its

properties, and a definition by way of a definite residue category 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 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 procedure 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 the entire algebra) to a bunch of graduate students

during the in depth week on the finish of the Spring semester. we've got retained

an crucial characteristic of those classes via 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 purposes and extensions of the theory.

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

(including with no remark a few of the 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 scholar within the direction, for sharing the task

of writing the notes. He corrected many mistakes and better the readability of the text.

Professor PaulMonsky supplied a number of 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 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. los angeles preferable 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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**Sample text**

In the first two examples, the linear term of the equation for the curve is just the tangent line to the curve at (0, 0). The lowest terms in C , D, E , and F respectively are Y 2 , Y 2 −X 2 = (Y −X )(Y +X ), 3X 2 Y −Y 3 = Y ( 3X −Y )( 3X +Y ), and −4X 2 Y 2 . In each case, the lowest order form picks out those lines that can best be called tangent to the curve at (0, 0). Let F be any curve, P = (0, 0). Write F = F m +F m+1 +· · ·+F n , where F i is a form in k[X , Y ] of degree i , F m = 0. We define m to be the multiplicity of F at P = (0, 0), write m = m P (F ).

X n )T = V (X s+1 , . . , X n ), m < s; show that Tm+1 , . . ∗ Let P = (a , . . , a ), Q = (b , . . , b ) be distinct points of An . The line through 1 n 1 n P and Q is defined to be {a 1 +t (b 1 −a 1 ), . . , a n +t (b n −a n )) | t ∈ k}. (a) Show that if L is the line through P and Q, and T is an affine change of coordinates, then T (L) is the line through T (P ) and T (Q). (b) Show that a line is a linear subvariety of dimension 1, and that a linear subvariety of dimension 1 is the line through any two of its points.

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. We will verify that ψ is one-to-one; the rest (which is easier) is left to the reader. If ψ(z) = 0, then G z = uF + vG H where u, v ∈ O . Choose S ∈ k[X , Y ] with S(P ) = 0, and Su = A, Sv = B , and Sz = C ∈ k[X , Y ]. Then G(C − B H ) = AF in k[X , Y ]. Since F and G have no common factors, F must divide C − B H , so C − B H = DF . Then z = (B /S)H + (D/S)F , or z = 0, as claimed. Property (5) is the hardest.