Algebraic Curves: An Introduction to Algebraic Geometry by William Fulton

By William Fulton

Preface
Third Preface, 2008

This textual content has been out of print for a number of years, with the writer maintaining copyrights.
Since I proceed to listen to from younger algebraic geometers who used this as
their first textual content, i'm completely happy now to make this variation on hand for gratis to anyone
interested. i'm such a lot thankful to Kwankyu Lee for creating a cautious LaTeX version,
which used to be the root 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 adjustments. I thank all who
have despatched corrections to prior types, specifically Grzegorz Bobi´nski for the most
recent and thorough checklist. it's inevitable that this conversion has brought some
new mistakes, and that i and destiny readers may be thankful in case you will ship any mistakes you
find to me at wfulton@umich.edu.

Second Preface, 1989

When this e-book first seemed, there have been few texts on hand to a amateur in modern
algebraic geometry. when you consider that 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 earlier 20 years have additionally noticeable a great deal of progress in our understanding
of the themes lined during this textual content: linear sequence on curves, intersection idea, 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 no leaving behind 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 assist 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 e-book and its
exercises is helping organize 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 manner into the subject.
There are numerous texts on an undergraduate point that provide a good remedy of
the classical conception of airplane curves, yet those don't organize the coed adequately
for smooth algebraic geometry. nevertheless, so much books with a latest approach
demand enormous historical past in algebra and topology, usually the equivalent
of a yr or extra of graduate learn. the purpose of those notes is to strengthen 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, resembling is usually lined in a one-semester path in modern
algebra; extra commutative algebra is built in later sections. Chapter
1 starts with a precis of the proof we'd like from algebra. the remainder of the chapter
is excited by uncomplicated homes of affine algebraic units; we now have given Zariski’s
proof of the real Nullstellensatz.

The coordinate ring, functionality box, and native earrings 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 examine of affine and projective varieties
is persisted in Chapters four and six, yet simply so far as beneficial for our examine of curves.

Chapter three considers affine aircraft curves. The classical definition of the multiplicity
of some degree 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 via its
properties, and a definition when it comes to a definite 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 kinds will realize that this cohomology is implicit in our proofs.)

In bankruptcy 7 the nonsingular version of a curve is developed by way of blowing
up issues, and the correspondence among algebraic functionality fields on one
variable and nonsingular projective curves is demonstrated. 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 direction taught to Juniors at Brandeis college in 1967–
68. The path used to be repeated (assuming the entire algebra) to a gaggle of graduate students
during the in depth week on the finish of the Spring semester. we have now retained
an crucial characteristic of those classes by way 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 functions and extensions of the theory.

From bankruptcy three on, ok denotes a set algebraically closed box. at any time when convenient
(including with out remark a number of the difficulties) we've assumed okay to
be of attribute 0. The minor alterations essential to expand the speculation 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 blunders and greater 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 best 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 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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Extra resources for Algebraic Curves: An Introduction to Algebraic Geometry

Example text

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.

9. It is easy to check that the top row is exact. It follows that dim(k[X , Y ]/I n ) + dim(k[X , Y ]/I m ) ≥ dim(Ker(ϕ)), with equality if and only if ψ is one-to-one, and that dim(k[X , Y ]/(I m+n , F,G)) = dim(k[X , Y ]/I m+n ) − dim(Ker(ϕ)). 3. 46 and arithmetic). This shows that I (P, F ∩ G) ≥ mn, and that I (P, F ∩ G) = mn if and only if both inequalities in the above string are equalities. , if I m+n ⊂ (F,G)O . The second is an equality if and only if ψ is one-to-one. Property (5) is therefore a consequence of: Lemma.

Consider the following diagram of vector spaces and linear maps: k[X , Y ]/I n ×k[X , Y ]/I m ψ / k[X , Y ]/I m+n ϕ / k[X , Y ]/(I m+n , F,G) /0 α O /(F,G) π  / O /(I m+n , F,G) /0 where ϕ, π, and α are the natural ring homomorphisms, and ψ is defined by letting ψ(A, B ) = AF + BG. 9. It is easy to check that the top row is exact. It follows that dim(k[X , Y ]/I n ) + dim(k[X , Y ]/I m ) ≥ dim(Ker(ϕ)), with equality if and only if ψ is one-to-one, and that dim(k[X , Y ]/(I m+n , F,G)) = dim(k[X , Y ]/I m+n ) − dim(Ker(ϕ)).

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