Download Algebraic Curves: An Introduction to Algebraic Geometry by William Fulton PDF

By William Fulton

Preface
Third Preface, 2008

This textual content has been out of print for numerous 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 satisfied now to make this version on hand at no cost to anyone
interested. i'm so much thankful to Kwankyu Lee for creating a cautious LaTeX version,
which used to be 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 past types, specifically Grzegorz Bobi´nski for the most
recent and thorough checklist. it's inevitable that this conversion has brought some
new error, and that i and destiny readers should be thankful should 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 on hand to a beginner in modern
algebraic geometry. given that then many introductory treatises have seemed, including
excellent texts via Shafarevich,Mumford,Hartshorne, Griffiths-Harris, Kunz,
Clemens, Iitaka, Brieskorn-Knörrer, and Arbarello-Cornalba-Griffiths-Harris.

The earlier 20 years have additionally obvious a great deal of development 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 e-book to mirror this
progress, however it doesn't look attainable to take action with out leaving behind its elementary
character and destroying its unique function: 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 during the publication and its
exercises is helping organize a reader for any of the texts pointed out above, that would 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 exceptional therapy of
the classical idea of airplane curves, yet those don't organize the scholar adequately
for glossy algebraic geometry. nevertheless, so much books with a contemporary approach
demand massive 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 improve the
theory of algebraic curves from the perspective of recent algebraic geometry, but
without over the top prerequisites.

We have assumed that the reader understands a few uncomplicated houses of rings,
ideals, and polynomials, reminiscent of is frequently coated in a one-semester direction 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 interested in easy houses of affine algebraic units; we've given Zariski’s
proof of the $64000 Nullstellensatz.

The coordinate ring, functionality box, and native earrings of an affine kind are studied
in bankruptcy 2. As in any sleek remedy of algebraic geometry, they play a fundamental
role in our instruction. the final research of affine and projective varieties
is persevered in Chapters four and six, yet purely so far as helpful for our research of curves.

Chapter three considers affine aircraft curves. The classical definition of the multiplicity
of some degree on a curve is proven to count in simple terms at the neighborhood ring of the curve at the
point. The intersection variety of aircraft curves at some degree is characterised by means of its
properties, and a definition by way of a undeniable residue classification 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 acquainted with 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 via blowing
up issues, and the correspondence among algebraic functionality fields on one
variable and nonsingular projective curves is confirmed. 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 college in 1967–
68. The direction was once repeated (assuming all of the algebra) to a bunch 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 way 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 a few of the difficulties) we now have 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 direction, for sharing the task
of writing the notes. He corrected many mistakes and more advantageous the readability of the text.
Professor PaulMonsky supplied numerous valuable 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 superior 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 l. a. 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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Additional resources for Algebraic Curves: An Introduction to Algebraic Geometry

Example text

Suppose F,G ∈ k[X 1 , . . , X n ] are forms of degree r , r + 1 respectively, with no common factors (k a field). Show that F +G is irreducible. 7. ∗ (a) Show that there are d + 1 monomials of degree d in R[X , Y ], and 1 + 2 + · · · + (d + 1) = (d + 1)(d + 2)/2 monomials of degree d in R[X , Y , Z ]. (b) Let V (d , n) = {forms of degree d in k[X 1 , . . , X n ]}, k a field. Show that V (d , n) is a vector space over k, and that the monomials of degree d form a basis. So dimV (d , 1) = 1; dimV (d , 2) = d + 1; dimV (d , 3) = (d + 1)(d + 2)/2.

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 . The intersection numbers should add when we take unions of curves: (6) If F = r F i i , and G = sj G j , then I (P, F ∩G) = i , j r i s j I (P, F i ∩G j ).

Conversely, for any polynomial f ∈ R[X 1 , . . , X n ] of degree d , write f = f 0 + f 1 + · · · + f d , where f i is a form of degree i , and define f ∗ ∈ R[X 1 , . . , X n+1 ] by setting d d −1 d f ∗ = X n+1 f 0 + X n+1 f 1 + · · · + f d = X n+1 f (X 1 /X n+1 , . . , X n /X n+1 ); f ∗ is a form of degree d . ) The proof of the following proposition is left to the reader: Proposition 5. (1) (F G)∗ = F ∗G ∗ ; ( f g )∗ = f ∗ g ∗ . r (2) If F = 0 and r is the highest power of X n+1 that divides F , then X n+1 (F ∗ )∗ = F ; ∗ ( f )∗ = f .

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