integer solution at all. Here we have:

Z ’ R = Z[X, Y ]/(X 2 ’ Y 2 ’ 5), Spec(R) ’ Spec(Z) .

We obtain an a¬ne scheme over Z. Now Spec(Z) is a one-dimensional base, the ¬bres

are one-dimensional curves, and Spec(R) is two-dimensional. It is an arithmetic surface.

We want to study the ¬bres in more detail. Let Y ’ X be a homomorphism of schemes

and p a point on the base scheme X. The topological ¬bre over p is just the usual pre-

image of the point p. But here we have to give the ¬bre the structure of a scheme. The

general construction is as follows. Represent the point p by its residue ¬eld k(p) and

a homomorphism of schemes Spec(k(p)) ’ X . Take the “¬bre product of schemes”

of the scheme Y with Spec(k(p)) over X. Instead of giving the general de¬nition let

34 MARTIN SCHLICHENMAIER

me just write this down in our a¬ne situation:

Spec(R) ← ’ ’ Spec(R k(p))

’’ R ’’’ R

’’ k(p)

¦Z Z

¦

¦ ¦ ¦ ¦

¦ ¦

Z ’’’

Spec(Z) ← ’ ’

’’ ’’

Spec(k(p)) k(p)

Both diagrams are commutative diagrams and are dual to each other.

Here we obtain for the generic point [0] the residue ¬eld k(0) = Q and as ¬bre the

Spec of

R Q = Q[X, Y ]/(X 2 ’ Y 2 ’ 5) .

Z

For the closed points [p] we get k(p) = Fp and as ¬bre the Spec of

Fp = Fp [X, Y ]/(X 2 ’ Y 2 ’ 5) .

R

Z

In the ¬bres over the primes we just do calculation modulo p. A point lying on a curve

in the plane is a singular point of the curve if both partial derivatives of the de¬ning

equation vanish at this point. Zero conditions for functions are always closed conditions.

Hence non-singularity is an open condition on the individual curve. In fact, it is even an

open condition with respect to the variation of the point on the base scheme. The curve

X 2 ’ Y 2 ’ 5 = 0 is a non-singular curve over Q. The openness principle applied to the

base scheme says that there are only ¬nitely many primes for which the ¬bre will become

singular. Here it is quite easy to calculate these primes. Let f (X, Y ) = X 2 ’ Y 2 ’ 5

‚f ‚f

be the de¬ning equation. Then ‚X = 2X and ‚Y = 2Y . For p = 2 both partial

derivatives vanish at every point on the curve (the ¬bre). Hence every point of the ¬bre

is a singular point. This says that the ¬bre over the point [(2)] is a multiple ¬bre. In

this case we see immediately (X 2 ’ Y 2 ’ 5) ≡ (X + Y + 1)2 mod 2. This special ¬bre

is Spec(F2 [X, Y ]/((X + Y + 1)2 ) which is a non-reduced scheme. For p = 2 the only

candidate for a singular point is (0, 0). But this candidate lies on the curve if and only

if 5 ≡ 0 mod p hence only for p = 5. In this case we get one singularity. Here we

calculate that (X 2 ’ Y 2 ’ 5) = (X + Y )(X ’ Y ) mod 5. Altogether we obtain that

nearly every ¬bre is a non-singular conic. Only the ¬bre over [(2)] is a double line and

the ¬bre over [(5)] is a union of two lines which meet at one point.

4. Other objects. In lecture 5 we already said that moduli problems (degenerations

etc.) can be conveniently be described as functors. It is not always possible to ¬nd a

scheme representing a certain moduli functor. To obtain a representing geometric object

it is sometimes necessary to enlarge the category of schemes by introducing more general

objects like algebraic spaces and algebraic stacks. It is quite impossible even to give the

basics of their de¬nitions. Here let me only say that in a ¬rst step it is necessary to

CONCEPTS OF MODERN ALGEBRAIC GEOMETRY 35

introduce a ¬ner topology on the schemes, the etale topology. With respect to the etale

topology one has more open sets. Schemes are “glued” together from a¬ne schemes

using algebraic morphisms. Algebraic spaces are objects where the “glueing maps” are

more general maps (etale maps). Algebraic stacks are even more general than algebraic

spaces. The typical situation where they occur is in connection with moduli functors.

Here one has a scheme which represents a set of certain objects. If one wants to have

only one copy for each isomorphy class of the objects one usually has to divide out a

group action. But not every orbit space of a scheme by a group action can be made to

a scheme again. Hence we indeed get new objects. This new objects are the algebraic

stacks.

Let me here only give a few references. More information on algebraic spaces you can

¬nd in the book of Artin [Ar-1] or Knutson [Kn]. For stacks the appendix of [Vi] gives

a very short introduction and some examples.

References

[Ar-1] Artin, M., Algebraic Spaces, Yale Mathematical Monographs 3, Yale University

Press, New Haven, London, 1971.

[Ar-2] Artin, M., Geometry of quantum planes, Azumaya algebras, actions, and mod-

ules, Proceedings Bloomington 1990, Contemp. Math. 124, AMS, Providence,

pp. 1-15.

[BGR] Borho, W., Gabriel, P., Rentschler, R., Primideale in Einh¨llenden au¬‚¨sbarer

u o

Lie-Algebren, Lecture Notes in Mathematics 357, Springer, Berlin, Heidelberg,

New York, 1973.

[EH] Eisenbud, D., Harris, J., Schemes: The language of modern algebraic geometry,

Wadsworth &Brooks, Paci¬c Grove, California, 1992.

[GoWa] Goodearl, K.R., War¬eld, R.B., An introduction to noncommutative noether-

ian rings, London Math. Soc. Student Texts 16, Cambridge University Press,

New York, Port Chester, Melbourne, Sydney, 1989.

[GH] Gri¬ths, Ph., Harris J., Principles of algebraic geometry, John Wiley, New

York, 1978.

[EGA I] Grothendieck, A., Dieudonn´, J. A., El´ments de g´om´trie alg´brique I, Springer,

e e ee e

Berlin, Heidelberg, New York, 1971.

36 MARTIN SCHLICHENMAIER

[EGA] Grothendieck, A., Dieudonn´, J. A., El´ments de g´om´trie alg´brique, Publi-

e e ee e

´

cations Math´matiques de l™Institute des Hautes Etudes Scienti¬ques 8, 11,

e

17, 20, 24, 28, 32.

[H] Hartshorne, R., Algebraic geometry, Springer, Berlin, Heidelberg, New York,

1977.

[Kn] Knutson, D., Algebraic Spaces, Lecture Notes in Mathematics 203, Springer,

Berlin, Heidelberg, New York, 1971.

[Ku] Kunz, E., Einf¨hrung in die kommutative Algebra und algebraische Geome-

u

trie, Vieweg, Braunschweig, 1979; Introduction to commutative algebra and

algebraic geometry, Birkh¨user, Boston, Basel, Stuttgart, 1985.

a

[Ma-1] Manin, Y. I., Quantum groups and noncommutative geometry, Universit´ de

e

Montr´al (CRM), Montr´al, 1988.

e e

[Ma-2] Manin, Y. I., Topics in noncommutative geometry, M. B. Porter Lectures,

Princeton University Press, Princeton, New Jersey, 1991.

[Ma-3] Manin, Y. I., Notes on quantum groups and quantum de Rham complexes,

MPI/91-60.

[Mu-1] Mumford, D., The red book of varieties and schemes, Lecture Notes in Math-

ematics 1358 (reprint), Springer, Berlin, Heidelberg, New York, 1988.

[Mu-2] Mumford, D., Picard groups of moduli problems, Arithmetical algebraic geom-

etry (Purdue 1963) (O. F. G., Schilling, ed.), Harper &Row,, New York, 1965,

pp. 33-38.

[R] Rosenberg, Alexander, L., The left spectrum, the Levitzki radical, and noncom-

mutative schemes, Proc. Natl. Acad. Sci. USA 87 (1990), no. 4, 8583-8586.

[Sch] Schlichenmaier, M., An introduction to Riemann surfaces, algebraic curves

and moduli spaces, Lecture Notes in Physics 322, Springer, Berlin, Heidelberg,

New York, 1989.

[Vi] Vistoli, A., Intersection theory on algebraic stacks and their moduli spaces,

Appendix, Invent. Math. 97 (1989), 613 “ 670.

[WZ] Wess, J., Zumino, B., Covariant di¬erential calculus on the quantum hyper-

plane, Nucl. Phys. B, Proceed. Suppl. 18B (1990), 302.