By Oscar Zariski

Moduli difficulties in algebraic geometry date again to Riemann's well-known count number of the $3g-3$ parameters had to make certain a curve of genus $g$. during this booklet, Zariski reviews the moduli house of curves of a similar equisingularity type. After constructing and reviewing the fundamental fabric, Zariski devotes one bankruptcy to the topology of the moduli area, together with an specific choice of the infrequent circumstances whilst the gap is compact. bankruptcy V appears to be like at particular examples the place the size of the customary part might be decided via relatively concrete equipment. Zariski's final bankruptcy matters the appliance of deformation thought to the moduli challenge, together with the decision of the measurement of the accepted part for a specific relations of curves. An appendix through Bernard Teissier reconsiders the moduli challenge from the viewpoint of deformation conception. He provides new proofs of a few of Zariski's effects, in addition to a typical development of a compactification of the moduli area

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In other words, all ideals of Vq are given by Therefore Vq is a discrete valuation ring. We have Vq C nq Iq Vq. If %E Ql[(pn] for a, b E Vq is in the intersection, then a is prime to q for all qlq; thus, a is prime to q. This implies Vq = nqlq V q, and Vq is integrally closed (Exercise 3). Since ( = (pn is defined by the relation 1 + (pn-l + (2 pn - + ... + ((p_l)pn-l = 0, 1 thepn-l(p-1) elements 1,(, ... ,(pn-l(p-l)-1 form a base of Z[(pn] over Z. Then we have n =n Vq = (nqZ(q))l Vq + (nqz(q))( + ...

L) = j(L) for the j-invariant defined by 3 j (w) = j (L) = ~. The following facts are known from the general theory of algebraic curves and are proven in this chapter. • • • • If ¢ : EL ---+ Eu is a holomorphic homomorphism with ¢(O) = 0, then there exists>. 39); Write a morphism ¢ : EL ---+ Eu as ¢(u) = (¢x(u), ¢y(u), 1) using the coordinates of the projective space p2. 35). The function j is an example of a modular function and g2 and g3 are examples of modular forms. Modular forms are a special kind of automorphic forms defined on a more general complex domain.

K U {oo} = P(K) (canonically). If K is not algebraically closed, take an algebraic closure K of K. Defining P(K) by the set of points in P(K) with coordinates in K, we have {Vlvaluation rings of K(x) trivial on K with K(P) = K} 9:"! P(K). 4 Algebraic Curves over a Field 41 The set {Vlvaluation rings of K(x) trivial on K} is called the set of closed points of P. They are associated with a maximal ideal of K[x] or K[X-l]. Of course, if V is associated with (t(x)) and t(x) -=I- x, then the same V is associated with x-deg(t)t(x) E K[x-l] because vp(x) = 0 if t(x) -=I- x.