Geometry of Algebraic Curves: Volume I by Enrico Arbarello, Maurizio Cornalba, Phillip Griffiths,

By Enrico Arbarello, Maurizio Cornalba, Phillip Griffiths, Joseph Daniel Harris

In contemporary years there was huge, immense job within the conception of algebraic curves. Many long-standing difficulties were solved utilizing the overall ideas built in algebraic geometry through the 1950's and 1960's. also, unforeseen and deep connections among algebraic curves and differential equations were exposed, and those in flip make clear different classical difficulties in curve idea. it kind of feels reasonable to assert that the idea of algebraic curves seems different now from the way it seemed 15 years in the past; particularly, our present nation of data repre­ sents an important boost past the legacy left through the classical geometers reminiscent of Noether, Castelnuovo, Enriques, and Severi. those books provide a presentation of 1 of the primary components of this contemporary task; specifically, the learn of linear sequence on either a set curve (Volume I) and on a variable curve (Volume II). Our target is to offer a entire and self-contained account of the extrinsic geometry of algebraic curves, which in our opinion constitutes the most geometric center of the hot advances in curve thought. alongside the way in which we will, in fact, talk about appli­ cations of the idea of linear sequence to a couple of classical issues (e.g., the geometry of the Riemann theta divisor) in addition to to a few of the present learn (e.g., the Kodaira size of the moduli area of curves).

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B-5. Let C c lfl>' (r ~ 3) be a smooth curve. Show that a general projection maps C biregularly to a smooth curve. B-6. Let C c lfl> 3 be a non-degenerate smooth curve. Show that a general projection with center p E lfl> 3 n: lfl>3 _ {p}-+ lfl>z maps C birationally to a plane curve having ordinary double points as its only singularities. (Hint: Show that the set of trisecant lines to C and the set of tangent lines to C describe two surfaces SandS'. ) 37 Exercises Note. It may be reassuring to the reader to know that the image of a complex analytic variety under a proper holomorphic map is again a complex analytic variety of the appropriate dimension.

On the product C 0 x Jlfy we define Riemann's theta function e(u, Z) = L e"J=1, me7L9 La where the bracket denotes the standard duality pairing (m, v) = m«v« on C 0 • Since Im Z > 0, the above series converges normally, and hence uniformly on compact sets, on C0 x ~ to a holomorphic function. One easily checks that the theta function satisfies the functional equations O(u O(u + ea, Z) = O(u, Z), + e0 +a• Z) = e- 2 "J=1e(u, Z), where Z = (Zap) and eh ... , e20 are the 2g column vectors in the period matrix Q.

Find the branch points and ramification points of n: C -> IP' 1 and show that the genus of C is equal to one. (ii) For any two points p, q E C find the complete linear series IP + ql. ) (iii) Find a map ry: C-> IP' 1 of degree two such that ry((l, 0)) = ry((O, 1)), and determine the branch points of rJ. (iv) Using A-3(i) and (ii) show that the curves in (a) and (b) are isomorphic, and find an explicit isomorphism. (i) A-5. Let C be the compact Riemann surface associated to the curve in (d) above, realized as a branched covering n: C .....

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