# Classification of Algebraic Varieties by Ciro Ciliberto, E. Laura Livorni, Andrew J. Sommese By Ciro Ciliberto, E. Laura Livorni, Andrew J. Sommese

Similar algebraic geometry books

Mathematical Aspects of Geometric Modeling

This monograph examines intimately definite recommendations which are priceless for the modeling of curves and surfaces and emphasizes the mathematical idea that underlies those rules. the 2 imperative issues of the textual content are using piecewise polynomial illustration (this topic appears to be like in a single shape or one other in each chapter), and iterative refinement, often known as subdivision.

Fractured Fractals and Broken Dreams: Self-Similar Geometry through Metric and Measure

Fractal styles have emerged in lots of contexts, yet what precisely is a trend? How can one make designated the constructions mendacity inside of items and the relationships among them? This publication proposes new notions of coherent geometric constitution to supply a clean method of this widespread box. It develops a brand new notion of self-similarity known as "BPI" or "big items of itself," which makes the sphere a lot more uncomplicated for individuals to go into.

Singularity Theory I

From the stories of the 1st printing of this publication, released as quantity 6 of the Encyclopaedia of Mathematical Sciences: ". .. My basic influence is of a very great booklet, with a well-balanced bibliography, suggested! "Medelingen van Het Wiskundig Genootschap, 1995". .. The authors supply right here an up-to-the-minute advisor to the subject and its major functions, together with a few new effects.

Extra resources for Classification of Algebraic Varieties

Sample text

The next dilogarithm variant we mention is the p-adic dilogarithm, studied by R. Coleman and other authors. We ﬁx a prime number p and deﬁne (p) Li2 (x) = n>0, p∤n xn . n2 (14) This function can be written as Li2 (x)−p−2 Li2 (xp ), so in the complex domain it is simply a combination of ordinary dilogarithms and of no independent interest, but because we have omitted the terms in (14) with p’s in the denominator, the power series converges p-adically for all p-adic numbers x with (p) valuation |x|p < 1.

The deﬁnition and the name, of course, come from the analogy with the Taylor series of the ordinary logarithm around 1, − log(1 − z) = ∞ n=1 zn n for |z| < 1 , which leads similarly to the deﬁnition of the polylogarithm Lim (z) = ∞ n=1 zn nm for |z| < 1, m = 1, 2, . . The relation 1 d Lim (z) = Lim−1 (z) (m ≥ 2) dz z is obvious and leads by induction to the extension of the domain of deﬁnition of Lim to the cut plane C [1, ∞); in particular, the analytic continuation of the dilogarithm is given by z Li2 (z) = − 1 0 log(1 − u) du u for z ∈ C [1, ∞) .

1]), has many other beautiful properties. In particular: (II) D(z), which is a real-valued function on C, can be expressed in terms of a function of a single real variable, namely D(z) = z 1 D +D 2 z¯ 1 − 1/z 1 − 1/¯ z +D 1/(1 − z) 1/(1 − z¯) (2) The Dilogarithm Function 11 which expresses D(z) for arbitrary complex z in terms of the function D(eiθ ) = ℑ[Li2 (eiθ )] = ∞ sin nθ . ) Formula (2) is due to Kummer. 6 4 (III) All of the functional equations satisﬁed by Li2 (z) lose the elementary correction terms (constants and products of logarithms) when expressed in terms of D(z).