Remember that a pair (X, [DELTA]) is called log surface if X is a normal

algebraic surface and [DELTA] is a boundary R-divisor on X such that [K.sub.X] + [DELTA] is R-Cartier (See [Fjn12, Definition 3.1]).

(4) We see that the locus is an

algebraic surface of third degree.

A real elliptic surface will be a morphism [[PI].sub.1] : Y [right arrow] [P.sup.1] defined over R, when Y is a real

algebraic surface such that over all but finitely many points in the basic curve, the fibre is a nonsingular curve of genus one.

Topics include algebraic curve theory,

algebraic surface theory, moduli space, automorphic forms, Mordell-Weil lattices and automorphisms of hyperkahler manifolds.

Let X be an

algebraic surface with isolated normal singularities, [pi]: V [right arrow] X its minimal resolution, and E the reduced exceptional divisor.

Let X be a projective

algebraic surface (over C) with an ample line bundle L.

Let X be a normal

algebraic surface and let [DELTA] be a boundary R-divisor on X such that [K.sub.X] + [DELTA] is R-Cartier.

Zariski, The theorem of Riemann-Roch for high multiples of an effective divisor on an

algebraic surface, Ann.

Goto, A Public-key Cryptosystem using

Algebraic Surfaces: Extended Abstract, PQCrypto Workshop Record, 2006.

Shigefumi Mori, 39, of Kyoto University, has devoted much of his career to pioneering methods of classifying certain kinds of surfaces defined by algebraic equations, thereby extending the classical theory of

algebraic surfaces to three dimensions.

Using this result, new methods for blending several

algebraic surfaces simultaneously are derived.