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Every irreducible complex algebraic curve is birational to a unique smooth projective curve, so the theory for curves is trivial. The case of surfaces was first investigated by the geometers of the Italian school around 1900; the contraction theorem of Guido Castelnuovo essentially describes the process of constructing a minimal model of any smooth projective surface.
In algebraic geometry, a Fano variety, introduced by Gino Fano (Fano 1934, 1942), is an algebraic variety that generalizes certain aspects of complete intersections of algebraic hypersurfaces whose sum of degrees is at most the total dimension of the ambient projective space.
Gneiss, a foliated metamorphic rock. Quartzite, a non-foliated metamorphic rock. Foliation in geology refers to repetitive layering in metamorphic rocks. [1] Each layer can be as thin as a sheet of paper, or over a meter in thickness. [1] The word comes from the Latin folium, meaning "leaf", and refers to the sheet-like planar structure. [1]
The species–area relationship or species–area curve describes the relationship between the area of a habitat, or of part of a habitat, and the number of species found within that area. Larger areas tend to contain larger numbers of species, and empirically, the relative numbers seem to follow systematic mathematical relationships. [ 1 ]
Deformation theory was famously applied in birational geometry by Shigefumi Mori to study the existence of rational curves on varieties. [2] For a Fano variety of positive dimension Mori showed that there is a rational curve passing through every point.
A 3-dimensional foliated chart with n = 3 and q = 1. The plaques are 2-dimensional and the transversals are 1-dimensional. A rectangular neighborhood in R n is an open subset of the form B = J 1 × ⋅⋅⋅ × J n, where J i is a (possibly unbounded) relatively open interval in the ith coordinate axis. If J 1 is of the form (a,0], it is said ...
Degree 1: they have 240 (−1)-curves corresponding to the roots of an E 8 root system. They form an 8-dimensional family. The anticanonical divisor is not very ample. The linear system |−2K| defines a degree 2 map from the del Pezzo surface to a quadratic cone in P 3, branched over a nonsingular genus 4 curve cut out by a cubic surface.
The Bombieri–Lang conjecture is an analogue for surfaces of Faltings's theorem, which states that algebraic curves of genus greater than one only have finitely many rational points. [ 8 ] If true, the Bombieri–Lang conjecture would resolve the Erdős–Ulam problem , as it would imply that there do not exist dense subsets of the Euclidean ...