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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]
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.
Animals mainly have bilateral or mirror symmetry, as do the leaves of plants and some flowers such as orchids. [30] Plants often have radial or rotational symmetry, as do many flowers and some groups of animals such as sea anemones. Fivefold symmetry is found in the echinoderms, the group that includes starfish, sea urchins, and sea lilies. [31]
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 ]
Primary succession is the beginning step of ecological succession where species known as pioneer species colonize an uninhabited site, which usually occurs in an environment devoid of vegetation and other organisms. In contrast, secondary succession occurs on substrates that previously supported vegetation before an ecological disturbance. This ...
Gröbner basis computation is one of the main practical tools for solving systems of polynomial equations and computing the images of algebraic varieties under projections or rational maps. Gröbner basis computation can be seen as a multivariate, non-linear generalization of both Euclid's algorithm for computing polynomial greatest common ...
The image of the 1-canonical map is called a canonical curve. A canonical curve of genus g always sits in a projective space of dimension g − 1. [3] When C is a hyperelliptic curve, the canonical curve is a rational normal curve, and C a double cover of its canonical curve.
Jacobians of curves are naturally equipped with a principal polarisation as soon as one picks an arbitrary rational base point on the curve, and the curve can be reconstructed from its polarised Jacobian when the genus is >. Not all principally polarised abelian varieties are Jacobians of curves; see the Schottky problem.