Kodaira dimensionIn algebraic geometry, the Kodaira dimension κ(X) measures the size of the canonical model of a projective variety X. Igor Shafarevich in a seminar introduced an important numerical invariant of surfaces with the notation κ. Shigeru Iitaka extended it and defined the Kodaira dimension for higher dimensional varieties (under the name of canonical dimension), and later named it after Kunihiko Kodaira. The canonical bundle of a smooth algebraic variety X of dimension n over a field is the line bundle of n-forms, which is the nth exterior power of the cotangent bundle of X.
Birational geometryIn mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational functions rather than polynomials; the map may fail to be defined where the rational functions have poles. A rational map from one variety (understood to be irreducible) to another variety , written as a dashed arrow X Y, is defined as a morphism from a nonempty open subset to .
K3 surfaceIn mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with а trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means a smooth proper geometrically connected algebraic surface that satisfies the same conditions. In the Enriques–Kodaira classification of surfaces, K3 surfaces form one of the four classes of minimal surfaces of Kodaira dimension zero. A simple example is the Fermat quartic surface in complex projective 3-space.
Cubic surfaceIn mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than affine space, and so cubic surfaces are generally considered in projective 3-space . The theory also becomes more uniform by focusing on surfaces over the complex numbers rather than the real numbers; note that a complex surface has real dimension 4.
Fano varietyIn algebraic geometry, a Fano variety, introduced by Gino Fano in , is a complete variety X whose anticanonical bundle KX* is ample. In this definition, one could assume that X is smooth over a field, but the minimal model program has also led to the study of Fano varieties with various types of singularities, such as terminal or klt singularities. Recently techniques in differential geometry have been applied to the study of Fano varieties over the complex numbers, and success has been found in constructing moduli spaces of Fano varieties and proving the existence of Kähler–Einstein metrics on them through the study of K-stability of Fano varieties.
ProvenceProvence (prəˈvɒ̃s, USalsoprəʊˈ-, UKalsoprɒˈ-, pʁɔvɑ̃s) is a geographical region and historical province of southeastern France, which extends from the left bank of the lower Rhône to the west to the Italian border to the east; it is bordered by the Mediterranean Sea to the south. It largely corresponds with the modern administrative region of Provence-Alpes-Côte d'Azur and includes the departments of Var, Bouches-du-Rhône, Alpes-de-Haute-Provence, as well as parts of Alpes-Maritimes and Vaucluse.
Characteristic classIn mathematics, a characteristic class is a way of associating to each principal bundle of X a cohomology class of X. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classes are global invariants that measure the deviation of a local product structure from a global product structure. They are one of the unifying geometric concepts in algebraic topology, differential geometry, and algebraic geometry.
Euler characteristicIn mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by (Greek lower-case letter chi). The Euler characteristic was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids.
Article (grammar)An article is any member of a class of dedicated words that are used with noun phrases to mark the identifiability of the referents of the noun phrases. The category of articles constitutes a part of speech. In English, both "the" and "a(n)" are articles, which combine with nouns to form noun phrases. Articles typically specify the grammatical definiteness of the noun phrase, but in many languages, they carry additional grammatical information such as gender, number, and case.
Characteristic (algebra)In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive identity the ring is said to have characteristic zero. That is, char(R) is the smallest positive number n such that: if such a number n exists, and 0 otherwise.
Characteristic polynomialIn linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The characteristic polynomial of an endomorphism of a finite-dimensional vector space is the characteristic polynomial of the matrix of that endomorphism over any base (that is, the characteristic polynomial does not depend on the choice of a basis).
River deltaA river delta is a landform shaped like a triangle, created by the deposition of sediment that is carried by a river and enters slower-moving or stagnant water. This occurs when a river enters an ocean, sea, estuary, lake, reservoir, or (more rarely) another river that cannot carry away the supplied sediment. It is so named because its triangle shape resembles the Greek letter Delta. The size and shape of a delta are controlled by the balance between watershed processes that supply sediment, and receiving basin processes that redistribute, sequester, and export that sediment.
Grammatical caseA grammatical case is a category of nouns and noun modifiers (determiners, adjectives, participles, and numerals) which corresponds to one or more potential grammatical functions for a nominal group in a wording. In various languages, nominal groups consisting of a noun and its modifiers belong to one of a few such categories. For instance, in English, one says I see them and they see me: the nominative pronouns I/they represent the perceiver and the accusative pronouns me/them represent the phenomenon perceived.
Aix-en-ProvenceAix-en-Provence (UKˌɛks_ɒ̃_prɒˈvɒ̃s, USˌeɪks_ɒ̃_proʊˈvɒ̃s,ˌɛks-), or simply Aix (medieval Occitan: Aics), is a city and commune in southern France, about north of Marseille. A former capital of Provence, it is the subprefecture of the arrondissement of Aix-en-Provence, in the department of Bouches-du-Rhône, in the region of Provence-Alpes-Côte d'Azur. The population of Aix-en-Provence is approximately 145,000. Its inhabitants are called Aixois or, less commonly, Aquisextains.
Provence wineProvence (Provençal) wine comes from the French wine-producing region of Provence in southeast France. The Romans called the area provincia nostra ("our province"), giving the region its name. Just south of the Alps, it was the first Roman province outside Italy. Wine has been made in this region for at least 2,600 years, ever since the ancient Greeks founded the city of Marseille in 600 BC. Throughout the region's history, viticulture and winemaking have been influenced by the cultures that have been present in Provence, which include the Ancient Greeks, Romans, Gauls, Catalans and Savoyards.
Nile DeltaThe Nile Delta (دلتا النيل, or simply الدلتا, ) is the delta formed in Lower Egypt where the Nile River spreads out and drains into the Mediterranean Sea. It is one of the world's largest river deltas—from Alexandria in the west to Port Said in the east, it covers of Mediterranean coastline and is a rich agricultural region. From north to south the delta is approximately in length. The Delta begins slightly down-river from Cairo. From north to south, the delta is approximately in length.
