subclass_of fano variety
Commutative ___MASK65___ is the branch of ___MASK65___ that studies commutative rings , their ideals , and modules over such rings . Both algebraic geometry and algebraic number theory build on commutative ___MASK65___ . Prominent examples of commutative rings include polynomial rings , rings of algebraic integers , including the ordinary integers formula_1 , and " p "- adic integers . In algebraic ___MASK15___ , a Fano ___MASK30___ , introduced in ( Fano 1934 , 1942 ) , is a complete ___MASK30___ 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 . In ___MASK21___ , a ___MASK99___ is an expression consisting of variables ( or indeterminates ) and coefficients , that involves only the operations of addition , subtraction , multiplication , and non - negative integer exponents . An example of a ___MASK99___ of a ___MASK23___ indeterminate is . An example in three variables is . In algebraic ___MASK15___ , the minimal ___MASK71___ program is part of the birational classification of algebraic varieties . Its goal is to construct a birational ___MASK71___ of any complex projective ___MASK30___ which is as simple as possible . The subject has its origins in the classical birational ___MASK15___ of surfaces studied by the Italian school , and is currently an active research area within algebraic ___MASK15___ . In algebraic ___MASK15___ , a projective ___MASK30___ over an algebraically closed field " k " is a subset of some projective " n "- space P over " k " that is the zero - locus of some finite family of homogeneous polynomials of " n " + 1 variables with coefficients in " k ", that generate a prime ideal , the defining ideal of the ___MASK30___ . Equivalently , an ___MASK26___ is projective if it can be embedded as a Zariski closed subvariety of P . ___MASK21___ ( from ___MASK55___  " máthma ",  knowledge , ___MASK89___ , learning ) is the ___MASK89___ of topics such as quantity ( numbers ), structure , space , and change . There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics . Algebraic ___MASK15___ is a branch of mathematics , classically studying zeros of multivariate polynomials . Modern algebraic ___MASK15___ is based on the use of abstract algebraic techniques , mainly from commutative ___MASK65___ , for solving geometrical problems about these sets of zeros . In ___MASK21___ , birational ___MASK15___ is a ___MASK49___ of algebraic ___MASK15___ the goal of which 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 ___MASK43___ may fail to be defined where the rational functions have poles .