View source for Birational geometry

The branch of algebraic geometry in which the main problem is the classification of algebraic varieties up to birational equivalence (cf. [[Birational mapping|Birational mapping]]). Over a fixed field of constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016490/b0164901.png" /> every class of birationally equivalent varieties defines a finitely generated field over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016490/b0164902.png" />, which is isomorphic to the field of rational functions on any variety in this class. Conversely, to each such field corresponds a class of birationally equivalent varieties — the model(s) of this field. Thus, the birational classification of algebraic varieties is equivalent to the classification (up to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016490/b0164903.png" />-isomorphism) of the finitely generated fields regular over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016490/b0164904.png" />.

The most general birational invariant is the dimension of an algebraic variety. For one-dimensional algebraic varieties — irreducible algebraic curves — each class of birational equivalence contains a non-singular model — a smooth projective curve, which is unique up to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016490/b0164905.png" />-isomorphisms. Accordingly, the birational classification of algebraic curves is reduced to the classification, up to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016490/b0164906.png" />-isomorphisms, of smooth projective curves, which leads to the [[Moduli problem|moduli problem]]. For dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016490/b0164907.png" /> the problem becomes much more complicated. The very existence of a smooth model constitutes a problem of [[Resolution of singularities|resolution of singularities]] of algebraic varieties, which at the time of writing (1986) has been positively solved only for surfaces and for varieties of arbitrary dimension over a field of characteristic zero. In such a case, if such models exist at all, there are infinitely many of them in the class of birationally equivalent varieties. The minimal models (cf. [[Minimal model|Minimal model]]) take a special place among such models. Their birational classification is often identical with classification up to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016490/b0164908.png" />-isomorphism, as in the case of curves. However, this is not true in the general case, even for (both rational and ruled) surfaces.

The principal results in the classification of algebraic surfaces were obtained by geometers of the Italian school [[#References|[1]]]. Only isolated results are so far (1986) available [[#References|[3]]], [[#References|[7]]], [[#References|[8]]] for varieties of dimensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016490/b0164909.png" />.

The principal discrete birational invariants of smooth complete algebraic varieties over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016490/b01649010.png" /> of characteristic zero include the arithmetic genus, the geometric genus, the multiple genus, the dimension of the space of regular differential forms, the Severi torsion, the fundamental group and the Brauer group.

One of the most important problems in birational geometry is the problem of rationality of algebraic varieties, i.e. the problem of the description of rational varieties — varieties which are birationally equivalent to a projective space, cf. [[Rational variety|Rational variety]].

If the field of constants is algebraically non-closed, the problems in birational geometry are closely connected with the arithmetic of algebraic varieties (cf. [[Algebraic varieties, arithmetic of|Algebraic varieties, arithmetic of]]). The important problem in such a case is that of describing the birational <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016490/b01649011.png" />-forms on a given variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016490/b01649012.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016490/b01649013.png" />, in particular if, say, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016490/b01649014.png" /> is a projective space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016490/b01649015.png" /> [[#References|[2]]]. An important part of this problem is the description of the group of birational transformations of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016490/b01649016.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> "Algebraic surfaces" ''Trudy Mat. Inst. Steklov.'' , '''75''' (1965) (In Russian) {{MR|}} {{ZBL|0154.33002}} {{ZBL|0154.21001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> Yu.I. Manin, "Cubic forms. Algebra, geometry, arithmetic" , North-Holland (1974) (Translated from Russian) {{MR|0460349}} {{ZBL|0277.14014}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> L. Roth, "Algebraic threefolds" , Springer (1955) {{MR|0076426}} {{ZBL|0066.14704}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> M. Baldassarri, "Algebraic varieties" , Springer (1956) {{MR|0082172}} {{ZBL|0995.14003}} {{ZBL|0075.15902}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> I.V. Dolgachev, V.A. Iskovskikh, "The geometry of algebraic varieties" ''J. Soviet Math.'' , '''5''' : 6 (1976) pp. 803–864 ''Itogi Nauk. i Tekhn. Algebra Topol. Geom.'' , '''12''' (1974) pp. 77–170</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> ''J. Soviet Math.'' , '''13''' : 6 (1980) ''Itogi Nauk. i Tekhn. Sovrem. Probl. Mat.'' , '''12''' (1979)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> S. Iitaka, "Algebraic geometry, an introduction to birational geometry of algebraic varieties" , Springer (1982) {{MR|}} {{ZBL|0491.14006}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>



====Comments====
A field extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016490/b01649017.png" /> is regular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016490/b01649018.png" /> is algebraically closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016490/b01649019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016490/b01649020.png" /> is linearly disjoint from the algebraic closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016490/b01649021.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016490/b01649022.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016490/b01649023.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016490/b01649024.png" /> is algebraically closed, then every extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016490/b01649025.png" /> is regular and there is an arrow-reversing equivalence of categories between the category of varieties and dominant rational mappings over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016490/b01649026.png" /> and the category of finitely generated field extensions, cf. [[#References|[8]]], Sect. I.4. For dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016490/b01649027.png" /> many new results have recently become available in the (birational) classification of algebraic varieties, cf. [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Conte (ed.) , ''Algebraic threefolds'' , ''Lect. notes in math.'' , '''947''' , Springer (1982) {{MR|0672612}} {{ZBL|0477.00008}} </TD></TR></table>