View source for Adjunction theory

In its basic form, the study of the interplay between an embedding of a projective manifold (cf. [[Projective algebraic set|Projective algebraic set]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a1104101.png" /> into [[Projective space|projective space]] and its canonical [[Bundle|bundle]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a1104102.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a1104103.png" /> is the cotangent bundle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a1104104.png" />. For simplicity, below the objects are considered over the complex numbers. The book [[#References|[a1]]] is a general reference with full coverage of the literature for the whole theory with its history. The book [[#References|[a4]]] is a fine reference on related material.

==Classical adjunction theory.==
To study a smooth projective [[Algebraic curve|algebraic curve]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a1104105.png" />, i.e., a compact [[Riemann surface|Riemann surface]], a major approach in the 19th century was to relate properties of the curve to properties of the canonical mapping of the curve, i.e., the mapping of the curve into projective space given by sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a1104106.png" />. To study a two-dimensional algebraic submanifold of projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a1104107.png" />, it was natural to try to reduce questions about the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a1104108.png" /> to the hyperplane sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a1104109.png" />, i.e., to the curves obtained by slicing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041010.png" /> with linear hyperplanes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041011.png" />. This led to the study of the adjoint bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041013.png" /> is the restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041014.png" /> of the hyperplane section bundle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041015.png" />, i.e., the line bundle on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041016.png" /> whose sections vanish on hyperplanes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041017.png" />. The restriction of the adjoint bundle to a hyperplane section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041019.png" /> is the canonical bundle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041020.png" />, and, except in a few trivial cases, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041021.png" /> is the only line bundle that has this property. Therefore, if the mapping associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041022.png" /> exists, it would tie together the canonical mappings of the smooth hyperplane sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041023.png" />. In the 19th century geometers, especially G. Castelnuovo and F. Enriques, used this rational mapping to study the embedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041024.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041025.png" />. The general procedure was to consider the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041026.png" />. A classical result was that adjunction terminates, i.e., there is a positive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041027.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041028.png" /> has no sections, if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041029.png" /> is birational (cf. [[Birational morphism|Birational morphism]]) to the product of an algebraic curve and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041030.png" />. A key point was that if the above <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041031.png" /> was the first positive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041032.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041033.png" /> has no sections, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041034.png" /> is one of only a "very short list of pairs" . Classically, it was not known if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041035.png" /> was spanned and, therefore, this procedure did not usually lead to a biregular classification. (A line bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041036.png" /> on an algebraic set is said to be spanned if global sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041037.png" /> surject onto the fibre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041038.png" /> at any point of the algebraic set.) There were also analogous classical procedures on threefolds due to G. Fano and U. Morin (see [[#References|[a5]]], [[#References|[a1]]]).

A.J. Sommese [[#References|[a6]]] started the modern study of the mapping (which he called the adjunction mapping) associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041039.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041040.png" /> is a very ample line bundle (cf. [[Ample vector bundle|Ample vector bundle]]) on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041041.png" />-dimensional projective manifold. The complete story with history of the adjunction mapping can be found in [[#References|[a1]]], Chapts. 8–11. The fundamental results of this theory are that except for a few special varieties, the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041042.png" /> can be replaced by a closely related pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041043.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041044.png" /> the blow-up of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041045.png" />-dimensional projective manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041046.png" /> at a finite set, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041047.png" /> an ample line bundle, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041048.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041049.png" /> very ample. Moreover, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041050.png" /> then, except for a few more examples, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041051.png" /> is numerically effective, or nef, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041052.png" /> for any effective curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041053.png" />. These results allow the classical birational results alluded to above to be both considerably extended and redone biregularly.

The major open question for this part of adjunction theory is to what extent the mapping associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041054.png" /> is well-behaved when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041055.png" /> is very ample, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041056.png" /> is smooth, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041057.png" />. For example, it is known when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041058.png" /> that, except for obvious counterexamples, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041059.png" />, and if the [[Kodaira dimension|Kodaira dimension]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041060.png" /> is non-negative, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041061.png" /> unless <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041062.png" /> is a quintic threefold with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041063.png" /> trivial. See [[#References|[a2]]], [[#References|[a1]]] for this and some further discussion of this problem.

==General adjunction theory.==
A more abstract approach to adjunction theory is to start with a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041064.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041065.png" /> an ample line bundle on a projective variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041066.png" /> having at worst mild singularities, e.g., terminal singularities. Then, assuming <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041067.png" /> is not nef, Kawamata's theorem asserts that there is a rational number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041068.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041069.png" />, called the nefvalue of the pair such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041070.png" /> is nef but not ample. By the Kawamata–Shokurov theorem, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041071.png" /> is spanned for all sufficiently large positive integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041072.png" />. The morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041073.png" />, from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041074.png" /> to a normal projective variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041075.png" />, associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041076.png" />, is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041077.png" /> for all sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041078.png" />. This mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041079.png" />, called the nefvalue morphism of the pair, has connected fibres, and at least one positive-dimensional fibre. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041080.png" />, one can express <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041081.png" /> as a very special [[Fibration|fibration]] of Fano varieties, e.g., if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041082.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041083.png" /> is a contraction of an extremal ray (see [[#References|[a3]]] and [[#References|[a1]]], Chapt. 6). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041084.png" />, then one can replace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041085.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041086.png" /> and repeat the procedure. This works well for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041087.png" /> (see [[#References|[a1]]], Chapts. 6, 7).

Define the spectral value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041088.png" /> of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041089.png" /> as the infimum of the positive rational numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041090.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041091.png" /> such that there is some positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041092.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041093.png" />. A major conjecture in this part of the theory is that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041094.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041095.png" /> is the nefvalue of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110410/a11041096.png" /> and the nefvalue morphism has a lower-dimensional image.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M.C. Beltrametti, A.J. Sommese, "The adjunction theory of complex projective varieties" , ''Expositions in Mathematics'' , '''16''' , De Gruyter (1995) {{MR|1318687}} {{ZBL|0845.14003}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M.C. Beltrametti, A.J. Sommese, "On the dimension of the adjoint linear system for threefolds" ''Ann. Scuola Norm. Sup. Pisa Cl. Sci. Ser. (4)'' , '''XXII''' (1995) pp. 1–24 {{MR|1315348}} {{ZBL|0857.14001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> M.C. Beltrametti, A.J. Sommese, J.A. Wiśniewski, "Results on varieties with many lines and their applications to adjunction theory (with an appendix by M.C. Beltrametti and A.J. Sommese)" K. Hulek (ed.) T. Peternell (ed.) M. Schneider (ed.) F.-O. Schreyer (ed.) , ''Complex Algebraic Varieties, Bayreuth 1990'' , ''Lecture Notes in Mathematics'' , '''1507''' , Springer (1992) pp. 16–38</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> T. Fujita, "Classification theories of polarized varieties" , ''London Math. Soc. Lecture Notes'' , '''155''' , Cambridge Univ. Press (1990) {{MR|1162108}} {{ZBL|0743.14004}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> L. Roth, "Algebraic threefolds with special regard to problems of rationality" , Springer (1955) {{MR|0076426}} {{ZBL|0066.14704}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> A.J. Sommese, "Hyperplane sections of projective surfaces, I: The adjunction mapping" ''Duke Math. J.'' , '''46''' (1979) pp. 377–401 {{MR|0534057}} {{ZBL|0415.14019}} </TD></TR></table>