Adjunction formula

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In algebraic geometry, the adjunction formula states that if are smooth algebraic varieties, and is of codimension 1, then there is a natural isomorphism of sheaves:

.

Examples

  • The genus-degree formula for plane curves: Let be a smooth plane curve of degree . Recall that if is a line, then and . Hence

. Since the degree of is , we see that:

.

  • The genus of a curve given by the transversal intersection of two smooth surfaces : let the degrees of the surfaces be . Recall that if is a plane, then and . Hence

and therefore .

e.g. if are a quadric and a cubic then the degree of the canonical sheaf of the intersection is 6, and so the genus of the interssection curve is 4.

Outline of proof and generalizations

The outline follows Fulton (see reference below): Let be a close embedding of smooth varieties, then we have a short exact sequence:

,

and so , where is the total chern class.

References

  • Intersection theory 2nd eddition, William Fulton, Springer, ISBN 0-387-98549-2, Example 3.2.12.
  • Prniciples of algebraic geometry, Griffiths and Harris, Wiley classics library, ISBN 0-471-05059-8 pp 146-147.
  • Algebraic geomtry, Robin Hartshorn, Springer GTM 52, ISBN 0-387-90244-9, Proposition II.8.20.