Contraction criteria for reducible rational curves with components of length one in smooth complex threefolds. (Q1880116)

Let \(C\) be a curve of irreducible components \(C_1,\dots,C_n\) in a smooth projective complex threefold \(X\). One assumes that \(C_i\) is rational nonsingular and \((K_X\cdot C_i)=0\), \(i=1,\dots,n\) (where \(K_X\) is the canonical class of \(X\)). Denote by \(\hat X\) the formal completion of \(X\) along \(C\). Under some additional technical assumption (i.e. each \(C_i\) has Kollár length \(1\)) the author discusses questions of the following type: when the curve \(C\) contracts to a (singular) point, when \(C\) deforms, and when \(C\) neither contracts, nor deforms in \(\hat X\). These questions are closely related with the deformation theory of the compound \(A_n\) singularity. The results of this paper generalize results due to \textit{H. B. Laufer} [Ann. Math. Stud. 100, 261--275 (1981; Zbl 0523.32007)], \textit{M. Reid} [in: Algebraic varieties and analytic varieties, Adv. Stud. Pure Math. 1, 131--180 (1983; Zbl 0558.14028)], \textit{J. Jiménez} [Duke Math. J. 65, 313--332 (1992; Zbl 0781.32025)], \textit{S. Katz} and \textit{D. Morrison} [J. Algebr. Geom. 1, 449--530 (1992; Zbl 0788.14036)], and others. The paper is part of the author's Ph.D dissertation supervised by S. Katz.