Birational geometry of terminal quartic 3-folds, I

By Alessio Corti, Massimiliano Di Mella. 2001

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In this paper, we study the birational geometry of certain examples of mildly singular quartic 3-folds. A quartic 3-fold is a special case of a Fano variety, that is, a variety X with ample anticanonical sheaf OX ( − KX ). Nonsingular Fano 3-folds have been studied extensively. Examples studied so far fall within two classes: either X is "close to being rational," and it then has very many biregularly distinct birational models as a Fano 3-fold, or, at the other extreme, X has a unique model. In this paper we construct examples of singular quartic 3-folds with exactly two birational

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/640f799efb54b403a8a859dbf31bd2f8f6223eba

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Citations in the corpus

Fano 3-folds in codimension 4, Tom and Jerry. Part I (2010, methodology): “tiates a 2-ray game on X 1, with the anticanonical model X 1 →Y and its ﬂop Y ←Y+ as ﬁrst step. In many cases, we know how to complete this to a Sarkisov link using Cox rings, in the spirit of [CPR], [CM], [BCZ] and [BZ]; we return to this in Part II. 4 Extended example The case g = 0 plus basket 1 ”
Kustin-Miller unprojection with complexes (2000, background): “uction of the birational involution of X 5. The construction continues by setting y′ = z+a3 x0 = −b5 z , thus unprojecting a diﬀerent plane x 0 = z + a 3 = 0. For details, consult [CPR], 4.4–4.9. See [CM] for a related use of the same algebra, to somewhat surprising eﬀect. 2.4 Projective case The ca”
Birational Geometry of 3-fold Mori Fibre Spaces (2003, background): “divisors having maximal vanishing at the singular point; see [ CM ], [BZ].”
Three-fold divisorial contractions to singularities of higher indices (2003, background): “Cartier at P, or equivalently (i,j) ∈ Z(n,−a). We classify faccording to the numerical data J:= {(rQ,vQ)}Q∈I. Theorem 2.3. fbelongs to one of the types in Table 4. Table 4. No J (a/n)E3 1 (6,3) 1/2 2 (7,3) 2/7 3 (8,3) 1/8 4 (4,2), (r,1) 1/r 5 (5,2), (2,1) 3/10 6 (5,2), (3,1) 2/15 7 (5,2), (4,1) 1/20”
Hyperelliptic and trigonal Fano threefolds (2004, background): “11) from the results of [22], [62], [54], [37], [16], [31]–[33], [60], [38], [52], [34], [28], [29], [61], [111], [5], [82], [63], [98], [64], [6], [102], [7], [59].”
Birational geometry of quartic 3-folds II: The importance of being -factorial (2004, methodology): “Remark 19. The Theorem is an important step in the direction of [ CM , Conjecture”