Examples Of Threefolds Research Articles

On examples of supersingular and rationally chain connected threefolds

In this work, we show that for a certain class of threefolds in positive characteristics, rational-chain-connectivity is equivalent to supersingularity. The same result is known for K3 surfaces with elliptic fibrations. And there are examples of threefolds that are both supersingular and rationally chain connected.

Examples of Threefolds with Kodaira Dimension 1 or 2

We construct three nonsingular threefolds X , X' and X'' with vanishing irregularities. X has Kodaira dimension =\kappa(X)=1 and its m -canonical transformation \varphi_{|mK_X|} has the following property: the minimum integer number m_0 , such that the dimension of the image \mathrm{dim} \varphi_{|mK_X|}(X)=\kappa(X)=1 for m \geq m_0 , is given by m_0=32 . X' and X'' have Kodaira dimension \kappa(X')=\kappa(X'')=2 and their m -canonical transformations have the properties: \mathrm{dim}\varphi_{|mK_{X'}|}(X')=\kappa(X')=2 if and only if m \geq 12 , \mathrm{dim} \varphi_{|mK_{X''}|}(X'')=\kappa(X'')=2 if and only if m = 9, 10 or m \geq 12 .

K3 surfaces with non-symplectic involution and compact irreducibleG2-manifolds

AbstractWe consider the connected-sum method of constructing compact Riemannian 7-manifolds with holonomyG2developed by the first named author. The method requires pairs of projective complex threefolds endowed with anticanonicalK3 divisors and the latterK3 surfaces should satisfy a certain ‘matching condition’ intertwining on their periods and Kähler classes. Suitable examples of threefolds were previously obtained by blowing up curves in Fano threefolds.In this paper, we give a large new class of suitable algebraic threefolds using theory ofK3 surfaces with non-symplectic involution due to Nikulin. These threefolds arenotobtainable from Fano threefolds as above, and admit matching pairs leading to topologically new examples of compact irreducibleG2-manifolds. ‘Geography’ of the values of Betti numbersb2,b3for the new (and previously known) examples of irreducibleG2manifolds is also discussed.

HODGE-THEORETIC OBSTRUCTION TO THE EXISTENCE OF QUATERNION ALGEBRAS

This paper gives a necessary criterion in terms of Hodge theory for representability by quaternion algebras of certain 2-torsion classes in the unramified Brauer group of a complex function field. This criterion is used to give examples of threefolds with unramified Brauer group elements which are the classes of biquaternion division algebras.

On algebraic threefolds whose hyperplane sections are Enriques surfaces

In this paper the following problem is solved: given a singular Fano variety X, find a smooth Enriques surface which is an ample Cartier divisor on X. The results obtained enable one to construct, using singular Fano varieties, examples of threefolds whose hyperplane sections are Enriques surfaces. They can be used in the classification of log-Fano varieties of (Fano) index 1.

Cycles on the generic abelian threefold

H Clemens and C Schoen gave examples of three-folds where the group of codimension two cycles modulo algebraic equivalence has infinite rank. This paper provides yet another example of the same phenomenon.