7,011 research outputs found
Union Closed Set Conjecture and Maximum Dicut in Connected Digraph
In this dissertation, we study the following two topics, i.e., the union closed set conjecture and the maximum edges cut in connected digraphs. The union-closed-set-conjecture-topic goes as follows. A finite family of finite sets is {\it union closed} if it contains the union of any two sets in it. Let . A union closed family of sets is {\it separating} if for any two distinct elements in , there is a set in containing one of them, but not the other and there does not exist an element which is contained in every set of it. Note that any union closed family is a poset with set inclusion as the partial order relation. A separating union closed family is {\it irreducible} ({\it normalized}) if is the minimum (maximum, resp.) with respect to the poset structure of . In the part of dissertation related to this topic, we develop algorithms to transfer any given separating union closed family to a/an normalized/irreducible family without changing its poset structure. We also study properties of these two extremal union closed families in connection with the {\it Union Closed Sets Conjecture} of Frankl. Our result may lead to potential full proof of the union closed set conjecture and several other conjectures. The part of the dissertation related to the maximum edge cuts in connected digraphs goes as follows. In a given digraph , a set of edges is defined to be a {\it directed cut} if there is a nontrivial partition of such that consists of all the directed edges from to . The maximum size of a directed cut in a given digraph is denoted by , and we let be the set of all digraphs such that or for every vertex in . In this part of dissertation, we prove that for any connected digraph , which provides a positive answer to a problem of Lehel, Maffray, and Preissmann. Additionally, we consider triangle-free digraphs in and answer their another question
Curves in the double plane
We study locally Cohen-Macaulay curves in projective three-space which are
contained in a double plane 2H, thus completing the classification of curves
lying on surfaces of degree two. We describe the irreducible components of the
Hilbert schemes of locally Cohen-Macaulay curves in 2H of given degree and
arithmetic genus. We show that these Hilbert schemes are connected. We also
discuss the Rao modules of these curves, and liaison and biliaison equivalence
classes.Comment: 20 page
Rationally cubic connected manifolds I: manifolds covered by lines
In this paper we study smooth complex projective polarized varieties (X,H) of
dimension n \ge 2 which admit a dominating family V of rational curves of
H-degree 3, such that two general points of X may be joined by a curve
parametrized by V, and such that there is a covering family of rational curves
of H-degree one. Our main result is that the Picard number of these manifolds
is at most three, and that, if equality holds, (X,H) has an adjuction theoretic
scroll structure over a smooth variety
Manifolds covered by lines and extremal rays
Let be a smooth complex projective variety and let H \in \pic(X) be an
ample line bundle. Assume that is covered by rational curves with degree
one with respect to and with anticanonical degree greater than or equal to
. We prove that there is a covering family of such curves whose
numerical class spans an extremal ray in the cone of curves \cone(X).Comment: Major revision, to appear in Canadian Mathematical Bulleti
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