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Topic: K3 surface

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	K3 surface - Wikipedia, the free encyclopedia
		A K3 surface is an important and interesting example of a compact complex surface (complex dimension 2 being real dimension 4).
		However, K3 surfaces first arose in algebraic geometry and it is in this context that they received their name — it is after three algebraic geometers, Kummer, Kähler and Kodaira, alluding also to the mountain peak K2 in the news when the name was given during the 1950s.
		As 4-dimensional real manifolds, all K3 surfaces are diffeomorphic to one another and so have the same Betti numbers: 1, 0, 22, 0, 1.
	en.wikipedia.org /wiki/K3_surface (403 words)

	K3 Surfaces (Site not responding. Last check: 2007-10-08)
		A K3 surface with genus g and basket of singularities B (which may be a basket type or in raw basket format [[r, a],...]).
		Return a new K3 surface that is the same as X but with the positive integer w included among the weights and all other data associated to the embedding adjusted as required.
		Return a new K3 surface that is the same as X but with the positive integer w removed from the weights, assuming it appears there and can be removed without destroying the property of the Hilbert numerator being a polynomial.
	www.math.lsu.edu /magma/text1195.htm (330 words)

	K3 surface -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		A Kummer surface is the quotient of a two-dimensional (Click link for more info and facts about abelian variety) abelian variety A by the action of a → −a.
		K3 manifolds play an important role in (Click link for more info and facts about string theory) string theory because they provide us with the second simplest compactification after the (Commonly the lowest molding at the base of a column) torus.
		Compactification on a K3 surface preserves one half of the original ((physics) a theory that tries to link the four fundamental forces) supersymmetry.
	www.absoluteastronomy.com /encyclopedia/k/k3/k3_surface.htm (400 words)

	Explain.html (Site not responding. Last check: 2007-10-08)
		The group of automorphisms of a K3 surface generates a tiling of a hyperbolic space of dimension n-1, where n is the Picard number for the K3 surface.
		The K3 surface is generated by a smooth (2,2,2) form in P
		The nef cone and dynamics on a K3 surface.
	www.nevada.edu /~baragar/Explain.html (254 words)

	Downregulation of Major Histocompatibility Complex Class I Molecules by Kaposi's Sarcoma-Associated Herpesvirus K3 and ...
		The cell surface levels of MHC class I molecules were assessed 48 h posttransfection by staining the cells with a W6/32 antibody for MHC class I or TU39 antibody for MHC class II (y axis) and gating the GFP-positive cell population (x axis) by flow cytometry.
		Cell surface levels of MHC class I molecules were assessed 48 h posttransfection by staining the cells with PE-conjugated pan-class I W6/32 antibody (y axis) and gating the GFP-positive cell population (x axis) by flow cytometry.
		The level of MHC class I surface expression was assessed 48 h posttransfection by staining the cells with a PE-conjugated W6/32 antibody for class I (y axis) and gating the GFP-positive cell population (x axis) by flow cytometry.
	jvi.asm.org /cgi/content/full/74/11/5300 (5441 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		In particular, he states that for quartic surfaces there are 4184 connected strata of the type we are considering, representing 2523 possible constellations of singularities; and that for sextic curves, there are 638 in each case.
		Since a quartic surface with only simple singularities is a K3-surface, and so finds its place in the moduli space of such surfaces, while one with a higher singularity is not, it is to be expected that stability in the Hilbert-Mumford sense will break down precisely when a higher singularity is present.
		Examples of such surfaces are surfaces $X$ as above, the intersections of 3-folds of degrees 2 and 3 in $P^{4}({\Bbb C})$, and intersections of sets of 3 hypersurfaces of degree 2 in $P^{5}({\Bbb C})$, assuming isolated singularities in each case.
	home.imf.au.dk /esn/preprints/006 (14107 words)

	Accessing the K3 Database
		The arbitrary integer assigned to the K3 surface X for the duration of the current session.
		The basket of singularities of the K3 surface X. Each singularity of the form oneover(r)(a, r - a) is listed in the basket as [r, a] with r, a coprime and r >= 2a.
		An integer, at least -1, that is one less that the number of 1s in the weights of the K3 surface X. In the classical theory, this is the genus of a polarising curve on a K3 of the family.
	www.math.niu.edu /help/math/magmahelp/text1102.html (695 words)

	The K3 Database
		Recall from Section K3 Surfaces the meaning of K3 surface in this context, and from Section Key Warning and Disclaimer the relationship between the Hilbert series, the weights and the (Hilbert) numerator.
		The `Number' of a K3 surface in the database and its `Index' may differ: the K3 surfaces of any fixed genus are numbered separately, while the index runs over the whole database.
		The K3 surface in the K3 database D with genus g >= - 1 and basket of singularities B (as a basket type or in raw sequence format).
	wwwmaths.anu.edu.au /research.groups/aat/htmlhelp/text1174.htm (1261 words)

	Abelian fibred holomorphic symplectic manifolds (Site not responding. Last check: 2007-10-08)
		In studying the moduli space of K3 surfaces, one typically looks at Kummer surfaces or quartics in P^3, as these are dense but also relatively easy to understand.
		There is evidence to suggest that if the 2n-dimensional irreducible holomorphic symplectic manifold X admits a non-trivial fibration, then the fibres must be abelian varieties and the base must be P^n (a large part of this has been proved by Matsushita).
		So the `right' generalization of an elliptic fibration on a K3 surface appears to be a fibration by n-dimensional abelian varieties over P^n, which we shall call an abelian fibration.
	www.math.sunysb.edu /~sawon/abelianHK.shtml (575 words)

	[No title]
		In the (g,R) case, the singular rank of a K3 X is the number of curves extracted by a minimal resolution.
		If a surface Y has a singularity p, one can make the projection and find another K3 as long as the resulting degree is positive --- this can all be predicted from the numerical types alone.
		However, it is a condition on a surface that it contain a line, so the images of such projections are rather special elements in their family.
	www.maths.warwick.ac.uk /~gavinb/threetalks/lecture2 (2921 words)

	Arithmetic and Geometry Seminar: Abstract (Site not responding. Last check: 2007-10-08)
		It follows from the classification theory of algebraic surfaces that there are at most finitely many rational curves in each linear system on a K3 surface.
		The number of rational curves in the primitive class of a K3 surface was calculated by E. Zaslow and S.T. Yau.
		To prove this, we need to degenerate a K3 surface to a union of rational surfaces and then study the corresponding limits of rational curves.
	www.math.ucsb.edu /~mckernan/98-99/chen.html (154 words)

	Building the K3 Database
		They will compute the Hilbert series of the candidate K3 surface from the data you give them and then will make some attempt to describe a ring having this series.
		A new K3 surface embedded in a K3 surface with an additional variable of weight w in the first case, or with such a variable omitted.
		A new K3 surface having the numerical invariants of X but with extra generators added to make it possible to realise the polarisations of the singularities of X as described by its basket.
	www.umich.edu /~gpcc/scs/magma/text1162.htm (696 words)

	Physical Association of the K3 Protein of Gamma-2 Herpesvirus 68 with Major Histocompatibility Complex Class I ...
		L-Ld cells and their K3 transfectants and B6/WT-3 cells and their K3 transfectants were analyzed for surface class I expression.
		Cell lysates from L-Ld and L-Ld+K3 were incubated with the indicated dilution of the HCMV peptide (YPHFMPTNL) or the CW3 peptide (RYLKNGKETL) for 2 h on ice prior to precipitation with MAb 64-3-7.
		The association of K3 with TAP and the peptide loading complex is intriguing and raises questions of functional significance.
	jvi.asm.org /cgi/content/full/76/6/2796 (5240 words)

	Kuga-Satake varieties and K3 surfaces (Site not responding. Last check: 2007-10-08)
		In Chapter 2 of the thesis we study the Kuga-Satake-Deligne correspondence.
		When the K3 surface is the desingularization of a double cover of the plane branched along six lines in general position, the abelian variety is isogenous to the product of a number of copies of a four dimensional abelian variety.
		We construct the Kuga-Satake-Deligne correspondence between this four dimensional abelian variety and the K3 surface in this case.
	www.imsc.ernet.in /~kapil/work/node3.html (146 words)

	UC Berkeley Mathematics
		Not much is known about the arithmetic of K3 surfaces in general.
		But still we don’t know of a single K3 surface whose set of rational points has been proved to be neither empty nor Zariski dense.
		Also, until recently, not even a single K3 surface was known with Neron-Severi rank 1 and infinitely many rational points.
	math.berkeley.edu /calendar-event572.html (171 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Title: K3 surfaces, vector bundles, and isogenies I,II Abstract: We will review the geometric properties of periods of K3 surfaces, period domains, and the resulting Torelli theorems.
		Then we will turn to isogenies between K3 surfaces, i.e., rational isomorphisms between their periods, and interpretations in terms of Fourier-Mukai transforms.
		We shall also explain how one K3 surface can be the moduli space of vector bundles on another K3 surface.
	math.rice.edu /~hassett/seminar/K3.html (69 words)

	LMS Proceedings Abstract, paper PLMS 1521 (Site not responding. Last check: 2007-10-08)
		Classification of real K3 surfaces $X$ with non-symplectic involution $\tau$ is considered.
		We show that the connected component of moduli of non-degenerate surfaces of this type is defined by the isomorphism class of the action of $\tau$ and the anti-holomorphic involution $\varphi$ in the homology lattice.
		As an application, we describe all real polarized K3 surfaces that are deformations of general real K3 double rational scrolls (the surfaces $V$ above).
	www.lms.ac.uk /publications/proceedings/abstracts/p1521a.html (176 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Using the Kodaira dimension and the fundamental group of X, we succeed in classifying algebraic surfaces which are dominable by C^2 except for certain cases in which X is an algebraic surface of Kodaira dimension zero and the case when X is rational without any logarithmic 1-form.
		More specifically, in the case when X is compact (namely projective), we need to exclude only the case when X is birationally equivalent to a K3 surface (a simply connected compact complex surface which admits a globally non-vanishing holomorphic 2-form) that is neither elliptic nor Kummer.
		With the exceptions noted above, we show that for any algebraic surface of Kodaira dimension less than 2, dominability by C^2 is equivalent to the apparently weaker requirement of the existence of a holomorphic image of C which is Zariski dense in the surface.
	celestial.eprints.org /cgi-bin/oaia2/arXiv.org?verb=GetRecord&identifier=oai:arXiv.org:math/9903193&metadataPrefix=oai_dc (205 words)

	Citebase - Algebraic surfaces holomorphically dominable by C^2
		Using the Kodaira dimension and the fundamental group of X, we succeed in classifying algebraic surfaces which are dominable by C 2 except for certain cases in which X is an algebraic surface of Kodaira dimension zero and the case when X is rational without any logarithmic 1-form.
		THEOREM 1.2 A projective surface X not birationally equivalent to a K3 surface is dominable by C 2 if and only if it has Kodaira dimension less than two and its fundamental group is a finite extension of an abelian group (of even rank four or less).
		DEFINITION 3.2 An elliptic fibration is a proper holomorphic map from a surface to a curve whose general fiber is an elliptic curve, i.e., a curve of genus one.
	citebase.eprints.org /cgi-bin/citations?archiveID=oai:arXiv.org:math/9903193 (7547 words)

	K3 Surface Database
		The computation of generic families of K3 surfaces embedded in weighted projective spaces of dimension at most
		The result is a collection of about 400 families of K3 surfaces.
		Another point is that these families exhibit large Gorenstein rings with as yet unknown structure.
	magma.maths.usyd.edu.au /magma/Features/node235.html (211 words)

	3 Utilities Puzzle: Water, Gas, Electricity
		A graph is a collection of nodes (also called vertices) and edges each connecting a pair of nodes.
		To visualize a graph, nodes may be thought of as points in space, plane, or another surface, while edges are represented by curves connecting the nodes.
		were planar, it would have a natural orientation inherited from the plane, hence could not be filled with a nonorientable surface.
	www.cut-the-knot.org /do_you_know/3Utilities.shtml (1380 words)

	[No title]
		The second will be the moduli space of principal bundles on an elliptic K3 surface.
		The main geometric objects involved are holomorphic symplectic surfaces which are the resolutions of simple singularities and Hilbert schemes of points on a surface which is a resolution of orbifold singularities.
		For a Riemann surface $X$ equipped with a projective structure, the space of differential operators on $X$ admits a simple description.
	www.math.sunysb.edu /~mde/seminarAY_99_00.html (1318 words)

	Untitled Document
		Any such group turns out to be isomorphic to a subgroup of the Mathieu group $M_{23}$ which has at least 5 orbits in its natural action on the set of 24 elements.
		None of the 3 proofs extends to the case of K3 surfaces over algebraically closed fields of positive characteristic $p$.
		For $p=2,3,5, 11$, we give examples of K3 surfaces over a field of characteristic $p$ whose automorphism group contains a finite symplectic subgroup not contained in Mukai's list.
	www.math.uga.edu /seminars_conferences/11.29.04.html (502 words)

	String theory - Wikipedia, the free encyclopedia
		This "extra dimension" is only visible within a relatively close range to the hose, just as the extra dimensions of the Calabi-Yau space are only visible at extremely small distances, and thus are not easily detected.
		A point on the hose's surface can be specified by two numbers, a distance along the hose and a distance along the circumference, just as points on the Earth's surface can be uniquely specified by latitude and longitude.
		In either case, we say that the object has two spatial dimensions.
	en.wikipedia.org /wiki/String_Theory (2656 words)

	A singular K3 surface related to sums of consecutive cubes (Site not responding. Last check: 2007-10-08)
		A singular K3 surface related to sums of consecutive cubes
		We study the surface arising from the diophantine equation $m^3+(m+1)^3+\dots+(m+k-1)^3=l^2$.
		It turns out that this is a $K3$ surface with Picard number 20.
	www.math.uiuc.edu /Algebraic-Number-Theory/0204 (77 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		With both the origins and the modern techniques in mind, I will use the spherical pendulum to illustrate physically some of the topology involved in the mirror symmetry of K3 surfaces, examples of which are hypersurfaces given by an equation of degree four.
		Specifically, the set of positions and velocities (or linear momenta) of a spherical pendulum (its phase space) breaks up into subsets with fixed energy and fixed angular momentum.
		Meanwhile, certain projections of a K3 surface to S^2 have level sets that are either tori or pinched tori.
	www.math.psu.edu /oldColloquium/040916.html (205 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Let $M$ be a K3 surface, $B$ a stable bundle on $M$, and $X$ the coarse moduli of stable deformations of $B$.
		A projectivization of a stable bundle on a K3 surface is hyperholomorphic, that is, holomorphic with respect to any complex structure induced by the hyperkaehler structure.
		Projectively hyperholomorphic bundles are stable; therefore, a pushforward of a projectively hyperholomorphic bundle is has stable cohomology.
	www.newton.cam.ac.uk /programmes/MTH/Verbitsky.html (157 words)

	hep-th/0506014--Fixing All Moduli for M-Theory on K3xK3 \| Physics Comments (Site not responding. Last check: 2007-10-08)
		The integral cohomology of 2-forms in a K3 surface, is isomorphic to the (unique) even self-dual (ESD) lattice of signature (3,19).
		For generic complex structure of K3, the Picard lattice is completely trivial, however for some suitable complex structure it can be nonempty.
		A K3 surface is said to be "attractive" whenever the rank of it's Picard lattice is 20.
	www.physcomments.org /node/316 (346 words)

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