User mzwang - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T05:43:07Z http://mathoverflow.net/feeds/user/3525 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/126397/what-is-the-definition-of-a-sufficiently-ample-line-bundle What is the definition of a sufficiently ample line bundle? MZWang 2013-04-03T14:49:35Z 2013-04-03T15:41:22Z <p>I find this concept in Kollar and Mori's book {\em Birational Geometry of Algebraic Varieties}, but cant search the precise definition in the book or google. Can you tell me the definition? Thanks for any comments or references.</p> http://mathoverflow.net/questions/121682/does-negative-semidefinite-intersection-matrix-of-bunch-of-curves-implies-zero-in Does negative semidefinite intersection matrix of bunch of curves implies zero intersection with other curves? MZWang 2013-02-13T07:18:06Z 2013-02-14T01:19:43Z <p>If we have the result as the title, then I can solve my real question. The original question was stated as follows.</p> <p>In a paper I found the following lemma：</p> <p>Let $S$ be a nonsingular projective surface, $R\in PicS$ a divisor with $R^2>0$ . Let </p> <p>$(E_{i})$ be the family of distinct curves such that $R\cdot E_{i}=0$.</p> <p>Then the $E_{i}$ are numerically independent.</p> <p>The proof just says that the result follows from Hodge Index Theorem. But I cant see how. HIT just assert that the intersection matrix $(E_i.E_j)$ is negative semidefinite. Why $E_i$ can't be numerically dependent?</p> <p>Any hint is welcome. Thanks a lot.</p> http://mathoverflow.net/questions/116528/preliminaries-to-bpvs-compact-complex-surfaces Preliminaries to BPV's compact complex surfaces MZWang 2012-12-16T14:59:15Z 2012-12-26T16:04:03Z <p>Where can I find a quick introduction to topological and analytic preliminaries to "Compact complex surfaces" by W. Barth, K. Hulek, C. Peters and A. Van de Ven？ Thanks for any references or comments. </p> http://mathoverflow.net/questions/98190/what-is-the-multiplicity-of-a-cartier-divisor-at-a-point What is the multiplicity of a Cartier divisor at a point？ MZWang 2012-05-28T12:45:30Z 2012-12-05T08:29:30Z <p>In the proof of nonvanishing theorem, people used this concept, but I cant see its definiton.</p> <p>The usual definition can be found in books which only deal with two dimensional case. I never see a definition on higher dimension. It seems that there are at least three descriptions concerned with multiplicity. </p> <p>Let $D$ be an effective Cartier divisor on a variety $X$, let $P\in D$. Pick a local equation $f$ of $D$.</p> <p>(1) Indeicating by Hartshorne exercise V.3.4, one can define the mulitiplicity of the Noetherian local ring $\mathscr{O}_{X,P}/f$ by Hilbert-Samuel polynomial.</p> <p>(2) One can define $\operatorname{mult}_P D:=\operatorname{ord}_P(f)=\max{n\in\mathbb{N};\ f\in \mathfrak{m}^n}$.</p> <p>(3) Set $\operatorname{mult}_PD=\min{C.D;\ C \mbox{ is a curve through P}}$.</p> <p>Does these three description coincides? Where can I find a detailed treatment of basic notions like these? Thank you!</p> http://mathoverflow.net/questions/109577/does-there-exist-a-non-effective-divisor-with-positive-degree Does there exist a non effective divisor with positive degree? MZWang 2012-10-14T02:37:33Z 2012-10-25T05:32:19Z <p>Suppose $X$ be a smooth projective curve over $\mathbb{C}$. Let $D$ be a divisor with $\deg D>0$ on $X$. Is it possible that $l(D)=0$, i.e. D is not linearly equivalent to an effective divisor? </p> http://mathoverflow.net/questions/110126/when-are-the-fibres-numerically-equivalent When are the fibres numerically equivalent? MZWang 2012-10-20T01:33:29Z 2012-10-20T03:29:45Z <p>Let $f:X\to Y$ be an algebraic fibre space, and $\dim Y=1$. What can I say about the numerical relation of the fibres?</p> http://mathoverflow.net/questions/106642/whats-the-meaning-of-pencils-in-birational-geometry What's the meaning of pencils in birational geometry? MZWang 2012-09-08T03:04:21Z 2012-09-08T17:32:57Z <p>I see in some books the authors call a one dimensional linear system a pencil, but in other books one call a linear system $|D|$ is not compsited with a pencil if $\dim \phi_{|D|}(X)\geq 2$ and even someone just say a pencil of curves etc. It seems these terms have different meanings. My question is the following. </p> <p>What is the common pointview about the notion of a pencil? What is the realtions among above terms?</p> http://mathoverflow.net/questions/106349/can-a-birational-morphism-between-smooth-varieties-be-dominated-by-smooth-blow-up Can a birational morphism between smooth varieties be dominated by smooth blow ups sequences MZWang 2012-09-04T14:32:18Z 2012-09-04T19:44:37Z <p>Suppose $f:X\rightarrow Y$ be an birational morphism between smooth varieties and D is a snc divisor on $Y$. Can we find a smooth blow ups sequence on $Y$ which dominates $f$ such that the preimage of D and all other subsequent preimages involved are snc divisors? Thanks for any comments or references.</p> http://mathoverflow.net/questions/104538/a-question-on-pull-back-of-a-nef-and-big-divisor A question on pull back of a nef and big divisor MZWang 2012-08-12T04:35:49Z 2012-08-12T04:35:49Z <p>In his book Higher-Dimensional Algerbraic Geometry, Debarre claimed that the pull back of a nef and big divisor under a generically finite morphism is still nef and big, but he only state the result and no proof. Can somebody tell me why or show me a reference? Thanks.</p> http://mathoverflow.net/questions/94901/characterization-of-big-divisors Characterization of big divisors MZWang 2012-04-23T02:08:26Z 2012-04-27T17:27:11Z <p>In Kollár and Mori's Birational Geometry of Algebraic Varieties, lemma2.60 claim some multiple of a big divisor induced birational morphism onto its image in a projective space. But the proof only show it can be written as a sum of an ample divisor and an effective divisor, and say the result is obvious for the latter. I try to find the details for the latter but have no clues. Thank you for any answer or comments. Furthermore the lemma assume the scheme is a projective variety, does it must be integral? Can the result be done for proper varieties?</p> <p>The second question: The authors also claim a divisor is big iff its birational pullback is big. I know when the varieties are integral normal and proper, this can be done by Zariski' main theorem and projective formula. Are these conditions necessary?</p> <p>Thank you!</p> http://mathoverflow.net/questions/93217/what-is-the-definition-of-exceptional-divisor What is the definition of exceptional divisor？ MZWang 2012-04-05T14:08:46Z 2012-04-05T19:07:58Z <p>Does this concept is defined for every birational morphism？ What is the precise meaning？ Thank you for your comments.</p> http://mathoverflow.net/questions/93212/tensor-product-of-reflexive-sheaves Tensor product of reflexive sheaves MZWang 2012-04-05T13:03:15Z 2012-04-05T15:53:47Z <p>If E and F are reflexive sheaves of rank one， is their tensor product $E\otimes F$ reflexive？ Thanks！</p> http://mathoverflow.net/questions/91803/definition-of-cw-complexes Definition of CW complexes MZWang 2012-03-21T09:01:24Z 2012-03-21T10:59:10Z <p>In Spanier's Algebraic Topology, he defined CW complexes assumed an additional strange condition: the cell must have the coherent topology with the characteristic map and the inclusion map of its boundary. What is the meaning of this condition? Some other author seems never use it, and they asked the space to be Hausdorff. Are the CW complexes Hausdorff in Spanier's way? Do the various definitions agree? Thanks!</p> http://mathoverflow.net/questions/88182/what-is-the-dimension-of-a-sheaf What is the dimension of a sheaf? MZWang 2012-02-11T07:24:39Z 2012-02-12T11:16:22Z <p>In the definition of smooth morphisms, Hartshorne use the notation $dim_{k(x)}(\Omega_{X/Y}\otimes k(x))$· But $\Omega_{X/Y}\otimes k(x)$ is a sheaf, what is the dimension? Thanks for any intepretation.</p> http://mathoverflow.net/questions/87242/lemma-of-hironaka-in-hartshorne-iii-9-12 Lemma of Hironaka in Hartshorne III.9.12 MZWang 2012-02-01T15:54:10Z 2012-02-11T07:16:28Z <p>In the proof, the author consider the normalization $\tilde{A}$ of $A$ and show $\tilde{A}/t \tilde{A}$ is a integral domain. He showed that the localizations at points of Spec $A$ are domains, but we know a non-domain ring can have integral localizations. How should I understand the proof? Thanks a lot.</p> http://mathoverflow.net/questions/83498/global-sections-of-tensor-product-of-pull-back-of-two-vector-bundles Global sections of tensor product of pull-back of two vector bundles MZWang 2011-12-15T06:00:02Z 2011-12-17T13:03:08Z <p>Suppose X,Y are two complex manifolds and E,F are vector bundles over them respective, what can I say about their global sections? Does this formula $\Gamma (X\times Y,p_1^{*}E\otimes p_2^{*}F)=\Gamma (X,E)\otimes\Gamma (Y,F)$ hold？</p> <p>If it holds，then we will get every holomorphic function of two complex variables will be the form $f_1(z_1)g_1(z_2)+\cdots+f_n(z_1)g_n(z_2)$. It seems plausibile.</p> http://mathoverflow.net/questions/81949/what-is-the-subscheme-associate-to-a-weil-divisor What is the subscheme associate to a Weil divisor? MZWang 2011-11-26T14:45:05Z 2011-11-26T14:45:05Z <p>I see some notation as D_red， I guess this mean consider divisor as a scheme then reduce it. So what is the subscheme? Thank your answer or references.</p> http://mathoverflow.net/questions/80984/what-is-the-meaning-of-canonical-isomorphism-in-the-cocycle-construction What is the meaning of canonical isomorphism in the cocycle construction? MZWang 2011-11-15T14:17:15Z 2011-11-15T14:17:15Z <p>In the definition of pull back of a smooth vector bundle, one usually use the cocycle description of vector bundle. But this method only decide the object up to isomorphism. Then the statement "the fibers of pull back is canonically ismorphic to the fibers of image points" seem cant give a precise meaning about canonically. Or there is some uniqueness statement about the cocycle description? Thx!</p> http://mathoverflow.net/questions/75581/an-irreducible-germ-of-holomorphic-function-at-origin-is-still-irreducible-around An irreducible germ of holomorphic function at origin is still irreducible around the origin? MZWang 2011-09-16T08:21:55Z 2011-09-16T12:57:58Z <p>This question comes from Huybrechts's book <em>Complex Geometry, An Introduction</em>. In proposition 1.1.35, the author claims that if $f$ is an irreducible holomorphic germ in $\mathcal{O}_{\mathbf{C}^n,0}$ at the origin of $\mathbf{C}^n$, then for any $z$ sufficiently close to the origin the holomorphic germ induced by $f$ in the local ring of $\mathbf{C}^n$ at z is irreducible. </p> <p>But the proof only shows the claim holds on the complement of a thin subset.</p> <p><strong>Question.</strong> Is the claim true or false? Can anyone give an answer or a reference?</p> http://mathoverflow.net/questions/73936/the-restriction-of-a-global-section-which-is-not-a-zero-divisor-is-still-an-non-z The restriction of a global section which is not a zero divisor is still an non-zero divisor? MZWang 2011-08-29T03:48:04Z 2011-08-29T12:13:39Z <p>Let X be a scheme. U is an open subscheme of X. Assume f is a global section on X which is not a zero divisor, then the restriction of f to U is still an non-zero divisor?</p> <p>If X is affine, the answer is obvious true. I don't know the answer for a general scheme.</p> <p>This is a question raised in the definition of sheaf of total fraction rings. Some author claim U|-> total fraction ring of sections over U is a presheaf, but I can't see the reason.</p> http://mathoverflow.net/questions/70503/stalks-of-structure-sheaf-of-fibre-product Stalks of structure sheaf of fibre product？ MZWang 2011-07-16T13:54:44Z 2011-07-18T07:35:24Z <p>What can I say about it？</p> <p>Can I say the stalks equal the tensor products of the corresponding factors stalks？</p> <p>Thanks！</p> http://mathoverflow.net/questions/70503/stalks-of-structure-sheaf-of-fibre-product/70536#70536 Answer by MZWang for Stalks of structure sheaf of fibre product？ MZWang 2011-07-17T02:43:11Z 2011-07-17T02:43:11Z <p>As Hartshorne chapter III.9.2 claim，an Ox-module （need not be quasi coherent) F's flatness is stable under base change. But the stalks is not the tensor products, how can I prove the claim?</p> http://mathoverflow.net/questions/67721/question-about-the-exact-sequences-of-sheaves-of-relative-differentials Question about the exact sequences of sheaves of relative differentials MZWang 2011-06-14T03:18:05Z 2011-06-14T10:19:56Z <p>Hartshorne gave the exact sequences in Chapter II.8 and just say they followed from the affine case. But this view should consider the compkex glueing construction. I am disturbed completely. How should I thunk this? Thanks!</p> http://mathoverflow.net/questions/67027/preliminaries-for-mumfords-abelian-varieties Preliminaries for Mumford's Abelian Varieties MZWang 2011-06-06T11:12:38Z 2011-06-06T12:06:25Z <p>I want to learn the book, but it seems that I should have some background on harmonic analysis, Lie Groups and measure theory. Can you give some references?</p> http://mathoverflow.net/questions/57320/what-is-the-meaning-of-independent What is the meaning of "independent"? MZWang 2011-03-04T05:07:08Z 2011-03-04T05:07:08Z <p>I often find some statements which say the word independent, but they are obviousy not. </p> <p>E.g. An exercise (in Atiyah and Macdonal's Introduction to Commutative Algebra) said given a commutative ring A, the ring $A(D(f))=A_f$ depends only on $D(f)$. But $A_f$ is clearly different with $A_g$ even though $D(f)=D(g)$. </p> <p>So what is the meaning? </p> http://mathoverflow.net/questions/46932/what-is-the-definition-of-product-of-ideal-sheaves What is the definition of product of ideal sheaves? MZWang 2010-11-22T11:13:14Z 2010-11-22T11:17:49Z <p>Each book on algebraic geometry write I^2 when it deal with nongsingular varieties, here I is a ideal sheaf. But no one give the definition. I guess it's the sheafification. It's right?</p> <p>Thanks.</p> http://mathoverflow.net/questions/126397/what-is-the-definition-of-a-sufficiently-ample-line-bundle/126405#126405 Comment by MZWang MZWang 2013-04-05T03:29:39Z 2013-04-05T03:29:39Z Thank you. That is what I want to find. http://mathoverflow.net/questions/121682/does-negative-semidefinite-intersection-matrix-of-bunch-of-curves-implies-zero-in/121696#121696 Comment by MZWang MZWang 2013-02-15T02:19:03Z 2013-02-15T02:19:03Z Why are the classes of $E_i$ orthogonal？ http://mathoverflow.net/questions/121682/does-negative-semidefinite-intersection-matrix-of-bunch-of-curves-implies-zero-in/121761#121761 Comment by MZWang MZWang 2013-02-15T02:16:07Z 2013-02-15T02:16:07Z Thank you very much. http://mathoverflow.net/questions/116528/preliminaries-to-bpvs-compact-complex-surfaces Comment by MZWang MZWang 2012-12-21T12:07:09Z 2012-12-21T12:07:09Z Dear Tim, you are right. I need some topological background such as why the singular cohomology coincides with the sheaf cohomology. Thanks for diverrietti's edit. http://mathoverflow.net/questions/110126/when-are-the-fibres-numerically-equivalent/110127#110127 Comment by MZWang MZWang 2012-10-20T14:52:37Z 2012-10-20T14:52:37Z Thank you. And I see the key concept. http://mathoverflow.net/questions/109577/does-there-exist-a-non-effective-divisor-with-positive-degree Comment by MZWang MZWang 2012-10-14T04:51:09Z 2012-10-14T04:51:09Z Thanks. Can you show me some examples or references? http://mathoverflow.net/questions/106642/whats-the-meaning-of-pencils-in-birational-geometry/106651#106651 Comment by MZWang MZWang 2012-09-10T12:16:29Z 2012-09-10T12:16:29Z Dear Sandor, thanks for your clarification. http://mathoverflow.net/questions/106642/whats-the-meaning-of-pencils-in-birational-geometry/106650#106650 Comment by MZWang MZWang 2012-09-10T12:15:14Z 2012-09-10T12:15:14Z @Jack: I prefer this explanation. Although I still have some confusion about a divisor composed of a pencil. http://mathoverflow.net/questions/106642/whats-the-meaning-of-pencils-in-birational-geometry/106651#106651 Comment by MZWang MZWang 2012-09-10T12:11:53Z 2012-09-10T12:11:53Z There are people still talking that's divisor composed of a pencil. I have some confusion with that. http://mathoverflow.net/questions/106349/can-a-birational-morphism-between-smooth-varieties-be-dominated-by-smooth-blow-up/106355#106355 Comment by MZWang MZWang 2012-09-05T02:27:00Z 2012-09-05T02:27:00Z Thanks a lot. You are very nice. http://mathoverflow.net/questions/87242/lemma-of-hironaka-in-hartshorne-iii-9-12 Comment by MZWang MZWang 2012-08-30T03:32:33Z 2012-08-30T03:32:33Z Julien, the problem is to show $\tilde{A}/t\tilde{A}$ is a domain. http://mathoverflow.net/questions/104538/a-question-on-pull-back-of-a-nef-and-big-divisor Comment by MZWang MZWang 2012-08-25T09:37:09Z 2012-08-25T09:37:09Z Thanks to Yusuf and Jason. http://mathoverflow.net/questions/98190/what-is-the-multiplicity-of-a-cartier-divisor-at-a-point Comment by MZWang MZWang 2012-05-29T00:32:41Z 2012-05-29T00:32:41Z Thanks to the above answers! http://mathoverflow.net/questions/16888/when-do-divisors-pull-back/18415#18415 Comment by MZWang MZWang 2012-05-07T01:46:53Z 2012-05-07T01:46:53Z Did not ask X to be integral？ Why should the pull back be a subsheaf of the sheaf of total quotients？ http://mathoverflow.net/questions/94901/characterization-of-big-divisors Comment by MZWang MZWang 2012-05-03T11:30:21Z 2012-05-03T11:30:21Z Dear Ottem, can you tell me how to understand the scheme structure on the image?