User stjc - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T23:29:24Z http://mathoverflow.net/feeds/user/15124 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/126928/relative-resolution-of-singularity relative resolution of singularity stjc 2013-04-09T02:32:49Z 2013-04-09T20:06:45Z <p>Is there a relative version of resolution of singularities (in characteristic 0)?</p> <p>For finite type morphism $f:X\rightarrow S$ where $S$ is a variety over a field $k$ with char $k=0$(or more generally excellent scheme with residue field are characteristic 0?). Are there modifications $i:S' \rightarrow S$, $j:X'\rightarrow X$, and morphism $f':X'\rightarrow S'$ s.t. $fj=if'$, and $f'$ smooth, $i$ proper, surjective. I'm not sure the above statement is a proper version of relative desingularities.</p> <p>Thanks</p> http://mathoverflow.net/questions/126489/is-there-non-simple-connected-projective-varietyover-c-with-trivial-etale-funda Is there non-simple-connected projective variety(over C) with trivial etale fundamental group? stjc 2013-04-04T08:38:51Z 2013-04-07T15:03:49Z <p>As the title. Geometrically, is there a projective complex manifold(or more generally an projective algebraic variety) accepting only infinite nontrivial cover(which may not be projective)? Thanks.</p> http://mathoverflow.net/questions/126225/how-to-define-intersection-of-coherent-sheaf-and-1-cycle How to define intersection of coherent sheaf and 1-cycle? stjc 2013-04-02T02:50:01Z 2013-04-02T05:43:26Z <p>Given a variety $X$, a coherent sheaf $F$ which may not have finite locally free resolution, and a curve（integral） $C$ in $X$. How can we define intersection $F.C$ properly?(I mean 'properly' by having good behavior on short exact sequence and have projection formula)</p> <p>Here is my attempt:</p> <ol> <li><p>One can define $F.C$ as degree of $F|_C$ on $C$, suggest by Riemann-Roch, namely $\chi(F|C)-rank(F|C)\chi(\mathcal{O}_C)$. However, in this setting, projection formula only hold for locally free sheaf。</p></li> <li><p>we take locally free resolution of $F$, and then pull them back（in fact we get $Li^\ast(F)$, where $i:C\rightarrow X$ is the inclusion） to $C$ and define intersection of $Li^\ast(F)$ with $C$. Where we have problem that $Li^\ast(F)$ may be unbounded.</p></li> </ol> <p>So both attempt fail.</p> http://mathoverflow.net/questions/114238/on-condition-when-the-push-forward-of-coherent-sheaf-is-locally-free On condition when the push-forward of coherent sheaf is locally free stjc 2012-11-23T14:05:35Z 2012-11-26T14:03:54Z <p>This is a result being widely used in the literature:</p> <p>$f:X\rightarrow Y$ proper morphism between Noetherian schemes. $F\in Coh(X)$ flat over $Y$, if $H^i(X_y,F_y)=const$, $y\in Y$, then $R^if_\ast F$ is locally free.</p> <p>The problem is that I can only find references (EGA or GTM 52, etc) of this result with a condition 'Y is reduced', which seems to be unavoidable if one try to prove it by using the canonical technique of 'Grothendieck complex'.</p> <p>However, the general case is crucial in many arguments.</p> <p>So my question is that: Is the general case true? When can I find the proof of the general case?</p> <p>Thanks!</p> http://mathoverflow.net/questions/90551/what-does-the-lefschetz-principle-in-algebraic-geometry-mean-exactly What does the Lefschetz principle (in algebraic geometry) mean exactly? stjc 2012-03-08T06:36:05Z 2012-03-10T16:03:25Z <p>This principle claims that every true statement about a variety over the complex number field $\mathbb{C}$ is true for a variety over any algebraic closed field of characteristic 0.</p> <p>But what is it mean? Is there some "statement" not allowed in this principle?</p> <p>Is there an analog in char p>0?</p> <p>Is there reference about this topic? I tried to find some but in vain.</p> <p>Thanks:)</p> http://mathoverflow.net/questions/89395/is-there-a-classification-of-surfacesmooth-and-projective-over-arbitrary-field Is there a classification of surface(smooth and projective) over arbitrary field? stjc 2012-02-24T13:25:42Z 2012-02-26T15:15:58Z <p>Is there a classification of surfaces(smooth and projective) over arbitrary field? Whether using the approach of Enriques or not. thanks</p> <p>P.S. By arbitrary I mean the field may not be algebraic closed, even not perfect, since as far as I know variety over perfect field is much like one over closed field. So Is there a treatment on the case of non-perfect case. Thanks</p> http://mathoverflow.net/questions/28595/does-there-exist-a-riemann-surface-corresponding-to-every-field-extension-any-ot/89393#89393 Answer by stjc for Does there exist a Riemann surface corresponding to every field extension? Any other hypothesis needed? stjc 2012-02-24T13:19:45Z 2012-02-24T13:19:45Z <p>For compact Riemann surface, the algebraic approach and the analytic approach is the same(Chow's lemma), so the algebraic answer is sufficient for you. i.e. For every extension finitely generated over \mathbb{C} which has transcendental degree 1(up to isomorphism), there is a unique Riemann surface you want.</p> http://mathoverflow.net/questions/126489/is-there-non-simple-connected-projective-varietyover-c-with-trivial-etale-funda Comment by stjc stjc 2013-04-07T11:50:56Z 2013-04-07T11:50:56Z @Misha Sorry, I have made the question precise. I only concern projective case. http://mathoverflow.net/questions/114238/on-condition-when-the-push-forward-of-coherent-sheaf-is-locally-free/114289#114289 Comment by stjc stjc 2012-11-25T12:51:13Z 2012-11-25T12:51:13Z This really helped, thanks http://mathoverflow.net/questions/89395/is-there-a-classification-of-surfacesmooth-and-projective-over-arbitrary-field/89412#89412 Comment by stjc stjc 2012-02-26T10:37:13Z 2012-02-26T10:37:13Z thanks a lot, I didn't make the question clear, but I want to know surface over non algebraic closed field in particular:) http://mathoverflow.net/questions/89395/is-there-a-classification-of-surfacesmooth-and-projective-over-arbitrary-field Comment by stjc stjc 2012-02-26T10:12:44Z 2012-02-26T10:12:44Z I mean a smooth surface X/k if the structure morphism X to k is smooth