User ruadhai dervan - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T22:39:31Z http://mathoverflow.net/feeds/user/22294 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/127829/when-constant-scalar-curvature-implies-einstein/127854#127854 Answer by Ruadhai Dervan for when constant scalar curvature implies Einstein? Ruadhai Dervan 2013-04-17T14:50:57Z 2013-04-17T21:24:21Z <p>As an example where this does hold, for $\omega$ a Kähler metric of constant scalar curvature with $\pi c_1(M) = \lambda [\omega]$, then $\omega$ is Kähler-Einstein. This is Proposition 2.12 in Tian's "Canonical metrics in Kähler Geometry".</p> http://mathoverflow.net/questions/121461/reference-for-notation-h0c-mk/121465#121465 Answer by Ruadhai Dervan for Reference for notation $H^0(C, mK)$ Ruadhai Dervan 2013-02-11T11:20:58Z 2013-02-11T13:57:27Z <p>I can't access the draft you've linked but (almost always) this means the vector space of global sections of the $m^{th}$ power of the canonical bundle. Note for a Riemann surface the canonical bundle is just the cotangent bundle, the dual of the tangent bundle. For the general theory of algebraic curves, see e.g. Geometry of Algebraic Curves I by Arabello et al.</p> http://mathoverflow.net/questions/95503/flow-of-a-hamiltonian-vector-field/95508#95508 Answer by Ruadhai Dervan for Flow of a Hamiltonian vector field Ruadhai Dervan 2012-04-29T13:35:23Z 2012-04-29T13:35:23Z <p>To answer the question of 'What does the flow of a Hamiltonian vector field correspond to', it's useful to think physically. Exactness means we have some function $H$ which corresponds to the vector field $X_H$. If you take a Darboux coordinate system, then the flow of the Hamiltonian vector field is exactly the solution to Hamilton's equations. Hamilton's principle basically states that the real trajectory of a physical system follows Hamilton's equations, where we interpret $H$ as something like an energy function (equivalent to minimising an energy functional). I hope this helps, it's probably more useful from a physical point of view than a pure mathematical one.</p> http://mathoverflow.net/questions/92423/general-topology-terminology-questions/92425#92425 Answer by Ruadhai Dervan for General topology terminology questions Ruadhai Dervan 2012-03-28T00:22:57Z 2012-03-28T00:22:57Z <p>In Geometric Invariant Theory, the study of quotients in algebraic geometry, some points are ignored in the quotient (by its construction) that would make the quotient non-Hausdorff. These points are called 'unstable'. Sometimes the set of all unstable points is called the 'unstable locus'. This is of course just a special case of your question, in a slightly different area, but perhaps the terminology is used elsewhere. A good reference for this if you're interested is <a href="http://arxiv.org/abs/math/0512411" rel="nofollow">these notes</a> by Richard Thomas.</p> http://mathoverflow.net/questions/91942/algebraic-geometry-definition-of-a-morphism/91948#91948 Answer by Ruadhai Dervan for Algebraic Geometry - Definition of a Morphism Ruadhai Dervan 2012-03-22T20:59:07Z 2012-03-22T20:59:07Z <p>A regular map $\phi: X \to Y$ of quasi-projective varieties is a continuous map with respect to the Zariski topology such that for $V \subset Y$ an open set and $f$ a regular function on $V$, we have $f\circ \phi$ is regular on $\phi^{-1}V$. This seems to me to be to be exactly what you would want and quite intuitive and understandable. </p> http://mathoverflow.net/questions/127829/when-constant-scalar-curvature-implies-einstein/127854#127854 Comment by Ruadhai Dervan Ruadhai Dervan 2013-04-17T14:52:22Z 2013-04-17T14:52:22Z P.S. If anyone would like to remove the e from &quot;Kaehler&quot; and insert an umlaut, they're more than welcome to, as I was unable to. http://mathoverflow.net/questions/125200/coboundaries-and-gluing-in-cech-cohomology-intuition Comment by Ruadhai Dervan Ruadhai Dervan 2013-03-21T20:32:06Z 2013-03-21T20:32:06Z Why is this tagged mathematical physics? http://mathoverflow.net/questions/120062/intersection-of-two-projective-submanifolds-in-pn-treatment-in-shafarevich-boo/120073#120073 Comment by Ruadhai Dervan Ruadhai Dervan 2013-01-28T10:06:12Z 2013-01-28T10:06:12Z This is how Shafarevich proves in my version of his book. http://mathoverflow.net/questions/120062/intersection-of-two-projective-submanifolds-in-pn-treatment-in-shafarevich-boo Comment by Ruadhai Dervan Ruadhai Dervan 2013-01-27T23:04:59Z 2013-01-27T23:04:59Z In my copy of the book 'Basic Algebraic Geometry I: Varieties in Projective Space' by Shafarevich, Theorem 6 is basically what your statement is - this is on page 76. This is the 'Second, Revised and Expanded Edition'. I can type it out for you, if you'd like. http://mathoverflow.net/questions/115220/how-do-fibers-of-the-functor-algebraic-varieties-to-complex-analytic-spaces-lo Comment by Ruadhai Dervan Ruadhai Dervan 2012-12-03T00:09:19Z 2012-12-03T00:09:19Z Could you link to the question you mention about non-injectivity please? Thanks. http://mathoverflow.net/questions/91942/algebraic-geometry-definition-of-a-morphism/91948#91948 Comment by Ruadhai Dervan Ruadhai Dervan 2012-03-22T23:50:47Z 2012-03-22T23:50:47Z This thread should be helpful, it's essentially the same question. <a href="http://mathoverflow.net/questions/1397/morphisms-of-quasi-projective-varieties" rel="nofollow" title="morphisms of quasi projective varieties">mathoverflow.net/questions/1397/&hellip;</a> http://mathoverflow.net/questions/91942/algebraic-geometry-definition-of-a-morphism/91948#91948 Comment by Ruadhai Dervan Ruadhai Dervan 2012-03-22T23:43:17Z 2012-03-22T23:43:17Z Yes, that is true. It would also be a nice definition. http://mathoverflow.net/questions/91942/algebraic-geometry-definition-of-a-morphism/91948#91948 Comment by Ruadhai Dervan Ruadhai Dervan 2012-03-22T21:12:52Z 2012-03-22T21:12:52Z Ah, it would be circular if you didn't know what a regular function was. A function $f: X \to \mathbb{A}^1$ on a quasi-projective variety $X$ is called regular if $\forall x\in X$ there is a neighbourhood $U$ of $x$ such that $f|_U$ is the quotient of two homogeneous polynomials $f|_U = F/G$ of the same degree such that $G$ has no zeroes on $U$. A good reference for this level of algebraic geometry is I.R. Shafarevich - Basic algebraic geometry I. – Ruadhai 0 secs ago