More upper/lower semi-continuous functions in (algebraic) geometry? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T13:47:38Z http://mathoverflow.net/feeds/question/42796 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/42796/more-upper-lower-semi-continuous-functions-in-algebraic-geometry More upper/lower semi-continuous functions in (algebraic) geometry? genshin 2010-10-19T14:51:00Z 2010-10-20T08:53:37Z <p>The notion of upper/lower semi-continuity is sometimes encountered in algebraic geometry.</p> <p>Here by upper semi-continuity one means a function on a topological space $f:X\rightarrow S$ with value in some ordered topological space (like the field of real numbers), such that $\lim\sup_{x\rightarrow y}f(x)\leq f(y)$. Intuitively, for points $x$ that are close to a given point $y$, may the value $f(x)$ "exceeds" $f(y)$, the difference should be "small" and "vanishes" as $x$ approaches $y$. And lower semi-continuity means the opposite, namely replace "$\leq$" by "$\geq$".</p> <p>Among the typical examples one thinks of the dimension function: let $k$ be a base field, and $X$ be a locally finite type $k$-scheme. The function $$x\in X \mapsto \dim_xX$$ is upper semi-continuous, where by $\dim_xX$ is understood to be the combinatory dimension of $X$ at $x$, namely the length of chains of inclusions of irreducible closed subschemes $$x\in X_1\subsetneq X_2\subsetneq \cdots\subsetneq X_d=X$$</p> <p>There seems to be a lot of examples of such upper/lower semi-continuous functions in geometry counting certain "discrete invariants", especially those related to stratifications of spaces. It'll be great to have the list extended in mathoverflow.</p> <p>I hope this question is well-posed...</p> http://mathoverflow.net/questions/42796/more-upper-lower-semi-continuous-functions-in-algebraic-geometry/42817#42817 Answer by J.C. Ottem for More upper/lower semi-continuous functions in (algebraic) geometry? J.C. Ottem 2010-10-19T18:39:48Z 2010-10-19T18:54:32Z <p>The semicontinuity theorem (Hartshorne III.11) states that the ranks of cohomology groups on the fibers of a morphism is a semicontinuous function. More precisely, given a projective morphism $f:X\to Y$ of noetherian schemes and a coherent sheaf $F$ on $X$, flat over Y, then the function $$h^i(y,F)=\dim_{k(y)}H^i(X_y,F_y)$$is upper semicontinuous as a function of $y$. Here $X_y$ denotes the fibre of $f$ over $y$. This is used widely in algebraic geometry. </p> http://mathoverflow.net/questions/42796/more-upper-lower-semi-continuous-functions-in-algebraic-geometry/42821#42821 Answer by Sándor Kovács for More upper/lower semi-continuous functions in (algebraic) geometry? Sándor Kovács 2010-10-19T19:30:53Z 2010-10-19T19:30:53Z <p>Hodge numbers are (upper) semi-continuous in a family of complex manifolds and constant for Kähler manifolds. Then again, this is a special case of the above, at least for projective families. See <a href="http://mathoverflow.net/questions/42709/question-about-hodge-number/42735#42735" rel="nofollow">this</a> for links and references for related stuff.</p> http://mathoverflow.net/questions/42796/more-upper-lower-semi-continuous-functions-in-algebraic-geometry/42848#42848 Answer by ABayer for More upper/lower semi-continuous functions in (algebraic) geometry? ABayer 2010-10-19T23:51:39Z 2010-10-19T23:51:39Z <p>The most basic semi-continuous function in algebraic geometry is, I think, the following: given a coherent sheaf $F$ on a scheme $X$ of (locally) finite type over $k$, the function</p> <p>$$x \mapsto \dim_{k(x)} F\mid_x = \dim_{k(x)} F \otimes k(x)$$</p> <p>is upper semi-continuous. It follows from Nakayama's Lemma.</p> http://mathoverflow.net/questions/42796/more-upper-lower-semi-continuous-functions-in-algebraic-geometry/42855#42855 Answer by Steven Sam for More upper/lower semi-continuous functions in (algebraic) geometry? Steven Sam 2010-10-20T00:57:35Z 2010-10-20T00:57:35Z <p>There are a lot of examples in the representation theory of finite-dimensional algebras.</p> <p>For instance, we can encode associative finite-dimensional algebras of given dimension $d$ into an algebraic variety and the map sending an algebra to its projective dimension is upper semicontinuous. Or for a fixed algebra, one can encode its modules or pairs of modules of given dimensions into an algebraic variety and functions like the dimension of the $i$th Ext group is upper semicontinuous.</p> <p>It might be kind of weird because these varieties don't parametrize isomorphism classes, but rather ways of putting a module or algebra structure on a fixed vector space, but these results tend to be useful in geometric approaches to finite-dimensional algebras.</p> http://mathoverflow.net/questions/42796/more-upper-lower-semi-continuous-functions-in-algebraic-geometry/42886#42886 Answer by quim for More upper/lower semi-continuous functions in (algebraic) geometry? quim 2010-10-20T08:53:37Z 2010-10-20T08:53:37Z <p>Seshadri constants (see arXiv:0810.0728) are lower semicontinuous. </p> <p>This fact is of course related to J.C. Ottem's example, but (a priori) it is not "the same". I believe that there is not a single coherent F giving the same stratification that the Seshadri constant gives (you could get it by a sequence of coherent F, or using non-coherent F, trivially.)</p>