# Questions tagged [semicontinuity]

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### Semicontinuity of Exts

Let $Z$ be a proper noetherian scheme over an algebraically closed field $k$, and $S$ a smooth scheme over $k$. Let $\cal E$, $\cal F$ be coherent sheaves on $Z\times S$, flat over $S$. Is it true ...
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### Right-continuity of covering number

Consider an ambient metric space $(\mathcal{X},\Vert\cdot\Vert_\infty)$. Let $\mathcal{B}_1 = \mathcal{B}_{\Vert\cdot\Vert_K}(0,1)\subseteq\mathcal{X}$ be the closed unit ball with respect to some ...
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### A question about a realcompact space and upper semicontinuous function

Nancy Dykes says in the proof of Theorem 3.4 in her article Generalizations of realcompact spaces that by a result of John Mack, if for every $p\in \beta X\setminus X$ there exists a nonnegative upper ...
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### A net of lower semicontinuous functions

Assume we have a non-decreasing net of lower semicontinuous functions $f_\alpha:[0,1]\to\mathbb{R}$ such that $\lim_\alpha f_\alpha\to f$ pointwise. Please is it true that one can extract a countable ...
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### Conditions ensuring that a compact set can be approximated by approximating its distance function

Let $\emptyset\neq K\subset Y$ be a closed subset of a compact metric space $(X,d)$ such that $K$ has at-least two points and such that $$d(Y,K):=\sup_{y\in Y}\,\inf_{k\in K}\,d(k,y)=:r>0.$$ ...
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### From convergence pointwise to convergence of the supremum for semicontinuous functions

Let $K\subset\mathbb{R}$ a compact set, and $(f_n)_{n\geq 1}$ and $f$ upper semicontinuous functions over $K$ (taking hence values in $\mathbb{R}\cup\{-\infty,+\infty\}$) such that for all $x\in K$, ...
244 views

### Semi-continuity in quasi-finite morphisms without properness

Let $f:U \to V$ be a flat, quasi-finite, surjective morphism between two affine varieties defined over $\mathbb{C}$. Assume that every closed fiber is reduced. Consider the function $\eta$ that sends ...
1 vote
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1 vote
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### Rank of a linear combination of linear operators

I asked this question a few days ago in MathExchange and received no satisfatory answer. I hope it is well suited for MathOverflow. Suppose I have two linear operators $X,\,Y$ on $\mathbb{C}^n$. Now ...
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1 vote
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### Upper semicontinuity in C(X)-algebras. Quotient norm question

upper semicontinuity in C(X)-algebras In the 5th paragraph of this post, I don't understand why there exists a vector b satisfying $||a+b||_A < ||q_x (a)||_A(x)$ By the definition of the quotient ...
482 views

### Lower semicontinuity of naive fiber size

I would like to present the following result in my algebraic geometry class, but it is seeming much harder than I would expect. Since my class is working with closed points over an algebraically ...
I'm looking for geometric applications of Ekeland's variational principle in order to see it at work in a context I'm familiar with. Let me recall the principle itself: Definition. Let $(X,d)$ be a ...
The notion of upper/lower semi-continuity is sometimes encountered in algebraic geometry. Here by upper semi-continuity one means a function on a topological space $f:X\rightarrow S$ with value in ...