This is Exercise 11.3.C from Vakil's notes.   Suppose $X$ is a closed subset of $\mathbb{P}_k^n$ of dimension at least $1$, and $H$ is a nonempty hypersurface in $\mathbb{P}_k^n$. Show that $H$ meets $X$.  (Hint: note that the affine cone over $H$ contains the origin in $\mathbb{A}_k^{n+1}$. Apply Krull's Principal Ideal Theorem to the cone over $X$.)  My attempt: Let $A=k[x_0,\dots,x_n]$,  $X=\operatorname{Proj}A/I$ and $H=\operatorname{Proj}A/(f)$. I suppose the hint suggests to consider $V(f)$ in $\operatorname{Spec}A/I$. Suppose $f$ is not a zero divisor of $A/I$, then every irreducible component of $\operatorname{Spec}A/I\cap \operatorname{Spec}A/(f)$ has codimension precisely $1$. There is an injection $\operatorname{Spec}A/I\setminus \{0\}\hookrightarrow \operatorname{Proj}A/I$, similiarly  $\operatorname{Spec}A/(f)\setminus \{0\}\hookrightarrow \operatorname{Proj}A/(f)$. If I can show that $\operatorname{Spec}A/I\cap \operatorname{Spec}A/(f)$ is not the maximal ideal corresponding to the origin. Then  I can conclude $X\cap H\neq \emptyset$.   Is this correct? Can any suggest how to proceed?
I'm having trouble with the exercise in the title, even with part (a), which asks to prove that if $X$ is a closed subset of $\mathbb{P}^n_k$ of dimension at least 1 and $H$ is a non-empty hypersurface, then $X$ and $H$ meet.   Following the hint, I've done the following: By assumption, $X$ is given by $V(I)$ for some homogenous prime ideal $I\subset k[x_0,\dots x_n]=:A$ and $H$ by $V(f)$ for some homogenous polynomial $f\in A$. We wish to prove that there is a homogenous prime ideal $\mathfrak{p}$ (that's not the irrelevant one) containing both $I$ and $f$.  Consider the affine cone $\mathrm{Spec}\ A/I$ of $X$ in $\mathrm{Spec}\ A$. Being homogenous, both $I$ and $f$ are contained in the ideal $(x_0,\dots x_n)$. By Krull's principal ideal theorem, every minimal prime ideal containing $f\mod I$ has codimension one. $X$ having dimension at least one implies that there are homogenous prime prime ideals $\mathfrak{q}\supsetneq \mathfrak{r}\supset I$. This implies that $(x_o,\dots x_n)$ has codimension at least two. Thus there are prime ideals containing $f\mod I$ that are contained in $(x_o,\dots x_n)$. However, I'd need to show that there are homogenous such prime ideals, right? I'd rather get hints than full solutions.
If nonempty, nonsingleton $Y$ is a proper convex subset of a simply ordered set $X$, then $Y$ is ray or interval? This is from Exercise 7 in p. 92 in Munkres's Topology.  Except for the trivial cases such as $Y$ is empty set or singleton, it seems if $Y$ is convex in an simply ordered set $X$ then $Y$ is interval or ray.  But I cannot start my proof because I cannot use $\sup$ or $\max$ functions.  What should I do? How can I undo git reset --hard HEAD~1? Is it possible to undo the changes caused by the following command? If so, how?  git reset --hard HEAD~1 What are the criteria for questions to be selected for Hot Network Questions? I really like the , where certain questions from the Stack Exchange Network are presented, along with a hotness rating that is described as &quot;arbitrary&quot; in its tooltip. Such questions also appear randomly on the sidebar on sites across the network, under the heading &quot;Hot Network Questions&quot;. How do these arbitrary hotness points work, and how are they used to select questions to appear in those lists? Cut points and connected space Right-Click menu awareness on Web Application The web application I am building has a custom right-click menu. I'm concerned that the user might not find this menu as normally right-clicking will only display the default browser's menu. Is there a standard way to make the user aware of custom right-click menus? Is there another way to show this information? How not to show Universal Access menu Icon in task bar in Ubuntu 18.04? Why do prisms work (why is refraction frequency dependent)? Where to place my local website starting with the 2.4.7 version of apache2? I recently installed Ubuntu 14.04, then I installed lamp-server and placed my webpages in /var/www directory, but when I opened localhost in the browser there was nothing. I think that happened because Zend updated Apache.  Whatever the reason may be, I want to know where to keep my php files so that I can access them from my browser. Received an email intended for another person I will be attending a summer school which will last 10 days.  I want to know: for applying for a Schengen visa, is the purpose of my trip considered as touristic or study? TimedRotatingFileHandler Changing File Name? How to add desktop shortcut for URL, ignoring web app manifest? Goal:  Have a desktop shortcut that opens  , ignoring the contained web app manifest that otherwise "overrides" the saved shortcut to  and with "display": "standalone".  I had this working previously on an older Android phone before a replacement / upgrade.  Prior to the upgrade, I was able to long-press and open individual articles in new tabs - as well as copy link shortcuts to specific articles, as well as all of the other features and functionality that Chrome has to offer.  Here is just a partial selection of references of other users complaining about the same:       Using  provides a temporary successful work-around.  However, I can't find a successful way to add a link for this to my home screen.  The "Add to Home screen" option within Chrome instead saves a link to  - and in such a way that it opens as a web application instead of a web page within Chrome:       On a desktop operating system, this would be a non-issue.  (Create shortcut, enter desired URL, done.)  Attempt 1: Provide my own Manifest  So I created my own manifest - editing  to change the start_url attribute to https://news.google.com/news/i#0, and the display attribute from standalone to browser.  I then created my own HTML page that simply contained the below - then self-hosted both on a local web server:  &lt;html&gt;         &lt;head&gt;                 &lt;Title&gt;Google News&lt;/Title&gt;                 &lt;link rel="manifest" href="/news/manifest.json"/&gt;         &lt;/head&gt;         &lt;body&gt;                 Test         &lt;/body&gt; &lt;/html&gt;   However, this doesn't work:  Despite the shortcut being saved with the desired name and icon, it links back to my self-hosted HTML page, and not the indicated start_url.  Unfortunately, it appears that cross-domain URLs are not allowed per the manifest intent - as per  and  .  Attempt 1b: MITM website  Within the confines of my own local network, I figured I could try to  (MITM) the website to temporarily provide a desired manfiest to use for addition to the home screen.  Unfortunately, the website uses  here, which appropriately blocks the attempt.  Regardless - this is way too much work for what should be a simple effort.  Attempt 2: Widget Bookmark  Unfortunately, it looks like the 1x1 bookmark widget was removed in Chrome 48, as per  .  The suggested work-around is to use the "Add to Home screen" feature within Chrome.  Goto 10 / see beginning.  Attempt 3: Use the default-provided "Internet" browser  Unfortunately, using "Add shortcut on Home screen" in the default-provided "Internet" browser produces the same result as Chrome.  Consideration 4: Use a custom app / widget  Could probably write and install a custom application or widget that could be added to the home screen to meet the goal.  Or don't re-invent the wheel, and use an existing one.  Some possibilities:   "More Shortcuts":  "Shortcut Customizer":    Conclusion  How can someone create a desktop shortcut to an arbitrary URL, ignoring a provided web app manifest? I want to use an old drive as a secondary hard drive without reformatting. But this old hard drive contains my old Vista installation which takes up gigs of space.  I tried changing ownership and it works to some extent but I can not delete anything in Windows/System32 folder.  Is there an easy way to change permissions so I can delete everything?    The solution is to use Takeown as Wil suggested and use cacls to take full control of the files. Since cacls doesn't seem to be recursive, I fired up cygwin and ran this command to recursively change all the files:  for d in $(find directory-of-interest -type d);do   cacls $d'/*' /e /G username:F; done   where directory-of-interest is in my case /cygdrive/g/Windows and username is my username.  I had to manually run it on directories that had spaces in its name. But after all this, I can now delete the directory. Warning: it seems to me that cacls can potentially royally screw things up if improperly used. How does the User Card popup work? We are experimenting with enhanced user card mouseover popups and have enabled them network wide. You can tell which users will trigger the popup because they have a 3-D drop shadowy look to their gravatar:    Hover your mouse over the gravatar and wait. Note that the user card mouseover popup will only be displayed if the following conditions are met:   Is the profile complete, and long enough? We enforce a minimum length requirement. The user's reputation must be >= 1k  If you have more links than are able to be displayed after truncation, those extra links will be shown in the bottom of the dialog. We order links at the bottom consistently -- if you include a twitter, google plus, linked in, or facebook those links are prioritized. It will be visible on a question and answer detail page, any page with a list of questions, and the /users list. It will not be shown when a question or answer is wiki'd. It will not be shown when excerpt summary is defined as &lt;!-- summary: &amp;nbsp; --&gt; we support explicitly specifying your user card excerpt summary using the following syntax:  &lt;!-- summary: This is my profile, there are others like it but this one is mine. --&gt;   The above example will result in:    This is now mostly complete.  The goal is allow people to show  off how awesome they are inline, without having to click through to the full user profile page. What's the 'correct' way of registering/installing an Assembly to the GAC? There seems to be lots of different ways to register assemblies with the GAC, as in, they 'work'.  However, what's the "proper" way of doing it?  In response to Lou Franco (and gacutil):  I'm using Gacutil for development, but it seems to me to be not the proper way to install it, since gacutil isn't included in the basic .NET utilities past .NET 1.1 - it's only a developer tool.  Additional: Gacutil (as seen in responses below) is not redistributable, and therefore should not be used in any app that you intend to give to people who are not developers.  AKA, customers.  See .  In response to using WIX:  WIX might be great and all, but how does it work under the hood?  What details makes the way WIX installs the assembly the right way to install it?  How does it look it up?  Is it a system/.NET call?  Is there some call in a dll buried somewhere in System32 that needs to be made?    (Edit: it looks like WIX uses MSI under the hood.  See my comments in the accepted answer.)  Final edit: It looks like the correct way to install an assembly to the GAC is using windows installer, and nothing else.  I'm going to give Wix a try.  Thanks all! Can I use a custom Splash screen? The default Splash screens are cool, but I'd like to add some of my own work to remind me of how awesome I am (no, not really).    Is it possible to customize/change the splash screen? Is it supposed to be a HTML or an HTML I've seen many people who say:     This is a HTML page.   Yet I've also seen many people who say:     This is an HTML page.   Are both usages equally correct?  Or, which is the grammatically correct one?       Possible Duplicates: Problem statement  Prove that if $A$ is an uncountable set and $B$ is a countable set, then $A\setminus B$ must be uncountable.  What I think  The statement does not mention $A$ and $B$ relationship. I think there are two possibilities:   If $A \cap B = \emptyset $, then $A\setminus B$ is trivially uncountable If $A \cap B = B$, then $B \subset A$ and as a bijection can not be made between $A\setminus B$ and $\mathbb{N}$, $A\setminus B$ is uncountable.   And there is where I'm stuck. How can I prove that a bijecton can't be made?  TIA. Deleted Ubuntu 20 alongside windows 10 and unable to install again Is the phrase "for free" correct?