{"version":"1.0","provider_name":"SOUL OF MATHEMATICS","provider_url":"https:\/\/soulofmathematics.com","author_name":"Rajarshi Dey","author_url":"https:\/\/soulofmathematics.com\/index.php\/author\/rajarshidey1729gmail-com\/","title":"ALGEBRAIC GEOMETRY - SOUL OF MATHEMATICS","type":"rich","width":600,"height":338,"html":"<blockquote class=\"wp-embedded-content\" data-secret=\"bd5WroJy6K\"><a href=\"https:\/\/soulofmathematics.com\/index.php\/algebraic-geometry\/\">ALGEBRAIC GEOMETRY<\/a><\/blockquote><iframe sandbox=\"allow-scripts\" security=\"restricted\" src=\"https:\/\/soulofmathematics.com\/index.php\/algebraic-geometry\/embed\/#?secret=bd5WroJy6K\" width=\"600\" height=\"338\" title=\"&#8220;ALGEBRAIC GEOMETRY&#8221; &#8212; SOUL OF MATHEMATICS\" data-secret=\"bd5WroJy6K\" frameborder=\"0\" marginwidth=\"0\" marginheight=\"0\" scrolling=\"no\" class=\"wp-embedded-content\"><\/iframe><script type=\"text\/javascript\">\n\/* <![CDATA[ *\/\n\/*! This file is auto-generated *\/\n!function(d,l){\"use strict\";l.querySelector&&d.addEventListener&&\"undefined\"!=typeof URL&&(d.wp=d.wp||{},d.wp.receiveEmbedMessage||(d.wp.receiveEmbedMessage=function(e){var t=e.data;if((t||t.secret||t.message||t.value)&&!\/[^a-zA-Z0-9]\/.test(t.secret)){for(var s,r,n,a=l.querySelectorAll('iframe[data-secret=\"'+t.secret+'\"]'),o=l.querySelectorAll('blockquote[data-secret=\"'+t.secret+'\"]'),c=new RegExp(\"^https?:$\",\"i\"),i=0;i<o.length;i++)o[i].style.display=\"none\";for(i=0;i<a.length;i++)s=a[i],e.source===s.contentWindow&&(s.removeAttribute(\"style\"),\"height\"===t.message?(1e3<(r=parseInt(t.value,10))?r=1e3:~~r<200&&(r=200),s.height=r):\"link\"===t.message&&(r=new URL(s.getAttribute(\"src\")),n=new URL(t.value),c.test(n.protocol))&&n.host===r.host&&l.activeElement===s&&(d.top.location.href=t.value))}},d.addEventListener(\"message\",d.wp.receiveEmbedMessage,!1),l.addEventListener(\"DOMContentLoaded\",function(){for(var e,t,s=l.querySelectorAll(\"iframe.wp-embedded-content\"),r=0;r<s.length;r++)(t=(e=s[r]).getAttribute(\"data-secret\"))||(t=Math.random().toString(36).substring(2,12),e.src+=\"#?secret=\"+t,e.setAttribute(\"data-secret\",t)),e.contentWindow.postMessage({message:\"ready\",secret:t},\"*\")},!1)))}(window,document);\n\/* ]]> *\/\n<\/script>\n","thumbnail_url":"https:\/\/i1.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2021\/03\/e289468a725435f2fa435acb6cdd41-unscreen.gif?fit=360%2C360&ssl=1","thumbnail_width":360,"thumbnail_height":360,"description":"The introduction of the digit 0 or the group concept was general nonsense too, and mathematics was more or less stagnating for thousands of years because nobody was around to take such childish steps\u2026 A. Grothendieck Algebraic geometry deals with curves or surfaces (or more abstract generalizations of these) which can be viewed both as geometric objects and as solutions of algebraic (specifically, polynomial) equations. Algebraic geometry sets out to answer these questions by applying the techniques of&nbsp;abstract algebra&nbsp;to the set of polynomials that define the curves (which are then called &#8220;algebraic varieties&#8221;).&nbsp;The mathematics involved is inevitably quite hard, although it is covered in degree-level courses. Other common questions in algebraic geometry concern points of special interest such as&nbsp;singularities,&nbsp;inflection points&nbsp;and&nbsp;points at infinity&nbsp;&#8211; we shall see these throughout the catalogue. More advanced questions in algebraic geometry concern relations between curves given by different equations and the topology of curves, and many other topics.&nbsp; Universal properties determine an object up to uniqueisomorphism Given some category that we come up with, we often will have ways of producing new objects from old. In good circumstances, such a definition can be made using the notion of a universal property. Informally, we wish that there were an object with some property. We first show that if it exists, then it is essentially unique, or more precisely, is unique up to unique isomorphism. Then we go about constructing an example of such an object to show existence. Explicit constructions are sometimes easier to work with than universal properties, but with a little practice, universal properties are useful in proving things quickly and slickly. Indeed, when learning the subject, people often find explicit constructions more appealing, and use them more often in proofs, but as they become more experienced, they find universal property arguments more elegant and insightful. Products were defined by a universal property. We have seen one important example of a universal property argument already in products. You should go back and verify that our discussion there gives a notion of product in any category, and shows that products, if they exist, are unique up to unique isomorphism. Initial, final, and zero objects. Here are some simple but useful concepts that will give you practice with universal property arguments. An object of a category C is an initial object if it has precisely one map to every object. It is a final object if it has precisely one map from every object. It is a zero object if it is both an initial object and a final object. Localization of rings and modules. Another important example of a definition by universal property is the notion of localization of a ring. We first review a constructive definition, and then reinterpret the notion in terms of universal property. A multiplicative subset S of a ring A is a subset closed under multiplication containing 1. We define a ring S\u22121A. Schemes: The underlying set, and topological space We will define a scheme to be the following data\u2022 The set: the points of the scheme\u2022 The topology: the open sets of the scheme\u2022 The structure sheaf: the sheaf of \u201calgebraic functions\u201d (a sheaf of rings) onthe scheme. A topological space with a sheaf of rings is called a ringed space. For the set, in the key example of complex varieties (roughly, things cut out in Cn by polynomials), we will see that the points are the \u201ctraditional points\u201d (n-tuples of complex numbers), plus some extra points that will be handy to have around. For the topology, we will require that \u201cthe subset where an algebraic function vanishes must be closed\u201d, and require nothing else. For the sheaf of algebraic functions (the structure sheaf), we will expect that in the complex plane, (3&#215;2+y2)\/(2x+4xy+1) should be an algebraic function on the open set consisting of points where the denominator doesn\u2019t vanish, and this will largely motivate our definition. Adapted from FOUNDATIONS OF ALGEBRAIC GEOMETRY No Copyright Infringement Intended"}