{"id":18091,"date":"2023-10-24T09:59:55","date_gmt":"2023-10-24T01:59:55","guid":{"rendered":"https:\/\/cantor.math.ntnu.edu.tw\/?p=18091"},"modified":"2023-11-01T16:23:45","modified_gmt":"2023-11-01T08:23:45","slug":"lecture20231108","status":"publish","type":"post","link":"https:\/\/cantor.math.ntnu.edu.tw\/index.php\/2023\/10\/24\/lecture20231108\/","title":{"rendered":"<span style=\"color:#3566BD\">[\u5c08\u984c\u6f14\u8b1b] <\/span>\u301011\u67088\u65e5\u3011Adeel Khan \/ The derived B\u00e9zout theorem"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"18091\" class=\"elementor elementor-18091\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-19a5332c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"19a5332c\" data-element_type=\"section\" data-e-type=\"section\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t\t\t\t<div class=\"elementor-background-overlay\"><\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-68eb95a1\" data-id=\"68eb95a1\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-4d878518 elementor-widget elementor-widget-heading\" data-id=\"4d878518\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">The derived B\u00e9zout theorem<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-51461cf0 elementor-widget elementor-widget-spacer\" data-id=\"51461cf0\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-4ccaf3ea elementor-widget elementor-widget-text-editor\" data-id=\"4ccaf3ea\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"color: #000080; font-family: \u5fae\u8edf\u6b63\u9ed1\u9ad4; font-size: 18px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: bold;\">\u6642\u3000\u9593\uff1a2023-11-08 14:00 (\u661f\u671f\u4e09) \/ \u5730\u3000\u9ede\uff1aS101 \/ \u8336\u3000\u6703\uff1aS205 (13:30)<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-4940900d elementor-widget elementor-widget-image\" data-id=\"4940900d\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"194\" height=\"245\" src=\"https:\/\/cantor.math.ntnu.edu.tw\/wp-content\/uploads\/2023\/10\/1121108Khan.jpg\" class=\"attachment-full size-full wp-image-18093\" alt=\"\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-26032a62 elementor-widget elementor-widget-text-editor\" data-id=\"26032a62\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div style=\"list-style: none; font-family: \u5fae\u8edf\u6b63\u9ed1\u9ad4; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; margin-top: 10px; font-size: 22px; line-height: 26px; text-align: center; color: #cc6633; font-weight: bold;\">Adeel Khan \u97d3\u5584\u745c<\/div><div style=\"list-style: none; font-family: \u5fae\u8edf\u6b63\u9ed1\u9ad4; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; margin-top: 10px; font-size: 18px; line-height: 22px; text-align: center; color: #cc9966; font-weight: bold;\">\u4e2d\u592e\u7814\u7a76\u9662 \u6578\u5b78\u7814\u7a76\u6240\u526f\u7814\u7a76\u54e1<\/div>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-5c79e75 elementor-widget elementor-widget-spacer\" data-id=\"5c79e75\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-5fd7cb59 elementor-widget elementor-widget-text-editor\" data-id=\"5fd7cb59\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"color: #000000;\">Given two smooth algebraic curves of degrees <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>d<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">d<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathdefault\">d<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>e<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">e<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathdefault\">e<\/span><\/span><\/span><\/span> in the complex projective plane, Bezout\u2019s theorem is a classical result asserting that the intersection consists of either infinitely many or exactly <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>d<\/mi><mo>\u22c5<\/mo><mi>e<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">d\\cdot e<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathdefault\">d<\/span><span class=\"mbin\">\u22c5<\/span><\/span><span class=\"base\"><span class=\"mord mathdefault\">e<\/span><\/span><\/span><\/span> points, counted with multiplicities. Generalizing this formula to intersections in higher-dimensional projective space was a task that fuelled much of 20th century algebraic geometry, eventually culminating in J.-P. Serre\u2019s solution in 1958.<\/span><\/p><p><span style=\"color: #000000;\">We will review some of this history before presenting a 21st century point of view on B\u00e9zout\u2019s theorem. Specifically, we will see that the modern language of derived algebraic geometry allows us to give a simple definition of intersection multiplicities, which unlike Serre\u2019s formula applies even to non-proper intersections, and yields a vast generalization of B\u00e9zout theorem for arbitrary intersections of projective hypersurfaces.<\/span><\/p><p><span style=\"color: #000000;\"><strong>\u500b\u4eba\u7db2\u9801 <a style=\"color: #000000;\" href=\"https:\/\/www.math.sinica.edu.tw\/www\/people\/websty1_20.jsp?owner=adeelkhan\">https:\/\/www.math.sinica.edu.tw\/www\/people\/websty1_20.jsp?owner=adeelkhan<\/a><\/strong><\/span><\/p><p><span style=\"color: #000000;\"><strong>\u66f4\u591a\u8cc7\u8a0a <a style=\"color: #000000;\" href=\"https:\/\/youtu.be\/oKPlH2Ybfsg?si=0LmjOxbSFRT2pc3t\">https:\/\/youtu.be\/oKPlH2Ybfsg?si=0LmjOxbSFRT2pc3t<\/a><\/strong><\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-76d951d3 elementor-widget elementor-widget-image\" data-id=\"76d951d3\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" width=\"1240\" height=\"758\" src=\"https:\/\/cantor.math.ntnu.edu.tw\/wp-content\/uploads\/2020\/08\/\u6f14\u8b1b\u6a19\u5c3e.jpg\" class=\"attachment-full size-full wp-image-7367\" alt=\"\" srcset=\"https:\/\/cantor.math.ntnu.edu.tw\/wp-content\/uploads\/2020\/08\/\u6f14\u8b1b\u6a19\u5c3e.jpg 1240w, https:\/\/cantor.math.ntnu.edu.tw\/wp-content\/uploads\/2020\/08\/\u6f14\u8b1b\u6a19\u5c3e-300x183.jpg 300w, https:\/\/cantor.math.ntnu.edu.tw\/wp-content\/uploads\/2020\/08\/\u6f14\u8b1b\u6a19\u5c3e-768x469.jpg 768w, https:\/\/cantor.math.ntnu.edu.tw\/wp-content\/uploads\/2020\/08\/\u6f14\u8b1b\u6a19\u5c3e-1024x626.jpg 1024w\" sizes=\"(max-width: 1240px) 100vw, 1240px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>The derived B\u00e9zout theorem \u6642\u3000\u9593\uff1a2023-11-08 14:00 (\u661f\u671f\u4e09) \/ [&hellip;]<\/p>\n","protected":false},"author":23,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"ocean_post_layout":"","ocean_both_sidebars_style":"","ocean_both_sidebars_content_width":0,"ocean_both_sidebars_sidebars_width":0,"ocean_sidebar":"","ocean_second_sidebar":"","ocean_disable_margins":"enable","ocean_add_body_class":"","ocean_shortcode_before_top_bar":"","ocean_shortcode_after_top_bar":"","ocean_shortcode_before_header":"","ocean_shortcode_after_header":"","ocean_has_shortcode":"","ocean_shortcode_after_title":"","ocean_shortcode_before_footer_widgets":"","ocean_shortcode_after_footer_widgets":"","ocean_shortcode_before_footer_bottom":"","ocean_shortcode_after_footer_bottom":"","ocean_display_top_bar":"default","ocean_display_header":"default","ocean_header_style":"","ocean_center_header_left_menu":"","ocean_custom_header_template":"","ocean_custom_logo":0,"ocean_custom_retina_logo":0,"ocean_custom_logo_max_width":0,"ocean_custom_logo_tablet_max_width":0,"ocean_custom_logo_mobile_max_width":0,"ocean_custom_logo_max_height":0,"ocean_custom_logo_tablet_max_height":0,"ocean_custom_logo_mobile_max_height":0,"ocean_header_custom_menu":"","ocean_menu_typo_font_family":"","ocean_menu_typo_font_subset":"","ocean_menu_typo_font_size":0,"ocean_menu_typo_font_size_tablet":0,"ocean_menu_typo_font_size_mobile":0,"ocean_menu_typo_font_size_unit":"px","ocean_menu_typo_font_weight":"","ocean_menu_typo_font_weight_tablet":"","ocean_menu_typo_font_weight_mobile":"","ocean_menu_typo_transform":"","ocean_menu_typo_transform_tablet":"","ocean_menu_typo_transform_mobile":"","ocean_menu_typo_line_height":0,"ocean_menu_typo_line_height_tablet":0,"ocean_menu_typo_line_height_mobile":0,"ocean_menu_typo_line_height_unit":"","ocean_menu_typo_spacing":0,"ocean_menu_typo_spacing_tablet":0,"ocean_menu_typo_spacing_mobile":0,"ocean_menu_typo_spacing_unit":"","ocean_menu_link_color":"","ocean_menu_link_color_hover":"","ocean_menu_link_color_active":"","ocean_menu_link_background":"","ocean_menu_link_hover_background":"","ocean_menu_link_active_background":"","ocean_menu_social_links_bg":"","ocean_menu_social_hover_links_bg":"","ocean_menu_social_links_color":"","ocean_menu_social_hover_links_color":"","ocean_disable_title":"default","ocean_disable_heading":"default","ocean_post_title":"","ocean_post_subheading":"","ocean_post_title_style":"","ocean_post_title_background_color":"","ocean_post_title_background":0,"ocean_post_title_bg_image_position":"","ocean_post_title_bg_image_attachment":"","ocean_post_title_bg_image_repeat":"","ocean_post_title_bg_image_size":"","ocean_post_title_height":0,"ocean_post_title_bg_overlay":0.5,"ocean_post_title_bg_overlay_color":"","ocean_disable_breadcrumbs":"default","ocean_breadcrumbs_color":"","ocean_breadcrumbs_separator_color":"","ocean_breadcrumbs_links_color":"","ocean_breadcrumbs_links_hover_color":"","ocean_display_footer_widgets":"default","ocean_display_footer_bottom":"default","ocean_custom_footer_template":"","ocean_post_oembed":"","ocean_post_self_hosted_media":"","ocean_post_video_embed":"","ocean_link_format":"","ocean_link_format_target":"self","ocean_quote_format":"","ocean_quote_format_link":"post","ocean_gallery_link_images":"off","ocean_gallery_id":[],"footnotes":""},"categories":[1,124],"tags":[],"class_list":["post-18091","post","type-post","status-publish","format-standard","hentry","category-news","category-speeches","entry"],"_links":{"self":[{"href":"https:\/\/cantor.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/posts\/18091","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/cantor.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/cantor.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/cantor.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/users\/23"}],"replies":[{"embeddable":true,"href":"https:\/\/cantor.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/comments?post=18091"}],"version-history":[{"count":12,"href":"https:\/\/cantor.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/posts\/18091\/revisions"}],"predecessor-version":[{"id":18213,"href":"https:\/\/cantor.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/posts\/18091\/revisions\/18213"}],"wp:attachment":[{"href":"https:\/\/cantor.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/media?parent=18091"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/cantor.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/categories?post=18091"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/cantor.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/tags?post=18091"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}