{"id":1195,"date":"2011-01-25T21:12:44","date_gmt":"2011-01-25T21:12:44","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1195"},"modified":"2011-01-25T21:12:44","modified_gmt":"2011-01-25T21:12:44","slug":"index-of-a-smooth-variety","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1195","title":{"rendered":"Index of a smooth variety"},"content":{"rendered":"

Let X be a smooth variety over a field k. The index<\/em> of X over k is the gcd of the degrees [\u03ba(x) : k] over all closed points x of X. The index is 1 if and only if X has a zero cycle of degree 1. If k is perfect, then the index of X is a birational invariant on smooth varieties over k: The reason is that given a nonempty open U of X and a closed point x in X you can find a curve C \u2282 X with x \u2208 C, and it is easy to move zero cycles on curves. (I think the birational invariance also holds over nonperfect fields, but I haven’t checked this.)<\/p>\n

Another birational invariant of a d-dimensional variety X over k is the gcd of the degrees of rational maps X —> P^d_k. This is the same as the gcd of closed subvarieties of P^n (any n) birational to X. Let’s temporarily call this the b-index<\/em>. Note that by taking inverse images of k-rational points on P^d_k we see that index | b-index for smooth X (if k is finite you have to look at points over finite extensions). I claim that in fact index = b-index at least over a perfect field. After shrinking X we may assume that X is affine, hence quasi-projective, so X \u2282 P^N_k for some N >> 0 having some (super large and super divisible) degree D. On the other hand, consider the blow up b : X’ —> X of X in x. Then the invertible sheaf b^*O_X(N)(-Exceptional) will be very ample and will embed X’ into a large projective space where it has degree N^dD – [\u03ba(x) : k]. This implies that b-index divides [\u03ba(x) : k] and we win.<\/p>\n","protected":false},"excerpt":{"rendered":"

Let X be a smooth variety over a field k. The index of X over k is the gcd of the degrees [\u03ba(x) : k] over all closed points x of X. The index is 1 if and only if … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-1195","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1195","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1195"}],"version-history":[{"count":5,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1195\/revisions"}],"predecessor-version":[{"id":1200,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1195\/revisions\/1200"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1195"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1195"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1195"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}