$D$-modules on spaces of rational maps. (May 2014)

• Record Type: Journal Article
• Authors: Barlev, Jonathan
• Is Part Of: Compositio mathematica. Volume 150:Number 5(2014:Sep.)
• Page Start: 835
• Page End: 876
• Publication Date: 2014-05
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X13007707
• Languages: English
• ISSNs: 0010-437X

Abstract

<abstract> <title>Abstract</title> <p>Let <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghjmhpw2d" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$X$]]></tex-math></alternatives></inline-formula> be an algebraic curve. We study the problem of parametrizing geometric structures over <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghjmhnnc7" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$X$]]></tex-math></alternatives></inline-formula> which are only generically defined. For example, parametrizing generically defined maps (rational maps) from <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghjmhpvm6" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$X$]]></tex-math></alternatives></inline-formula> to a fixed target scheme <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghjmhpwk5" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$Y$]]></tex-math></alternatives></inline-formula>. There are three methods for constructing functors of points for such moduli problems (all originally due to Drinfeld), and we show that the resulting functors are equivalent in the fppf Grothendieck topology. As an application, we obtain three presentations for the category of <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghjmhpws8" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$D$]]></tex-math></alternatives></inline-formula>-modules 'on' <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghjmhnnqd" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$B(K)\backslash G(\mathbb{A})/G(\mathbb{O})$]]></tex-math></alternatives></inline-formula>, and we combine results about this category coming from the different presentations.</p> </abstract>