Stable Sheave Moduli of Rank 2 with Chern Classes c 1 = -1; c2 = 2; c3 = 0 on Q3

• Main Author: A. D. Uvarov
• Format: Article
• Language: English
• Published: Yaroslavl State University 2012-01-01
• Series: Modelirovanie i Analiz Informacionnyh Sistem
• Subjects: компактификация; схема модулей; когерентный пучок ранга
• Online Access: http://mais-journal.ru/jour/article/view/16
• ISSN: 1818-1015; 2313-5417

<p align="LEFT">In this paper we consider the scheme <em><span style="font-family: CMMI12;">M</span><span style="font-family: CMMI8; font-size: xx-small;"><span style="font-family: CMMI8; font-size: xx-small;">Q</span></span></em><span style="font-family: CMR12;">( 2;</span><em><span style="font-family: CMSY10;">¡</span></em><span style="font-family: CMR12;">1</span><em><span style="font-family: CMMI12;">; </span></em><span style="font-family: CMR12;">2</span><em><span style="font-family: CMMI12;">; </span></em><span style="font-family: CMR12;">0 ) of stable torsion free sheaves </span>of rank 2 with Chern classes <em><span style="font-family: CMMI12;">c</span></em><span style="font-family: CMR8; font-size: xx-small;"><span style="font-family: CMR8; font-size: xx-small;">1 </span></span><span style="font-family: CMR12;">= -</span><span style="font-family: CMR12;">1, </span><em><span style="font-family: CMMI12;">c</span></em><span style="font-family: CMR8; font-size: xx-small;"><span style="font-family: CMR8; font-size: xx-small;">2 </span></span><span style="font-family: CMR12;">= 2, </span><em><span style="font-family: CMMI12;">c</span></em><span style="font-family: CMR8; font-size: xx-small;"><span style="font-family: CMR8; font-size: xx-small;">3 </span></span><span style="font-family: CMR12;">= 0 on a smooth 3-dimensional </span>projective quadric <em><span style="font-family: CMMI12;">Q</span></em><span style="font-family: CMR12;">. The manifold </span><em><span style="font-family: CMMI12;">M</span><span style="font-family: CMMI8; font-size: xx-small;"><span style="font-family: CMMI8; font-size: xx-small;">Q</span></span></em><span style="font-family: CMR12;">(-</span><span style="font-family: CMR12;">1</span><em><span style="font-family: CMMI12;">; </span></em><span style="font-family: CMR12;">2) of moduli bundles of rank 2 with Chern </span>classes <em><span style="font-family: CMMI12;">c</span></em><span style="font-family: CMR8; font-size: xx-small;"><span style="font-family: CMR8; font-size: xx-small;">1 </span></span><span style="font-family: CMR12;">= -</span><span style="font-family: CMR12;">1, </span><em><span style="font-family: CMMI12;">c</span></em><span style="font-family: CMR8; font-size: xx-small;"><span style="font-family: CMR8; font-size: xx-small;">2 </span></span><span style="font-family: CMR12;">= 2 on </span><em><span style="font-family: CMMI12;">Q </span></em><span style="font-family: CMR12;">was studied by Ottaviani and Szurek in 1994. In 2007 the </span>author described the closure <em><span style="font-family: CMMI12;">M</span><span style="font-family: CMMI8; font-size: xx-small;"><span style="font-family: CMMI8; font-size: xx-small;">Q </span></span></em><span style="font-family: CMR12;">(-</span><span style="font-family: CMR12;">1</span><em><span style="font-family: CMMI12;">; </span></em><span style="font-family: CMR12;">2) in the scheme </span><em><span style="font-family: CMMI12;">M</span><span style="font-family: CMMI8; font-size: xx-small;"><span style="font-family: CMMI8; font-size: xx-small;">Q</span></span></em><span style="font-family: CMR12;">(2;</span><em><span style="font-family: CMSY10;">¡</span></em><span style="font-family: CMR12;">1</span><em><span style="font-family: CMMI12;">; </span></em><span style="font-family: CMR12;">2</span><em><span style="font-family: CMMI12;">; </span></em><span style="font-family: CMR12;">0). In this paper we </span>prove that in <em><span style="font-family: CMMI12;">M</span><span style="font-family: CMMI8; font-size: xx-small;"><span style="font-family: CMMI8; font-size: xx-small;">Q</span></span></em><span style="font-family: CMR12;">(2;</span><em><span style="font-family: CMSY10;">¡</span></em><span style="font-family: CMR12;">1</span><em><span style="font-family: CMMI12;">; </span></em><span style="font-family: CMR12;">2</span><em><span style="font-family: CMMI12;">; </span></em><span style="font-family: CMR12;">0) there exists a unique irreducible component diferent from </span>M<span style="font-family: CMMI8; font-size: xx-small;"><span style="font-family: CMMI8; font-size: xx-small;">Q </span></span><span style="font-family: CMR12;">(</span><em><span style="font-family: CMSY10;">¡</span></em><span style="font-family: CMR12;">1</span><em><span style="font-family: CMMI12;">; </span></em><span style="font-family: CMR12;">2) which is a rational variety of dimension 10.</span></p>