On the Jacobian locus in the Prym locus and geodesics.

Let L, L' be two line bundles on I C i , let 0 -> L -> F -> L' -> 0 be a short exact sequence associated to HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign=" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi> </mi><mo> </mo><msubsup><mi mathvariant="normal">E</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">t</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi class="MJX-tex-caligraphic" mathvariant="script">O</mi><mi>C</mi></msub></mrow><mn>1</mn></msubsup><mo stretchy="false">(</mo><msup><mi class="MJX-tex-caligraphic" mathvariant="script">L</mi><mo>'</mo></msup><mo>,</mo><mi class="MJX-tex-caligraphic" mathvariant="script">L</mi><mo So dim Z (P) <= 2 g + 2 d - 1 and dim Z (P) HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign=" rowspacing="4pt" columnspacing="1em"><mtr><mtd><msubsup><mi class="MJX-tex-mathit" mathvariant="italic"> </mi><mn>0</mn><mn>0</mn></msubsup></mtd></mtr></mtable></math> ht <= dim Z (P) - 1. So M b b O SB C ' b sb and, since HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign=" rowspacing="4pt" columnspacing="1em"><mtr><mtd><msubsup><mi> </mi><mi>b</mi><mn>2</mn></msubsup><mo>=</mo><msub><mi class="MJX-tex-caligraphic" mathvariant="script">O</mi><msubsup><mi>C</mi><mi>b</mi><mo>'</mo></msubsup></msub><mo>,</mo></mtd></mtr></mtable></math> ht we conclude that M b b. But now h 0 (C ' b,M b) >= 2, h 0 (C ' v, b) = 0 and h 0 (C ' b, M b) = h 0 (C ' b, b) = 0 leads to a contradiction. [Extracted from the article]