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<pre style="text-align:center"><b style="mso-bidi-font-weight:normal"><span style="font-size:22.0pt;font-family:"Times New Roman","serif";
mso-ansi-language:EN-US" lang="EN-US">“On a conjecture of Donagi and Morrison”<o:p></o:p></span></b></pre>
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EN-US" lang="EN-US"><o:p> </o:p></span></i></b></p>
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EN-US" lang="EN-US"><o:p> </o:p></span></i></b></p>
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normal"><i style="mso-bidi-font-style:normal"><span
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EN-US" lang="EN-US"><o:p> </o:p></span></i></b></p>
<p class="MsoNormal" style="text-align:justify"><b
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normal"><i style="mso-bidi-font-style:normal"><span
style="font-family:"Times New
Roman","serif";mso-ansi-language:EN-US"
lang="EN-US">Abstract:<o:p></o:p></span></i></b></p>
<p class="MsoNormal" style="text-align:justify"><i
style="mso-bidi-font-style:
normal"><span style="font-family:"Times New
Roman","serif";
mso-ansi-language:EN-US" lang="EN-US">Let S be a K3 surface
and C a genus g curve lying on
it. The Donagi-Morrison conjecture concerns complete linear
series A of
type g<sup>r</sup><sub>d</sub> on C such that d≤ g – 1 and
the Brill–Noether
number is negative. It predicts that such an A is contained
in another
linear series which is cut out from a line bundle on S and
satisfies some
numerical conditions, meaningful from a Brill–Noether theory
viewpoint. We
provide counterexamples to the conjecture already for r = 2.
We then study a
slight modification of the conjecture, which is known to
hold for r = 1, 2.
After exhibiting counterexamples to the modified conjecture
for r ≥ 3, we prove
an optimal result on its range of validity. This is done by
using coherent
systems and generalized Lazarsfeld–Mukai bundles. The
conjecture turns out to
be strictly connected with existence of some special secant
varieties to the
curve C and its deformations.<b
style="mso-bidi-font-weight:normal"><o:p></o:p></b></span></i></p>
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mso-ansi-language:EN-US" lang="EN-US"><o:p> </o:p></span></i></p>
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style="font-size:16.0pt;font-family:"Times New
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EN-US" lang="EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal" style="text-align:justify"><span
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font-family:"Times New Roman","serif"">Tutti
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font-family:"Times New
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<pre class="moz-signature" cols="72">--
--------------------------------------------------------------------------------
Valeria Giuliani
Scuola Normale Superiore
Servizio alla Didattica e Allievi
Tel. 050-509260
Piazza dei Cavalieri, 7
56126 Pisa
E-Mail: <a class="moz-txt-link-abbreviated" href="mailto:v.giuliani@sns.it">v.giuliani@sns.it</a>
E-Mail: <a class="moz-txt-link-abbreviated" href="mailto:classi@sns.it">classi@sns.it</a>
Le informazioni contenute nella presente e-mail e nei relativi allegati possono essere riservate e sono, comunque, destinate esclusivamente al destinatario in indirizzo.
E’ vietata, pertanto, la diffusione, distribuzione e/o copiatura di tali informazioni da parte di qualsiasi soggetto diverso dal destinatario.
Chiunque abbia ricevuto o letto questa e-mail per errore o senza esserne legittimato č invitato a darne immediatamente notizia al mittente tramite fax o e-mail e a distruggerla.
Grazie.
</pre>
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