Moduli of Vector Bundles on Curves in Positive Characteristics

Let X be a projective curve of genus 2 over an algebraically closed field of characteristic 2. The Frobenius map on X induces a rational map on the moduli scheme of rank-2 bundles. We show that up to isomorphism, there is only one (up to tensoring by an or

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aMODULIOFVECTORBUNDLESONCURVESINPOSITIVECHARACTERISTICSKIRTIJOSHIANDEUGENEZ.XIAAbstract.LetXbeaprojectivecurveofgenus2overanalge-braicallyclosed eldofcharacteristic2.TheFrobeniusmaponXinducesarationalmaponthemodulischemeofrank-2bundles.Weshowthatuptoisomorphism,thereisonlyone(uptotensoringbyanordertwolinebundle)semi-stablevectorbundleofrank2(withdeterminantequaltoathetacharacteristic)whoseFrobeniuspull-backisnotsemi-stable.TheindeterminacyoftheFrobeniusmapatthispointcanberesolvedbyintroducingHiggsbundles.1.IntroductionandResultsLetXbeasmoothprojectivecurveofgenus2overanalgebraicallyclosed eldkofcharacteristicp>0.Let beitscanonicalbundle.De nethe(absolute)Frobeniusmorphism[3,4]F:X →Xwhichmapslocalsectionsf∈OXtofp.AsXissmooth,Fisa( nite) atmap.LetJ0,J1bethemodulischemesofisomorphismclassesoflinebun-dlesofdegree0and1,respectively.ChooseathetacharacteristicLθ∈J1.DenotebySO(resp.Sθ)themodulischemeofS-equivalenceclassesofsemi-stablevectorbundlesofrank2anddeterminantOX(resp.Lθ)onX[8].WestudytheFrobeniuspull-backsofthebundlesinSOandSθ.ThegeometryofSθhasbeenstudiedextensivelyby

Bhosle[1].

TheoperationofFrobeniuspull-backhasatendencytodestabilizebundles[9].Inparticular,themapV →F (V)isrationalonthemodulischeme.

TheFrobeniusdestabilizesonly nitemanybundlesinSO(seeThe-orem3.2).ForanyV∈SO,Proposition3.3givesanecessary

and

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