17
Semi-Ample Asymptotic Syzygies Juliette Bruce University of Wisconsin - Madison

Semi-Ample Asymptotic Syzygiesjarod/wagon/julietteBruce.pdf · ASYMPTOTIC SYZYGIES 0 S(X,Ld) F0 F1 ··· ··· Fr d " 0 minimal graded free resolution βp,q(X,Ld)=# minimal generators

  • Upload
    others

  • View
    0

  • Download
    0

Embed Size (px)

Citation preview

Page 1: Semi-Ample Asymptotic Syzygiesjarod/wagon/julietteBruce.pdf · ASYMPTOTIC SYZYGIES 0 S(X,Ld) F0 F1 ··· ··· Fr d " 0 minimal graded free resolution βp,q(X,Ld)=# minimal generators

Semi-Ample Asymptotic Syzygies

Juliette Bruce

University of Wisconsin - Madison

Page 2: Semi-Ample Asymptotic Syzygiesjarod/wagon/julietteBruce.pdf · ASYMPTOTIC SYZYGIES 0 S(X,Ld) F0 F1 ··· ··· Fr d " 0 minimal graded free resolution βp,q(X,Ld)=# minimal generators

ASYMPTOTIC SYZYGIES • Asymptotic syzygies is the study of the algebraic Betti numbers of

a variety as the positivity of the embedding increases.

n dimensional smooth projective variety

sequence of very ample line bundles

X PH0 (Ld ) ! Prd

Page 3: Semi-Ample Asymptotic Syzygiesjarod/wagon/julietteBruce.pdf · ASYMPTOTIC SYZYGIES 0 S(X,Ld) F0 F1 ··· ··· Fr d " 0 minimal graded free resolution βp,q(X,Ld)=# minimal generators

• To this we associate:

ASYMPTOTIC SYZYGIES

n dimensional smooth projective variety

sequence of very ample line bundles

S (X,Ld ) =!

k∈ZH0 (X,kLd )

• which we think of as a module over

X PH0 (Ld ) ! Prd

S = SymH0 (Ld ) ! C!x0,x1, . . . ,xrd

"

Page 4: Semi-Ample Asymptotic Syzygiesjarod/wagon/julietteBruce.pdf · ASYMPTOTIC SYZYGIES 0 S(X,Ld) F0 F1 ··· ··· Fr d " 0 minimal graded free resolution βp,q(X,Ld)=# minimal generators

ASYMPTOTIC SYZYGIES

0 S (X,Ld )!F0 F1 · · · · · · Frd

"0

minimal graded free resolution

βp,q (X,Ld ) = #!minimal generatorsof Fp of degree q

"= number of syzygies of degree q

and homological degree p

how do these vary as a function of d?

Page 5: Semi-Ample Asymptotic Syzygiesjarod/wagon/julietteBruce.pdf · ASYMPTOTIC SYZYGIES 0 S(X,Ld) F0 F1 ··· ··· Fr d " 0 minimal graded free resolution βp,q(X,Ld)=# minimal generators

ASYMPTOTIC SYZYGIES

• It is often useful to place the Betti numbers into a table:

notice the wonky change of coordinates

�(X,d) =

0 1 2 · · · p · · ·0 �0,0 �1,1 �2,2 · · · · · ·1 �0,1 �1,2 �2,3 · · · · · ·...

......

.... . . · · ·

q �p,p+q · · ·...

......

......

......

<latexit sha1_base64="wVQOMFKr+6qz9w7PFIL5KdXevK8=">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</latexit><latexit sha1_base64="tu0ggFyP+iqUgtVZ5e2H1BY+A5I=">AAAEgXicfVPLbtNAFHXSAMW8WliyGVFjtcKN7LAAqUKqYNMFSEX0EakTRePxJLEyD3dm3BIGr/knVqz4B5b8CePESfOi17Lm3HvnnnPnFWc0VToM/9TqG407d+9t3ncfPHz0+MnW9tMzJXKJySkWVMh2jBShKSenOtWUtDNJEIspOY+HH8r8+RWRKhX8RI8y0mGoz9NeipG2oe527S+MiUa77SDZgwfgHYAHro30U240inOKZGGA/I4nVrh+6Ed+y/cgToRWnu9l3o0DIAQuAHBQdgOsWScEPvDGGl0TBmHhAX/qRkFUurN0K2hV/oRvHgLfcq8xqxDNCywyRjPGqcLr2xUsnQevqtVMEfg/TNbQ2F1wS55LSzH9pg1kAcheXRarFcCd6d4I3I7WdVpxQcKT2fl1t3bCZjg2sAqiCuwcxs2fv343Px3bS/EDJgLnjHCNKVLqIgoz3TFI6hRTUrgwVyRDeIj65MJCjhhRHTO+jAV4aSMJ6Alpf67BODpfYRBTasRiO5MhPVDLuTK4NqfT4bd9nCzIG5JjRIOYBcOslFNBWRfHgi51qXtvOyblWa4Jx5MmezkFWoDyVYAklQRrOrIAYZnadQI8QBJhbd+OCzm5xoIxZHcVfhmxwkCREYm0kOXaTRmynIngZHk1esCKcb0eECEJM9VYmJMKLGQpsSqF+TgeXNceXbR8UKvgrNWMwmb02Z7he2dim85z54Wz60TOG+fQOXKOnVMH14/qvH5d/9rYaOw1wkZrMrVeq2qeOQvWOPgHqmJckw==</latexit><latexit sha1_base64="tu0ggFyP+iqUgtVZ5e2H1BY+A5I=">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</latexit><latexit sha1_base64="ozR91TASVxOLaa2mc0XxGqzLxlk=">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</latexit>

Page 6: Semi-Ample Asymptotic Syzygiesjarod/wagon/julietteBruce.pdf · ASYMPTOTIC SYZYGIES 0 S(X,Ld) F0 F1 ··· ··· Fr d " 0 minimal graded free resolution βp,q(X,Ld)=# minimal generators

FIRST RESULTS - CURVES

X PH0 (Ld ) ! Prd

smooth projective curve of genus g

line bundle of degree d

Theorem (Castelnuovo et al.).

1. If d � 2g +1 then Ld is defines an embedding into Prd.

2. If d � 2g +2 then IX is generated by quadrics.

<latexit sha1_base64="r87Jy830T3qY+2MBVl2WqFYeXgE=">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</latexit><latexit sha1_base64="r87Jy830T3qY+2MBVl2WqFYeXgE=">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</latexit><latexit sha1_base64="r87Jy830T3qY+2MBVl2WqFYeXgE=">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</latexit><latexit sha1_base64="r87Jy830T3qY+2MBVl2WqFYeXgE=">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</latexit>

Page 7: Semi-Ample Asymptotic Syzygiesjarod/wagon/julietteBruce.pdf · ASYMPTOTIC SYZYGIES 0 S(X,Ld) F0 F1 ··· ··· Fr d " 0 minimal graded free resolution βp,q(X,Ld)=# minimal generators

FIRST RESULTS - CURVES• Part (1) tells us the Betti table of X eventually looks like:

correspond to the generators of the defining ideal

• Part (2) tells us the Betti table of X eventually looks like:

0 1 2 · · · p · · · rd0 1 - - · · · - · · ·1 - �1,2 �2,3 · · · �p,p+1 · · ·2 - �1,3 �2,4 · · · �p,p+2 · · ·

<latexit sha1_base64="qNCPAlSMOeQDszxr3xMJ1qWtrsk=">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</latexit><latexit sha1_base64="qNCPAlSMOeQDszxr3xMJ1qWtrsk=">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</latexit><latexit sha1_base64="qNCPAlSMOeQDszxr3xMJ1qWtrsk=">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</latexit><latexit sha1_base64="qNCPAlSMOeQDszxr3xMJ1qWtrsk=">AAAD03icfVJLb9QwEHYaHiU82sKRi8UuERLpKkkrwbGCCxekIrptpWa1chxv11o/ItsBLSYXxJV/w5+BK/wQnN3saptWjDTyPPx9M+NxXjKqTRz/8rb8W7fv3N2+F9x/8PDRzu7e41MtK4XJEEsm1XmONGFUkKGhhpHzUhHEc0bO8tnbJn/2iShNpTgx85KMOLoUdEIxMi403vPeZzm5pMIalFcMqdpC9RUvpQ7COEzCNOxnuJBG98N+2d9w1NgWdR9mGQwgzKZNC9CJc2IYwsTpfqsrzMLd9B32BnEMyQqYE4PGNonSur/hp9FB66+JV6kyKl8mLtkp4zjTdfUV6UGH9PB/pGmHdDU4EcX69ca7vXgQLwReN5LW6IFWjt0CeFZIXHEiDGZI64skLs3IImUoZqQOskqTEuEZuiQXzhSIEz2yi8XX8LmLFHAilVNh4CK6ibCIaz3nubvJkZnqbq4J3pgzdPZlHxdXyltSYcSinEezsimnowaX55J1ujST1yNLRVkZIvCyyUnFoJGw+YGwoIpgw+bOQFhRNyfEU6QQNu6fBpkgn7HkHLlXzT7OeW0zWRKFjFTN7LYJOc5CCtKdxkx5vcCbKZGKcNuetT1pjSBwC0q667hunKaDJB4kHw57R2/aVW2Dp+AZeAES8AocgXfgGAwB9n56v70/3l9/6Fv/m/99eXXLazFPwBXxf/wDdfwkig==</latexit>

0 1 2 · · · p · · · rd0 1 - - · · · - · · ·1 - �1,2 �2,3 · · · �p,p+1 · · ·2 - - �2,4 · · · �p,p+2 · · ·

<latexit sha1_base64="A3QwHLrc0RuoZeyBvW5QzMHGz14=">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</latexit><latexit sha1_base64="A3QwHLrc0RuoZeyBvW5QzMHGz14=">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</latexit><latexit sha1_base64="A3QwHLrc0RuoZeyBvW5QzMHGz14=">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</latexit><latexit sha1_base64="A3QwHLrc0RuoZeyBvW5QzMHGz14=">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</latexit>

Page 8: Semi-Ample Asymptotic Syzygiesjarod/wagon/julietteBruce.pdf · ASYMPTOTIC SYZYGIES 0 S(X,Ld) F0 F1 ··· ··· Fr d " 0 minimal graded free resolution βp,q(X,Ld)=# minimal generators

FIRST RESULTS - CURVES

Theorem (Green, 1984). Let X be a smooth curve. If degLd = d then

lim

d!1⇢2

(X;Ld ) = 0.<latexit sha1_base64="ie/bbcppBxH0KT22VZfI3OSCtLk=">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</latexit><latexit sha1_base64="ie/bbcppBxH0KT22VZfI3OSCtLk=">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</latexit><latexit sha1_base64="ie/bbcppBxH0KT22VZfI3OSCtLk=">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</latexit>

the percentage of non-zero syzygies

in row q

⇢q (X;Ld ) B#n

p 2 N�

� �p,p+q (X,Ld ) , 0o

rd<latexit sha1_base64="2s0A2rRg5bh05sb4P1h+WCwhIFU=">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</latexit><latexit sha1_base64="wY7rxEWSVj7KttCAqh00FJGxrIc=">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</latexit><latexit sha1_base64="wY7rxEWSVj7KttCAqh00FJGxrIc=">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</latexit><latexit sha1_base64="AqtXqitE5/eGzdvwxJG6N8Afr08=">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</latexit>

Page 9: Semi-Ample Asymptotic Syzygiesjarod/wagon/julietteBruce.pdf · ASYMPTOTIC SYZYGIES 0 S(X,Ld) F0 F1 ··· ··· Fr d " 0 minimal graded free resolution βp,q(X,Ld)=# minimal generators

FIRST RESULTS - CURVES

Theorem (Green, 1984). Let X be a smooth curve. If degLd = d then

lim

d!1⇢2

(X;Ld ) = 0.<latexit sha1_base64="ie/bbcppBxH0KT22VZfI3OSCtLk=">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</latexit><latexit sha1_base64="ie/bbcppBxH0KT22VZfI3OSCtLk=">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</latexit><latexit sha1_base64="ie/bbcppBxH0KT22VZfI3OSCtLk=">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</latexit>

the percentage of non-zero syzygies

in row q

⇢q (X;Ld ) B#n

p 2 N�

� �p,p+q (X,Ld ) , 0o

rd<latexit sha1_base64="2s0A2rRg5bh05sb4P1h+WCwhIFU=">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</latexit><latexit sha1_base64="wY7rxEWSVj7KttCAqh00FJGxrIc=">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</latexit><latexit sha1_base64="wY7rxEWSVj7KttCAqh00FJGxrIc=">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</latexit><latexit sha1_base64="AqtXqitE5/eGzdvwxJG6N8Afr08=">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</latexit>

as the degree of the line bundle grows

the percentage of non-zero syzygies

in the second row

Page 10: Semi-Ample Asymptotic Syzygiesjarod/wagon/julietteBruce.pdf · ASYMPTOTIC SYZYGIES 0 S(X,Ld) F0 F1 ··· ··· Fr d " 0 minimal graded free resolution βp,q(X,Ld)=# minimal generators

FIRST RESULTS - AMPLE GROWTH

Theorem (Ein-Lazarsfeld, 2012). Let n � 2 and fix 1 q n. If Ld+1 � Ld is

constant and ample then

lim

d!1⇢q (X;Ld ) = 1

<latexit sha1_base64="DFTTJh9o/qhvCOiGDsnBxoUP3Rw=">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</latexit><latexit sha1_base64="DFTTJh9o/qhvCOiGDsnBxoUP3Rw=">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</latexit><latexit sha1_base64="DFTTJh9o/qhvCOiGDsnBxoUP3Rw=">AAADf3icbVLbbtNAELUbLsVcmsIjLytiRIEkssMDoAqpAiGBlIcimrYoG0Xr9ThZZS/O7gZwLf8I/8QH8DesnTyQlJGsmZ2bz5yZJOfM2Cj64++1bty8dXv/TnD33v0HB+3Dh+dGrTSFEVVc6cuEGOBMwsgyy+Ey10BEwuEiWXyo4xffQRum5JktcpgIMpMsY5RY55q2f+MEZkyWdi6q8Ucme0NyRbTJgKddNIjiwSQYgkWhxDNYDkJEZIoy9hOFMeawREvUKBn20ecMhcNpmb6Mq16tqxAxg6iSxhJpm0Iicg7IzkGiAI8DzJlwiQhbhTCTmS0qhI8R1nM1LZeV65zZo8tj1HTDms3m9rlLcCnv1ioO8CTAINMGfhAE03Yn6keNoOtGvDE63kZOp4f+L5wquhIgLeXEmHEc5XZSEm0Z5VAFeGUgJ3RBZjB2piQCzKRseK/QU+dxdCjtPjdi4/23oiTCmEIkLlMQOze7sdr535hli6seTbd+X4J0MDWx26jqPs302PGqNLhXwaFMIWOS1Tt2EQk/NsEXDVPl2fq1039FCe8morvI63FMt8aVJIrvsGCzN5OSyXxlQdI1CdmKI7fE+sBQyjRQywtnEKodBIronGhCrTvDBgtVQrhzKPHXwmHBKq+nUrrmtqxdrmeqJNTbjHd3d904H/TjqB9/GXRO3m/2uu899p54R17svfZOvE/eqTfyqP/K/+YnPm35rWetfitap+75m5pH3pa03v4F0ycdew==</latexit>

the percentage of non-zero syzygies

asymptotically syzygies occur in every possible

degree

Page 11: Semi-Ample Asymptotic Syzygiesjarod/wagon/julietteBruce.pdf · ASYMPTOTIC SYZYGIES 0 S(X,Ld) F0 F1 ··· ··· Fr d " 0 minimal graded free resolution βp,q(X,Ld)=# minimal generators

FIRST RESULTS - AMPLE GROWTH

Theorem (Ein-Lazarsfeld, 2012). Let n � 2 and fix 1 q n. If Ld+1 � Ld is

constant and ample then

lim

d!1⇢q (X;Ld ) = 1

<latexit sha1_base64="5Z1e0lSueR5DoanoHsnsuLjjO0o=">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</latexit><latexit sha1_base64="5Z1e0lSueR5DoanoHsnsuLjjO0o=">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</latexit><latexit sha1_base64="5Z1e0lSueR5DoanoHsnsuLjjO0o=">AAAD9XicbVJLbxMxEN40PMryauHIxaJBKpBEu0GFShVSBUKiUg9F9CXFUeT1zm6seO2N7dCm1v4Ubogrv4cz/4IT9iZCJGUk2+OZz+NvHknJmTZR9LOx1rxx89bt9Tvh3Xv3Hzzc2Hx0quVUUTihkkt1nhANnAk4McxwOC8VkCLhcJaM33v/2RdQmklxbGYlDAqSC5YxSowzDTd+4QRyJqwZFVX/AxOdQ3JFlM6Ap23Ui+LeIDwEg1oC5zDptRARKcrYJWrFmMMETVB9iFYXHWTI4pqQTQFKCoppqFDrcGjTl3HV8WfVQkwjKoU2RJg6GClK7mBmBAKFuB9izgqHRNhIhJnIzKxCeA9hNZJDO6ncd5nZPt9DdTisWD4yzx3AQd7OjzjEgxCDSOucQifDja2oG9WCrivxQtkKFnI03Gz8xqmk0wKEoZxo3Y+j0gwsUYZRxzbEUw0loWOSQ9+pghSgB7ZOvkLPnMUVSSq3XJK19d8XlhRaz4rEIQtiRnrV543/9Rk2vurQdOl7C8LRVMQss/Jx6vSxK6xU4G4zDq4xGRPMd955BFwsnC/qUtnj+W0l/pQS3k6K9rj06ei255Ukkq9UwWS7A8tEOTUg6LwI2ZQj10U/dihlCqjhM6cQqhwFiuiIKEKNG86aC5VF4QbC4s8zxwXL0mclla+t9SYXM5UCQlwnAdcmzao8qWzU3d1pu+b2/LbzqlqGU6JYkgARuQIQf59EHrzrt9eVn5Z4dTauK6e9bhx140+9rf13i7lZD54ET4PtIA7eBPvBx+AoOAlo46AhG5eNWfOi+bX5rfl9Dl1rLN48Dpak+eMPLNNKNQ==</latexit>

L1

L2

L3

the ample cone of X

Page 12: Semi-Ample Asymptotic Syzygiesjarod/wagon/julietteBruce.pdf · ASYMPTOTIC SYZYGIES 0 S(X,Ld) F0 F1 ··· ··· Fr d " 0 minimal graded free resolution βp,q(X,Ld)=# minimal generators

NEW RESULTS - SEMI-AMPLE GROWTH

the percent of possible degrees with non-zero syzygies

the main specific asymptotic behavior

Theorem (Juliette Bruce). Let X = Pn⇥Pmand fix an index 1 q n+m. There

exists constants Ci,j and Di,j such that

⇢q (X;O (d1

,d2

)) � 1 �X

i+j=q0in0jm

0BBBBB@Ci,j

di1

dj2

+

Di,j

dn�i1

dm�j2

1CCCCCA � O

lower ord.

terms

!.

<latexit sha1_base64="Ht68iE/t8Snj28kDpX4AtbYQGJg=">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</latexit><latexit sha1_base64="Ht68iE/t8Snj28kDpX4AtbYQGJg=">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</latexit><latexit sha1_base64="Ht68iE/t8Snj28kDpX4AtbYQGJg=">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</latexit><latexit sha1_base64="uMbNSA+2tRCv1dDA+977awhQHn4=">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</latexit><latexit sha1_base64="dfsOO1GLUToy5Fxhxm4XGrlBxAI=">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</latexit><latexit sha1_base64="dfsOO1GLUToy5Fxhxm4XGrlBxAI=">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</latexit><latexit sha1_base64="aH72pniJf4OkFJCBIRLgtgxMesQ=">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</latexit><latexit sha1_base64="Ht68iE/t8Snj28kDpX4AtbYQGJg=">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</latexit><latexit sha1_base64="Ht68iE/t8Snj28kDpX4AtbYQGJg=">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</latexit><latexit sha1_base64="Ht68iE/t8Snj28kDpX4AtbYQGJg=">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</latexit><latexit sha1_base64="Ht68iE/t8Snj28kDpX4AtbYQGJg=">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</latexit><latexit sha1_base64="Ht68iE/t8Snj28kDpX4AtbYQGJg=">AAAEo3icbVPbbhs3EF0rSptu0zaXx7wQtQIktSRo/ZIEQgDD9kPQFrCT2IkBUxa43JGWNi8yyY2lEvtf+Y6+9W86e3FiKyEgcXgOd+bMhelCCudHo/82One6d3/48d5P8c/3f/n1twcPH31wprAcjrmRxp6kzIEUGo698BJOFhaYSiV8TC/2Kv7jJ7BOGH3kVwuYKDbXYiY48whNH278S1OYCx18rsrTPwspwHsgu7bgMIn/Bk96J6+pYj5P03BYngVdUi8UuJuYKnuE6YzMxBJ3InQGS9JLqIRLcknqTW+p3pAc5WCBwBLzcoQb7TzTaPX2pkH0z1svvf3rkyt4TnzOfExPY2pzMw2XJbqb+WcnY1Ir4EyGgxbLpkk/m25TK+a5f369jekcLumYJHQ8oK5Q04D/KUbmF0Fsnb9GgZSMapGi1foVOW8QVbYhYjqzjIdWbxkw5JnAmGfn5VZD7d+kgh6IsqKDGiD2RdKAjg8ah231MRUrllhaWPogzRVYYmw2LFFIg3mwypUUdPblbuNsGNNJXONVB+NqTR9sjoajepFvjaQ1NqN2HeIUfKaZ4YUC7blkzp0mo4WfBGa94BLKmBYOFlgwNodTNDXDAZiEevpK8hQR7L2x+NOe1OjNLwJTzq1Uijerjrl1rgK/y3lx8c+AZ7fCB9Ao0zJ/W1Xlp86f+hyMBTytJIQMZkKLatKR0XDVkn/UtQpHzWmN4phT2DPWSMnsai14gfPWT1X/YlHl6vrNIzByrUR+9nIShF4UHjRvKjQrJPGGVG+QZMIC93KFBuMW9XHCc4bTg112tRpulMK3EOj7FQqlZlGlbGxV+FBB6DMzGupeJ+ud/db4sD1MRsPk7fbmzm7b9XvRk+j36FmURC+inehNdBgdR7zzqjPt5B3Rfdr9q/uue9Rc7Wy03zyObq3u5H9yGY6v</latexit><latexit sha1_base64="Ht68iE/t8Snj28kDpX4AtbYQGJg=">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</latexit>

Page 13: Semi-Ample Asymptotic Syzygiesjarod/wagon/julietteBruce.pdf · ASYMPTOTIC SYZYGIES 0 S(X,Ld) F0 F1 ··· ··· Fr d " 0 minimal graded free resolution βp,q(X,Ld)=# minimal generators

NEW RESULTS - SEMI-AMPLE GROWTH

• My result does not require an assumption of ample growth.

O(0,1)

O(1,0)

ample cone Pn ×Pm

Definition. A line bundle L is semi-ample if |kL| is base point free for some k.

Ld+1 −Ld = O(1,0)

is not ample, it’s semi-ample

Page 14: Semi-Ample Asymptotic Syzygiesjarod/wagon/julietteBruce.pdf · ASYMPTOTIC SYZYGIES 0 S(X,Ld) F0 F1 ··· ··· Fr d " 0 minimal graded free resolution βp,q(X,Ld)=# minimal generators

NEW RESULTS - SEMI-AMPLE GROWTH

⇢2

⇣P1 ⇥P5

; O (d1

,d2

)

⌘� 1 � 20

d22

� 60

d1

d32

� 5

d1

d2

� 120

d42

� O

lower ord.

terms

!

<latexit sha1_base64="xpgIm/rb9kYX9U6TM2LsU0Ahux8=">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</latexit><latexit sha1_base64="PMAhx7oYrbPxJ/jxWmVO4Ft069I=">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</latexit><latexit sha1_base64="q5S2Poig/WeokkDtLnbHAIfBxdU=">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</latexit>

limd1!1

⇢2⇣P1 ⇥P5; O (d1,d2)

⌘� 1 � 20

d22� 120

d42.

<latexit sha1_base64="k+4qmrKRkJbWK+8UTbVUywM3BUM=">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</latexit><latexit sha1_base64="k+4qmrKRkJbWK+8UTbVUywM3BUM=">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</latexit><latexit sha1_base64="k+4qmrKRkJbWK+8UTbVUywM3BUM=">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</latexit><latexit sha1_base64="OALYu+/l/NIJE9YsdsfcAU4yHd4=">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</latexit><latexit sha1_base64="ExH3BhcW3WUzuiSlrxy+vYuaqyQ=">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</latexit><latexit sha1_base64="ExH3BhcW3WUzuiSlrxy+vYuaqyQ=">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</latexit><latexit sha1_base64="8DaAqB4ViBBUm0euklT7c6NZmk8=">AAADKHicbVFNb9NAEF2brzZ8pXDkYhEhFSkNtgWiKJcKLtwIomkrdRNrvV7Hq+yHWU9AYeVfw41fww31yi9h7RjRpoxk7fObfTM789JS8ArC8MLzb9y8dfvOzm7v7r37Dx729x6dVHplKJtSLbQ5S0nFBFdsChwEOysNIzIV7DRdvmvyp1+YqbhWx7Au2UySheI5pwQclfS/Y8FlYrMkwqAxVzms6wCbQicxFiyHfSwJFGlqJ/XcRjUGLll1iXs1xuOg/adE2A/1RuTKDTNXwfBFAc//HmO8YJ/xOHKKg0aVG0JtHNY2m8dJ7NqO/9FRyyfx/GU9SvqDcBS2EVwHUQcGqItJsudJnGm6kkwBFaSqzqOwhJklBjgVrO7hVcVKQpdkwc4dVMQNNbPtNuvgmWOyINfGfQqClr2ssERW1Vqm7mYzd7Wda8j/5oAvvx3Q7Ep7y1ZuccNUDpdl064abnarxdYrIT+cWa7KFTBFN4/MVyIAHTS2Bhk3jIJYO0Co4W7OgBbEbRKc+T2s2FeqpSQqs/jTWtYW65IZAto0s9uGcjUzrdj2NFDIutVDwbRh0nZnbY870Os5g6JtO66Dk3gUhaPoYzg4ettZtYOeoKdoH0XoNTpC79EETRH1dr0X3qH3xv/h//R/+Rebq77XaR6jK+H//gPnFwQq</latexit><latexit sha1_base64="k+4qmrKRkJbWK+8UTbVUywM3BUM=">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</latexit><latexit sha1_base64="k+4qmrKRkJbWK+8UTbVUywM3BUM=">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</latexit><latexit sha1_base64="k+4qmrKRkJbWK+8UTbVUywM3BUM=">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</latexit><latexit sha1_base64="k+4qmrKRkJbWK+8UTbVUywM3BUM=">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</latexit><latexit sha1_base64="k+4qmrKRkJbWK+8UTbVUywM3BUM=">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</latexit><latexit sha1_base64="k+4qmrKRkJbWK+8UTbVUywM3BUM=">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</latexit>

• If n = 1 and m = 5 then my result shows:

• In particular, if is fixed thend2<latexit sha1_base64="6f0nfdrkh1zuc7aN+OpcJK8iGyc=">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</latexit><latexit sha1_base64="6f0nfdrkh1zuc7aN+OpcJK8iGyc=">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</latexit><latexit sha1_base64="6f0nfdrkh1zuc7aN+OpcJK8iGyc=">AAACpXicbVHLbtNAFJ2YVzGvFJZsLCIEixDZVSXCrioL2CAVSNJKsWWNxzfNKPOwZq5bwshbvoYt/At/w9j1gqZcaTRH595zn0UluMU4/jMIbt2+c/fe3v3wwcNHj58M958urK4NgznTQpuzgloQXMEcOQo4qwxQWQg4LTbvW//pBRjLtZrhtoJM0nPFV5xR9FQ+jFKEb9jlWX75cJy5d8l4Oh1PDxtX5u6gafLhKJ7EnUU3QdKDEentJN8fyLTUrJagkAlq7TKJK8wcNciZgCZMawsVZRt6DksPFZVgM9e10EQvPVNGK238Uxh17L8KR6W1W1n4SElxbXd9LflfH/LN9zesvFbeQc2oGBdyvKnacnbc6opCi50ucTXNHFdVjaDYVZOrWkSoo3anUckNMBRbDygz3M8ZsTU1lKHffJgquGRaSqpKl37dysalugJDUZt2dtdSPmepFexOg2vZdHpcgzYgXf83btaDMPQHSnbPcRMsDiZJPEk+H46OjvtT7ZHn5AV5TRLylhyRj+SEzAkjP8hP8ov8Dl4Fn4JZsLgKDQa95hm5ZkH+FwgE1DY=</latexit><latexit sha1_base64="6f0nfdrkh1zuc7aN+OpcJK8iGyc=">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</latexit>

neither 0 nor 1

Page 15: Semi-Ample Asymptotic Syzygiesjarod/wagon/julietteBruce.pdf · ASYMPTOTIC SYZYGIES 0 S(X,Ld) F0 F1 ··· ··· Fr d " 0 minimal graded free resolution βp,q(X,Ld)=# minimal generators

NEW RESULTS - SEMI-AMPLE GROWTH

⇢2

⇣P1 ⇥P5

; O (d1

,d2

)

⌘� 1 � 20

d22

� 60

d1

d32

� 5

d1

d2

� 120

d42

� O

lower ord.

terms

!

<latexit sha1_base64="xpgIm/rb9kYX9U6TM2LsU0Ahux8=">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</latexit><latexit sha1_base64="PMAhx7oYrbPxJ/jxWmVO4Ft069I=">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</latexit><latexit sha1_base64="q5S2Poig/WeokkDtLnbHAIfBxdU=">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</latexit>

limd1!1

⇢2⇣P1 ⇥P5; O (d1,d2)

⌘� 1 � 20

d22� 120

d42.

<latexit sha1_base64="k+4qmrKRkJbWK+8UTbVUywM3BUM=">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</latexit><latexit sha1_base64="k+4qmrKRkJbWK+8UTbVUywM3BUM=">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</latexit><latexit sha1_base64="k+4qmrKRkJbWK+8UTbVUywM3BUM=">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</latexit><latexit sha1_base64="OALYu+/l/NIJE9YsdsfcAU4yHd4=">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</latexit><latexit sha1_base64="ExH3BhcW3WUzuiSlrxy+vYuaqyQ=">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</latexit><latexit sha1_base64="ExH3BhcW3WUzuiSlrxy+vYuaqyQ=">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</latexit><latexit sha1_base64="8DaAqB4ViBBUm0euklT7c6NZmk8=">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</latexit><latexit sha1_base64="k+4qmrKRkJbWK+8UTbVUywM3BUM=">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</latexit><latexit sha1_base64="k+4qmrKRkJbWK+8UTbVUywM3BUM=">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</latexit><latexit sha1_base64="k+4qmrKRkJbWK+8UTbVUywM3BUM=">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</latexit><latexit sha1_base64="k+4qmrKRkJbWK+8UTbVUywM3BUM=">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</latexit><latexit sha1_base64="k+4qmrKRkJbWK+8UTbVUywM3BUM=">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</latexit><latexit sha1_base64="k+4qmrKRkJbWK+8UTbVUywM3BUM=">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</latexit>

• If n = 1 and m = 5 then my result shows:

• In particular, if is fixed thend2<latexit sha1_base64="6f0nfdrkh1zuc7aN+OpcJK8iGyc=">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</latexit><latexit sha1_base64="6f0nfdrkh1zuc7aN+OpcJK8iGyc=">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</latexit><latexit sha1_base64="6f0nfdrkh1zuc7aN+OpcJK8iGyc=">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</latexit><latexit sha1_base64="6f0nfdrkh1zuc7aN+OpcJK8iGyc=">AAACpXicbVHLbtNAFJ2YVzGvFJZsLCIEixDZVSXCrioL2CAVSNJKsWWNxzfNKPOwZq5bwshbvoYt/At/w9j1gqZcaTRH595zn0UluMU4/jMIbt2+c/fe3v3wwcNHj58M958urK4NgznTQpuzgloQXMEcOQo4qwxQWQg4LTbvW//pBRjLtZrhtoJM0nPFV5xR9FQ+jFKEb9jlWX75cJy5d8l4Oh1PDxtX5u6gafLhKJ7EnUU3QdKDEentJN8fyLTUrJagkAlq7TKJK8wcNciZgCZMawsVZRt6DksPFZVgM9e10EQvPVNGK238Uxh17L8KR6W1W1n4SElxbXd9LflfH/LN9zesvFbeQc2oGBdyvKnacnbc6opCi50ucTXNHFdVjaDYVZOrWkSoo3anUckNMBRbDygz3M8ZsTU1lKHffJgquGRaSqpKl37dysalugJDUZt2dtdSPmepFexOg2vZdHpcgzYgXf83btaDMPQHSnbPcRMsDiZJPEk+H46OjvtT7ZHn5AV5TRLylhyRj+SEzAkjP8hP8ov8Dl4Fn4JZsLgKDQa95hm5ZkH+FwgE1DY=</latexit>

neither 0 nor 1

• Syzygies in the setting of semi-ample growth exhibit behavior that is different from previous cases!

• curves (Green) • ample growth (Ein-Lazarsfeld)

Page 16: Semi-Ample Asymptotic Syzygiesjarod/wagon/julietteBruce.pdf · ASYMPTOTIC SYZYGIES 0 S(X,Ld) F0 F1 ··· ··· Fr d " 0 minimal graded free resolution βp,q(X,Ld)=# minimal generators

APPROACH OF PROOF• The proof generalizes the monomial methods of Ein, Erman,

Lazarsfeld to explicitly produce non-trivial syzygies.

this requires an Artinian reduction, but there are no

monomial regular sequences

• If n = 2 and m = 4 the regular sequence we work with is:!xd1

0 yd20

xd10 yd2

1 C xd11 yd2

0

xd10 yd2

2 C xd11 yd2

1 C xd12 yd2

0

xd10 yd2

3 C xd11 yd2

2 C xd12 yd2

1

xd10 yd2

4 C xd11 yd2

3 C xd12 yd2

2

xd11 yd2

4 C xd12 yd2

3

xd12 yd2

4

"

is not in this ideal, whilexd1−10 xd1−11 x3d1+22 y3d2−10 yd2−11 yd2−12 y33

xd1−10 xd1−11 x3d1+22 y3d20 yd2−11 yd2−12 y33

is in this ideal…

Page 17: Semi-Ample Asymptotic Syzygiesjarod/wagon/julietteBruce.pdf · ASYMPTOTIC SYZYGIES 0 S(X,Ld) F0 F1 ··· ··· Fr d " 0 minimal graded free resolution βp,q(X,Ld)=# minimal generators

• I exploit the fact that this regular sequence has a lot of symmetries:

• These symmetries enable me to use spectral sequence arguments to deeply understand this ideal.

i.e. homogenous with respect to a grading

the sum of the indices is constant

the power on each

!xd1

0 yd20

xd10 yd2

1 C xd11 yd2

0

xd10 yd2

2 C xd11 yd2

1 C xd12 yd2

0

xd10 yd2

3 C xd11 yd2

2 C xd12 yd2

1

xd10 yd2

4 C xd11 yd2

3 C xd12 yd2

2

xd11 yd2

4 C xd12 yd2

3

xd12 yd2

4

"

xis 0 mod d1

APPROACH OF PROOF