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❚♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✉❜s❡ts ♦❢ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡ ❉❛r✐♦ ❙♣✐r✐t♦ ✭❥♦✐♥t ✇♦r❦ ✇✐t❤ ●✐✉❧✐♦ P❡r✉❣✐♥❡❧❧✐✮ ❯♥✐✈❡rs✐tà ❞✐ ❘♦♠❛ ❚r❡ ❆▲❛◆❚ ✺ ✕ ❏♦✐♥t ❈♦♥❢❡r❡♥❝❡s ♦♥ ❆❧❣❡❜r❛✱ ▲♦❣✐❝ ❛♥❞ ◆✉♠❜❡r ❚❤❡♦r② ✷✼ ❏✉♥❡ ✷✵✶✽ ❉❛r✐♦ ❙♣✐r✐t♦ ❚♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✉❜s❡ts ♦❢ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡

■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ❩❛r✐s❦✐ s♣❛❝❡ ❛♥❞ t❤❡ ❩❛r✐s❦✐ t♦♣♦❧♦❣② ❲❡ s❤❛❧❧ ❛❧✇❛②s ❝♦♥s✐❞❡r ❛ ❞♦♠❛✐♥ D ❛♥❞ ❛ ✜❡❧❞ K ❝♦♥t❛✐♥✐♥❣ D ✳ ❩❛r ( K | D ) ✐s t❤❡ s❡t ♦❢ ✈❛❧✉❛t✐♦♥ ❞♦♠❛✐♥s ♦❢ K ❝♦♥t❛✐♥✐♥❣ D ✳ ■❢ K ✐s t❤❡ q✉♦t✐❡♥t ✜❡❧❞ ♦❢ D ✱ ✇❡ s❡t ❩❛r ( K | D ) = ❩❛r ( D ) ✱ ❛♥❞ ✐ts ❡❧❡♠❡♥ts ❛r❡ t❤❡ ✈❛❧✉❛t✐♦♥ ♦✈❡rr✐♥❣s ♦❢ D ✳ ❚❤❡ ❩❛r✐s❦✐ t♦♣♦❧♦❣② ♦♥ ❩❛r ( K | D ) ✐s ❣❡♥❡r❛t❡❞ ❜② t❤❡ s❡ts B ( x ✶ , . . . , x n ) := { V ∈ ❩❛r ( K | D ) | x ✶ , . . . , x n ∈ V } . ❊①❝❡♣t tr✐✈✐❛❧ ❝❛s❡s✱ t❤❡ ❩❛r✐s❦✐ t♦♣♦❧♦❣② ✐s ♥♦t T ✶ ✭✐✳❡✳✱ ♥♦t ❛❧❧ ♣♦✐♥ts ❛r❡ ❝❧♦s❡❞✮❀ ✐♥ ♣❛rt✐❝✉❧❛r✱ ✐t ✐s ♥♦t ❍❛✉s❞♦r✛✳ ❚❤❡ ♦♥❧② ❝❧♦s❡❞ ♣♦✐♥ts ❛r❡ t❤❡ ♠✐♥✐♠❛❧ ❡❧❡♠❡♥ts✳ ❉❛r✐♦ ❙♣✐r✐t♦ ❚♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✉❜s❡ts ♦❢ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡

■♥tr♦❞✉❝t✐♦♥ ❙♣❡❝tr❛❧ s♣❛❝❡s ❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ ✐s s♣❡❝tr❛❧ ✐❢ ✐t ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ Spec( R ) ❢♦r s♦♠❡ r✐♥❣ R ✳ ❙♣❡❝tr❛❧ s♣❛❝❡s ❝❛♥ ❜❡ ❝❤❛r❛❝t❡r✐③❡❞ t♦♣♦❧♦❣✐❝❛❧❧② ❬❍♦❝❤st❡r✱ ✶✾✻✾❪✳ ❩❛r ( K | D ) ✐s ❛ s♣❡❝tr❛❧ s♣❛❝❡✿ ♠♦r❡ ♣r❡❝✐s❡❧②✱ ✇❡ ❝❛♥ ✜♥❞ ✭❡①♣❧✐❝✐t❧②✮ ❛♥ ♦✈❡rr✐♥❣ ❑r ( K | D ) ♦❢ K [ X ] s✉❝❤ t❤❛t ❩❛r ( K | D ) ≃ Spec( ❑r ( K | D )) ✳ ❚❤❡ ❝♦♥str✉❝t✐❜❧❡ t♦♣♦❧♦❣② ♦♥ ❛ s♣❡❝tr❛❧ s♣❛❝❡ X ✐s t❤❡ ❝♦❛rs❡st t♦♣♦❧♦❣② ✇❤❡r❡ t❤❡ ♦♣❡♥ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡ts ♦❢ X ❛r❡ ❜♦t❤ ♦♣❡♥ ❛♥❞ ❝❧♦s❡❞✳ X ❝♦♥s ✐s ❛ s♣❡❝tr❛❧ s♣❛❝❡ t❤❛t ✐s ❛❧s♦ ❍❛✉s❞♦r✛✳ ❚❤❡ ❝❧♦s❡❞ s❡t ♦❢ X ❝♦♥s ❛r❡ s♣❡❝tr❛❧ ✭✐♥ t❤❡ st❛rt✐♥❣ t♦♣♦❧♦❣②✮✳ ❉❛r✐♦ ❙♣✐r✐t♦ ❚♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✉❜s❡ts ♦❢ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡

■♥tr♦❞✉❝t✐♦♥ ❲❤② t❤❡ ❩❛r✐s❦✐ t♦♣♦❧♦❣②❄ ❙t✉❞② ♦❢ r❡s♦❧✉t✐♦♥ ♦❢ s✐♥❣✉❧❛r✐t✐❡s ✭❩❛r✐s❦✐✮✳ ❨♦✉ ♥❡❡❞ t❤❡ ❝♦♠♣❛❝t♥❡ss ♦❢ ❩❛r ( K | D ) ✳ ❙t✉❞② ♦❢ ✐♥t❡rs❡❝t✐♦♥ ♦❢ ✈❛❧✉❛t✐♦♥ r✐♥❣s✳ ■❢ X ⊆ ❩❛r ( D ) ✐s ❛ ❝♦♠♣❛❝t s✉❜s❡t✱ ❡❛❝❤ V ∈ X ✐s ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧✱ ❛♥❞ � V ∈ X m V � = ( ✵ ) ✱ t❤❡♥ � V ∈ X V ✐s ❛ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❇é③♦✉t ❞♦♠❛✐♥ ✭✐✳❡✳✱ ❛❧❧ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✐❞❡❛❧s ❛r❡ ♣r✐♥❝✐♣❛❧✮ ❬❖❧❜❡r❞✐♥❣✱ ✷✵✶✼❪✳ ❙t✉❞② ♦❢ ❤♦❧♦♠♦r♣❤② ❛♥❞ r❡❛❧ ❤♦❧♦♠♦r♣❤② r✐♥❣s✳ ■❢ D ✐s ❛ Prü❢❡r ❞♦♠❛✐♥✱ t❤❡♥ ❩❛r ( D ) ≃ Spec( D ) ✳ ■❢ D ✐s ❛♥② ❞♦♠❛✐♥✱ t❤❡ ♠❛♣ ❩❛r ( D ) − → Spec( D ) ✱ V �→ m V ∩ D ✐s ❛ ❝❧♦s❡❞ ❝♦♥t✐♥✉♦✉s s✉r❥❡❝t✐♦♥✳ ■❢ D ✐s ❛♥② ❞♦♠❛✐♥✱ t❤❡ ♠❛♣ P �→ D P ❡♠❜❡❞s Spec( D ) ✐♥ t❤❡ s❡t ♦❢ ♦✈❡rr✐♥❣s ♦❢ D ✱ ❡♥❞♦✇❡❞ ✇✐t❤ t❤❡ ❩❛r✐s❦✐ t♦♣♦❧♦❣②✳ ❉❛r✐♦ ❙♣✐r✐t♦ ❚♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✉❜s❡ts ♦❢ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡

❈♦♠♣❛❝t♥❡ss ◆♦♥✲❝♦♠♣❛❝t s✉❜s♣❛❝❡s ♦❢ ✈❛❧✉❛t✐♦♥ r✐♥❣s ❲❤✐❧❡ ❩❛r ( D ) ✐s ❛❧✇❛②s ❝♦♠♣❛❝t✱ t❤❡ s❛♠❡ ❞♦❡s ♥♦t ❤❛♣♣❡♥ ❢♦r s✉❜s❡ts✳ ▲❡t V ❜❡ ❛ ♠✐♥✐♠❛❧ ❡❧❡♠❡♥t ♦❢ ❩❛r ( D ) ✳ ■❢ ❩❛r ( D ) \ { V } ✐s ❝♦♠♣❛❝t✱ t❤❡♥ V ✐s t❤❡ ✐♥t❡❣r❛❧ ❝❧♦s✉r❡ ♦❢ D [ x ✶ , . . . , x n ] M ❢♦r s♦♠❡ x ✶ , . . . , x n ∈ K ❛♥❞ M ∈ Max( D [ x ✶ , . . . , x n ]) ✳ ❚❤❡ ♣r♦♦❢ ✉s❡s t❤❡ ✐♥t❡❣r❛❧ ❝❧♦s✉r❡ ♦❢ ✐❞❡❛❧s ❛♥❞ ❛ ❝r✐t❡r✐♦♥ ❜❛s❡❞ ♦♥ s❡♠✐st❛r ♦♣❡r❛t✐♦♥s✳ ❚❤✐s ❝❛♥♥♦t ❤❛♣♣❡♥ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛s❡s✿ D ✐s ◆♦❡t❤❡r✐❛♥ ❛♥❞ dim( V ) ≥ ✷❀ dim( V ) > ✷ dim( D ) ❀ D ✐s ❧♦❝❛❧ ❛♥❞ � { P | P ∈ X } = ( ✵ ) ❢♦r s♦♠❡ ❢❛♠✐❧② X ♦❢ ♥♦♥③❡r♦ ✐♥❝♦♠♣❛r❛❜❧❡ ♣r✐♠❡ ✐❞❡❛❧s✳ ❉❛r✐♦ ❙♣✐r✐t♦ ❚♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✉❜s❡ts ♦❢ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡

❈♦♠♣❛❝t♥❡ss ❲❤❡♥ ❩❛r ( K | D ) ✐s ◆♦❡t❤❡r✐❛♥ ❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ ✐s ◆♦❡t❤❡r✐❛♥ ✐❢ ❛❧❧ ✐ts s✉❜s❡ts ❛r❡ ❝♦♠♣❛❝t✳ ■❢ R ✐s ❛ ◆♦❡t❤❡r✐❛♥ r✐♥❣✱ Spec( R ) ✐s ❛ ◆♦❡t❤❡r✐❛♥ s♣❛❝❡✳ ■❢ D = F ✐s ❛ ✜❡❧❞✱ t❤❡♥ ❩❛r ( K | F ) ✐s ❛ ◆♦❡t❤❡r✐❛♥ s♣❛❝❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ trdeg F K ≤ ✶ ❛♥❞✱ ✐❢ X ∈ K ✐s tr❛♥s❝❡♥❞❡♥t❛❧ ♦✈❡r F ✱ t❤❡♥ ❡✈❡r② ✈❛❧✉❛t✐♦♥ ♦♥ F [ X ] ❡①t❡♥❞s t♦ ✜♥✐t❡❧② ♠❛♥② ✈❛❧✉❛t✐♦♥s ♦❢ K ✳ ■❢ D ✐s ❧♦❝❛❧✱ t❤❡♥ ❩❛r ( D ) ❝❛♥ ❜❡ ◆♦❡t❤❡r✐❛♥ ♦♥❧② ✐❢ D ✐s ❛ ♣s❡✉❞♦✲✈❛❧✉❛t✐♦♥ ❞♦♠❛✐♥ ✭P❱❉✮✳ D ✐s ❛ P❱❉ ✐❢ ✐ts ♠❛①✐♠❛❧ ✐❞❡❛❧ M ✐s t❤❡ ♠❛①✐♠❛❧ ✐❞❡❛❧ ♦❢ ❛ ✈❛❧✉❛t✐♦♥ ♦✈❡rr✐♥❣✳ ■❢ ❩❛r ( D ) ✐s ◆♦❡t❤❡r✐❛♥✱ t❤❡♥ ❩❛r ( D ) \ ❩❛r ♠✐♥ ( D ) ✐s ❧✐♥❡❛r❧② ♦r❞❡r❡❞✳ ❩❛r ( D ) ✐s ◆♦❡t❤❡r✐❛♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ Spec( D ) ❛♥❞ ❩❛r ( D M ) ❛r❡ ◆♦❡t❤❡r✐❛♥ ❢♦r ❡✈❡r② M ∈ Max( D ) ✳ ❉❛r✐♦ ❙♣✐r✐t♦ ❚♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✉❜s❡ts ♦❢ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡

❈♦♠♣❛❝t♥❡ss ❖✈❡rr✐♥❣s ♦❢ ◆♦❡t❤❡r✐❛♥ ❞♦♠❛✐♥s ▲❡t D ❜❡ ❛ ◆♦❡t❤❡r✐❛♥ r✐♥❣ ✇✐t❤ dim( D ) ≥ ✷✱ ❛♥❞ q✉♦t✐❡♥t ✜❡❧❞ K ✳ ❲✐t❤ t❤❡ s❛♠❡ ♠❡t❤♦❞s ❛s ❛❜♦✈❡✱ t❤❡ s♣❛❝❡ ∆ ♦❢ ◆♦❡t❤❡r✐❛♥ ✈❛❧✉❛t✐♦♥ ♦✈❡rr✐♥❣s ♦❢ D ✐s ♥♦t ❝♦♠♣❛❝t✳ ❈♦♥s✐❞❡r t❤❡ s♣❛❝❡ ❖✈❡r ( D ) ♦❢ ♦✈❡rr✐♥❣s ♦❢ D ✭✐✳❡✳✱ r✐♥❣s ❜❡t✇❡❡♥ D ❛♥❞ K ✮ ✇✐t❤ t❤❡ ❩❛r✐s❦✐ t♦♣♦❧♦❣②✳ ❚❤❡ s❡t ♦❢ ♦✈❡rr✐♥❣s ♦❢ D t❤❛t ❛r❡ ◆♦❡t❤❡r✐❛♥ ✐s ❝♦♠♣❛❝t ✭✐t ❤❛s ❛ ♠✐♥✐♠✉♠✮ ❜✉t ✐t ✐s ♥♦t ❛ s♣❡❝tr❛❧ s♣❛❝❡✳ ◆♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❝♦♠♣❛❝t✿ { T ∈ ❖✈❡r ( D ) | T ✐s ❛ ♣r✐♥❝✐♣❛❧ ✐❞❡❛❧ ❞♦♠❛✐♥ } ❀ { T ∈ ❖✈❡r ( D ) | T ✐s ❛ ❉❡❞❡❦✐♥❞ ❞♦♠❛✐♥ } ❀ { T ∈ ❖✈❡r ( D ) | T ✐s ◆♦❡t❤❡r✐❛♥ ✇✐t❤ dim( T ) ≤ ✶ } ✳ ❉❛r✐♦ ❙♣✐r✐t♦ ❚♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✉❜s❡ts ♦❢ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡