Toric Criterion

Intro

conjecture

complexity

complexity
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absolute complexity
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conjecture

Results

history
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toric for surface germ
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threefold toric
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Precise theorem in BMSZ18 Main Thm A
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Remarks

Cox ring of MDS
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Sketch

step 1

Replace by dlt model.

dlt modification
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step 2

Replace by MDS.

step 3

Show that the cox ring is a polynomial ring.

step 4

Deduct to original pair.

Cox ring

recall

local complexity
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pic 1 complexity
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Cox Ring of Toric
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need to show $r=\rho$

A local version

Lemma local version
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from local version to easy global version, using the projective cone and apply local version
r=1 lemma
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Step 3.1

r=1 implies $\rho =1$

By some mmp and lemma above r=1 case
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induction on r

P1 bundle over MDS
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smoothness of Cox ring

Cox lc if X CY type MDS
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CY type iff lc cox
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universal torsor
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MDS

fano type is MDS
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Fano and CY type
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small complexity implies fano type
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less numerical dimension
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vertical component
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relative version

formally toric
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local formally toric
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