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Home , Birational geometry, Minimal model program

:CONTRACT YOUR BAD CURVES! ANDREA FANELLI,DILETTA MARTINELLI SUPERVISOR:DR PAOLO CASCINI

INTRODUCTION:OUR GYM SURFACES:ITALIAN VINTAGE TRAINING THREEFOLDS:THE 80’S,FLIPPING PLASTIC SURGERY

Costumers: PROJECTIVE VARIETIES over C (geomet- Finally someone who needs an effective workout! In Everyone knows that: the bigger that?!?!? K < 0 ric objects locally defined as zero loci of polynomials, early 20th century, many Italian surfaces put on some you are, the harder is to contract X Problem 3. If we contract a subva- embedded in some ). weight, due to fat meals and their mammas’ tiramisu’! bad curves! riety in high codimension, we come They ended up with tons of new curves! up with a new variety where neither Our goal: classify them up to birational equivalence makes sense! (we are allowed to cut off some useless and adipose Here’s the new phenomenon in dimension 2: we can K = 0 Solution: There is only one thing subvarieties, keeping essential information on them, X blow up points and obtain infinitely many new sur- to do: THE CURVE. This is such us their function fields). faces, all birational to the one we started with! a totally different step of the MMP: EXTREMAL μ-1l μ-1l 3 RAYS Main Tool: the KX , which is intrin- 2 starting with X we need to con- + E struct a new variety X (HOW?!?!) sic of our variety and can be useful to detect how the -1 μ l 1 training is going. It is our fat detector! and a birational map μ So now we know our enemy! We P l + l 1 A fat 3-fold 2 l f∶ X⋯ → X Our Training Program (MMP): start with a variety X, 3 start contracting them as for sur- make it sweat and come up with a MINIMAL MODEL faces, BUT... C Xmin. But every blow-up leaves a (fat and superfluous) trace: It took many decades to understand Problem 2. When you contract the cutting off the curve which neg- a rigid (-1)-curve. how to attach the problem for vari- atively intersects KX and replace rays, your variety can become sin- + eties of dimension 3 or bigger! gular! it with a new curve C such that Theorem 2 (Castelnuovo’s Contractibility Criterion). If + Problem 1. (−1)-curves mean noth- Solution: We need to enlarge the (KX+ ⋅ C ) > 0: this is the plastic a non-singular surface S contains a (−1)-curve, we can con- ing on a 3-fold. category we work with ; terminal surgery we need! tract it via a birational morphism φ∶ S → T to obtain another Solution: “rigid” curves are the singularities (the mildest singulari- We expect to end up with a MIN- non-singular surface T . MMP ones which intersect negatively ties you can imagine!) IMAL model (with NEF canonical K And here’s our personalised program for surfaces: the canonical bundle X (extremal bundle) also in this case, but there is rays) ; Mori’s Cone theorem, Well... Fair enough... Even with a priori no reason why the flipped

Start which describes the cone of effec- some stretch marks our variety variety should be “skinnier” than X S: non- tive curves on our variety: looks great! OH NO... What is ! So: "Do flips terminate?" singular surface MMPTODAY:BCHM’S LIPOSUCTION FLOWCHART:THE TRAINING TABLE CURVES:TOO SKINNY FOR US! Is there a S ∶= (− ) no c This is the (conjectural) training program for varieties 1 -curve Smin End In 2010, Birkar, Cascini, Hacon and M Kernan devel- Even curves can easily get very complicated! E on S? in arbitrary dimension! C is the category of varieties oped a new revolutionary surgical technique: cooking with terminal singularities. yes up the canonical bundle they can Castelnuovo’s Theorem φ∶ S → T ● construct flips; the contrac- Start S ∶= T tion of E ● come up with a minimal model for a large class of T : nons- X ∈ C ing surface varieties, called of GENERAL TYPE. It requires a lot of work and many techniques, but the yes X ∶= KX nef? End result is outstanding! Xmin After a finite number of steps, the routine ends, since But our work is rather easy for curves: just need to re- no the contraction φ actually simplifies the structure of S Cone and move the bad points! Contraction and we get a MINIMAL surface in perfect shape. FUTURE:FUSION WORKOUT Theorem φ∶ X → Y Theorem 1. Every curve is birationally equivalent to a One can also see that the canonical bundle of Smin has a very interesting property: its intersection with curves What’s left? Tons of varieties still waiting for your help: X is a unique nonsingular projective curve. + dim Y X ∶= X X ∶= Y < yes Mori fiber they really need to loose weight! dim X? End is positive (KSmin is NEF). space Algebraic curves (= Riemann Surfaces) are classified Here’s some open problems: no by their genus g (number of holes) into 3 classes: ● minimal models for many varieties of special type;

REFERENCES:THE COACHING STAFF ● Y ∈ C codim E(φ) termination of flips in dimension higher than 4; no g = 0 g = 1 ≥ 2? RATIONAL ( ), ELLIPTIC ( ) AND CURVES ● Debarre, Higher-Dimensional Algebraic , Springer-Verlag; ● (minimal models are even OF GENERAL TYPE (g ≥ 2) ● Hacon, Kovács, Classification of Higher Dimensional Algebraic Vari- better than expected... they got ripped!); yes eties, Birkhäuser; ● MMP in positive characteristic (this is hardcore!); flip X / X+ ∈ C ● Kollár, Mori, of Algebraic Varieties, Cambridge ● moduli spaces of varieties of general type (yes, they University Press; " z ● Matsuki, Introduction to the Mori Program, Springer-Verlag, Berlin. are in good shape now... what about a contest?) Y ∈~ C