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# «September 29, 2012 Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 1 / 22 Introduction This is work with Melanie ...»

Cutting and pasting in algebraic geometry

Ravi Vakil

September 29, 2012

Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 1 / 22

Introduction

This is work with Melanie Matchett Wood, arXiv:1208.3166. I have a few

hard copies here.

Goals

Fun background: Grothendieck ring of varieties

Conjectures and speculations

Baby case: Points on a line (polynomials in one variable)

Hypersurfaces and conﬁguration spaces on general X

Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 2 / 22 Varieties

Complex aﬃne varieties VarC :

{x : f1 (x) = f2 (x) = · · · = 0} ⊂ Cn =: An.

C

Complex varieties: glue these together, with algebraic gluing maps. E.g.:

CPn.

Vark is “similar”. But An is not just k n ; it is a richer version.

k Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 3 / 22 Cutting and pasting varieties The Grothendieck ring of varieties (“cutting and pasting”) Generated by [X ] where X is variety (up to isomorphism) Additive structure: If Z ⊂ X closed, complement U open, then [X ] = [U] + [Z ].

Multiplicative structure: [X ][Y ] = [X × Y ].

Example: [pt] = 1.

Important deﬁnition L = [A1 ].

Example: [P2 ] = L2 + L + 1.

Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 4 / 22 Cutting and pasting varieties Analog in “usual geometry” (think: ﬁnite CW-complexes, manifolds) [R] = [R0 ] + [0] + [R0 ] so [R] = −1 and [Rn ] = (−1)n.

Get [X ] = χc, hence the Grothendieck ring is (only) Z.

Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 5 / 22 Cutting and pasting varieties Analog in “usual geometry” (think: ﬁnite CW-complexes, manifolds) [R] = [R0 ] + [0] + [R0 ] so [R] = −1 and [Rn ] = (−1)n.

Get [X ] = χc, hence the Grothendieck ring is (only) Z.

Back to algebraic geometry:

If k = Fq : “point-counting” map # : K (Var) → Z.

Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 5 / 22 Cutting and pasting varieties If k = C, we have a map χc (·) : K (Var) → Z.

Example: [CP2 ] = [C2 ] + [C] + 1 implies χ(CP2 ) = h0 − h1 + h2 − h3 + h4 = 3.

Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 6 / 22 Cutting and pasting varieties If k = C, we have a map χc (·) : K (Var) → Z.

Example: [CP2 ] = [C2 ] + [C] + 1 implies χ(CP2 ) = h0 − h1 + h2 − h3 + h4 = 3.

The fact that a geometric object is algebraic imposes more structure than you might even hope. If X is smooth and compact, we can recover each hi (X, Q) (Deligne’s theory of weights).

Example: h0 = h2 = h4 = 2.

Even just h0 is striking: the number of components is identiﬁable.

(Even more so: Hodge structures.) Without smoothness or compactness, you can still make predictions.

(Motivating example: [C∗ ] = L − 1, from which we can “see” h0 = 1, h1 = 1.) Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 6 / 22 Conjectures/questions/speculations Conjectures/questions/speculations.

If X and Y can both be cut up into the same pieces, then [X ] = [Y ].

E.g. two P1 -bundles over P1 have the same class.

Cut-and-paste conjecture (Larsen-Lunts, 2003) If X and Y are varieties with [X ] = [Y ], then X is cut-and-pastable to Y.

Theorem (Liu-Sebag) If /C, true for (i) smooth projective surfaces, (ii) varieties with ﬁnite number of rational curves.

Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 7 / 22 Conjectures/questions/speculations There are many clues that nature wants us to invert the aﬃne line L.

K (Var)L = K (Var)[1/L]

–  –  –

Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 8 / 22 Conjectures/questions/speculations

Main character of this talk: The motivic zeta function of X :

ZX (t) = [Symn X ]t n ∈ K (Var)[[t]].

Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 9 / 22 Conjectures/questions/speculations

Main character of this talk: The motivic zeta function of X :

ZX (t) = [Symn X ]t n ∈ K (Var)[[t]].

/k = Fq, point counting gives map ZX (t) → ζX (t) (the Weil zeta function, essentially generating function for solutions over all ﬁnite ﬁelds).

–  –  –

Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 9 / 22 Conjectures/questions/speculations Question (Kapranov, 2001) Is ZX (t) rational?

If X is a curve with rational point: yes.

If yes, then it would imply rationality of the Weil zeta function ζX (t) (Dwork, 1960).

Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 10 / 22 Conjectures/questions/speculations Question (Kapranov, 2001) Is ZX (t) rational?

If X is a curve with rational point: yes.

If yes, then it would imply rationality of the Weil zeta function ζX (t) (Dwork, 1960).

This is a strong prediction even over C: if we knew the ﬁrst few Symn X, we would know them all, by “cutting-and-pasting”.

Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 10 / 22 Conjectures/questions/speculations Question (Kapranov, 2001) Is ZX (t) rational?

If X is a curve with rational point: yes.

If yes, then it would imply rationality of the Weil zeta function ζX (t) (Dwork, 1960).

This is a strong prediction even over C: if we knew the ﬁrst few Symn X, we would know them all, by “cutting-and-pasting”.

Big question: Where in between K (Var) and Z does the zeta function start being rational? (In K (Vark )L ?) Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 10 / 22 Motivic stabilization of symmetric powers

–  –  –

True for (stably) rational varieties, and curves.

Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 11 / 22 Motivic stabilization of symmetric powers

–  –  –

True for (stably) rational varieties, and curves.

True once you “specialize to point-counting” (Dwork).

Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 11 / 22 Motivic stabilization of symmetric powers

–  –  –

True for (stably) rational varieties, and curves.

True once you “specialize to point-counting” (Dwork).

Motivated by Weil conjectures.

ζX (t) = (H 1 )(H 3 ) · · · (H 2d−1 )/(1 − t) · · · (1 − q dim X t) Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 11 / 22 Motivic stabilization of symmetric powers

–  –  –

True for (stably) rational varieties, and curves.

True once you “specialize to point-counting” (Dwork).

Motivated by Weil conjectures.

ζX (t) = (H 1 )(H 3 ) · · · (H 2d−1 )/(1 − t) · · · (1 − q dim X t) Topological motivation (Dold-Thom theorem — think Sym∞ X ) and consequences.

Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 11 / 22 Motivic stabilization of symmetric powers

–  –  –

True for (stably) rational varieties, and curves.

True once you “specialize to point-counting” (Dwork).

Motivated by Weil conjectures.

ζX (t) = (H 1 )(H 3 ) · · · (H 2d−1 )/(1 − t) · · · (1 − q dim X t) Topological motivation (Dold-Thom theorem — think Sym∞ X ) and consequences.

True once you specialize to “Hodge structures” (Cheah).

Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 11 / 22 Motivic stabilization of symmetric powers

–  –  –

True for (stably) rational varieties, and curves.

True once you “specialize to point-counting” (Dwork).

Motivated by Weil conjectures.

ζX (t) = (H 1 )(H 3 ) · · · (H 2d−1 )/(1 − t) · · · (1 − q dim X t) Topological motivation (Dold-Thom theorem — think Sym∞ X ) and consequences.

True once you specialize to “Hodge structures” (Cheah).

Theorem (Litt ’12): contradicts cut-and-paste conjecture, and “L not zero-divisor” conjecture.

Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 11 / 22 Baby Example: Points on a line (“no topology”) Consider n unordered points on a line = degree n monic polynomials =

–  –  –

discriminant: ∆1n−2 2 ⊂ An : locus with double root (or worse).

∆1n−m m ⊂ An : locus with m-fold root (or worse).

arithmetic: k = Fq. Probability of having m-fold root: 1/q m−1 (interesting exercise).

Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 12 / 22 Points on a line topology: k = C.

Theorem (Arnol’d) hi (Cn \ ∆1n−m m ) = 1 if i = 0, 2m − 1, and 0 otherwise.

Thus ∆1n−m m has only one nonzero (compactly-supported) cohomology group.

algebraic geometry: ∆1n−m m cut-and-pastes to An−m+1 (interesting exercise).

We should have predicted Arnol’d result.

Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 13 / 22 Points on a line Notice: Stabilization as n → ∞.

Arithmetic: stupid. Topology: stupid. But also: there is a map.

Theorem (Arnold; Galatius) The space of complex polynomials with no m-fold root has two nonzero cohomology groups: h0 = h2m−1 = 1.

Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 14 / 22 Points on a line More complicated discriminants: ∆v ⊂ An where v n.

Example: v = 1n−5 23 Then [∆v ]/[An ] = 1/L3.

Notice:

Stabilizes (by n = 5) Closed strata are better-behaved than open strata.

Arithmetic: 1/q 3. (Immediate from alg. geom. but easy like ∆1n−m m !) Topology: explicit prediction on cohomology.

Remark (hint of further structure): if v = 1n−7 223, the probability is no longer a power of L (i.e. q)!

Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 15 / 22 Conﬁguration spaces on general X

–  –  –

Theorem (Church) ∆0 1n stabilizes topologically (Inv.), know “what to”. (Randall-Williams v too) Theorem (V-Wood) (with hypotheses) ∆0 1n, ∆v 1n stabilize, can say what to, in terms of v motivic zeta-values. (Moral: ∆ is better) Hence conjecture: ∆v 1n stabilizes topologically.

Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 16 / 22 Hypersurfaces

Five motivations for theorem on hypersurfaces:

1) baby example: points on a line

2) probability n ∈ Z square-free Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 17 / 22 Hypersurfaces

Five motivations for theorem on hypersurfaces:

1) baby example: points on a line

2) probability n ∈ Z square-freeis 1/ζ(2) 3) Theorem (Poonen) Given smooth projective variety X in Pn over Fq. Probability that a random degree d hypersurface cuts X smoothly?

Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 17 / 22 Hypersurfaces

Five motivations for theorem on hypersurfaces:

1) baby example: points on a line

2) probability n ∈ Z square-freeis 1/ζ(2) 3) Theorem (Poonen) Given smooth projective variety X in Pn over Fq. Probability that a random degree d hypersurface cuts X smoothly?

–  –  –

Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 17 / 22 Hypersurfaces 4) Theorem (Vasiliev, ICM talk) Space of smooth divisors on Cn has two nonzero cohomology groups.

5) Severi varieties: Fix a smooth projective surface X, and ample line bundle L. Degree ∆1n−m m in P|L|? Gottsche conjecture (Tzeng;

Kool-Shende-Thomas): there is structure, related to modular forms. What about the limit motive?

Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 18 / 22 Hypersurfaces

–  –  –

Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 19 / 22 Hypersurfaces

Remarks:

(a) feature: even over C, proof uses ﬁnite ﬁelds (b) m = 0 get 1/ZX (1/Ld+1 ) — parallel to Poonen, but logically independent, and completely diﬀerent proof.

(c) closed form in terms of zeta-values and Sym? X.

(d) Suggests: enhanced version of Vassiliev’s theorem, Poonen’s theorem, etc. (Example: rational homology type of space of smooth divisors in |L⊗n | stabilizes.) (e) Question: (geometry) ZX (L? ) ↔??? (topology)?

Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 20 / 22 Back to points on X

–  –  –

(Same as “hypersurface” formula, except d + 1 has been replaced by 2d.) Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 21 / 22 Conclusion

In this area, there are three ways of looking at the world:

–  –  –

This talk is about a series of results in the central area that is cognate to things we know in the other two areas, although the proofs and methods are quite diﬀerent.

Discriminant (complements) stabilize as the problem goes to ∞, and the limit can be explicitly described in terms of the motivic zeta function.

Thank you!

Ravi Vakil (Stanford) Cutting and pasting in algebraic geometry September 29, 2012 22 / 22

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