Smoothability of Genus 6 Petri General Curves Aaron Landesman - - PowerPoint PPT Presentation

Smoothability of Genus 6 Petri General Curves

Aaron Landesman (Harvard University) David Zureick-Brown (Emory University) Joint Mathematics Meetings Seattle, WA January 8, 2016

Smoothability

Definition

A scheme is smoothable if it can be exhibited as the special fiber of a flat family, whose general member is smooth. “No one knows what is in the closure of the locus of smooth curves...That is, we may not know how to tell whether a given singular one-dimensional scheme C ⊂ Pr is smoothable” – Eisenbud and Harris, 3264 & All That

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Main Result

Definition

A Gorenstein canonically embedded curve is Petri-general if it has a simple g − 2 secant.

Theorem (L–, Zureick-Brown)

Genus 6 Petri-general curves are smoothable.

Remark

Little claimed to show this, but his proof has a serious error.

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Introducing the Petri Scheme

1 The Hilbert scheme – too big 2 The moduli space of curves – too small 3 The Petri scheme – just right Aaron Landesman and David Zureick-Brown Smoothability of Genus 6 Petri General Curves 4 / 21

Bonus Result: A Generalization of Mukai’s Description

Theorem (L–, Zureick-Brown)

Let C be a Petri-general curve of genus 6 and let G(2, 5) ֒ → P(5

2)−1 be the

Pl¨ ucker embedding of the Grassmannian. Then, there exist hyperplanes H1, H2, H3, H4 and a quadric hypersurface Q in P9 so that C = G(2, 5) ∩ H1 ∩ H2 ∩ H3 ∩ H4 ∩ Q if and only if β1,3 = 0.

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Petri’s Theorem

Theorem (Noether–Enriques–Babbage–Petri)

Let C be a smooth nonhyperelliptic canonical curve of genus at least 3. The ideal IC is generated by quadrics if and only if C is neither trigonal nor a plane quintic.

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Examples of Petri’s Theorem

Example

1 Genus 3 - plane quartics. 2 Genus 4 - complete intersection of a cubic and a quadric. 3 Genus 5 - complete intersection of three quadrics 4 Genus 6 - six quadrics, not complete intersection 5 Higher genus - massively overdetermined

Remark

But, there is a uniform description for all canonical curves using Grobner bases!

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Initial Ideals

Definition

Let ≺ be an ordering on k[x1, . . . , xn] and let f ∈ k[x1, . . . , xn] be a homogeneous polynomial. The initial term in≺(f ) of f is the largest monomial in the support of f with respect to the ordering ≺.

Definition

Let I be a homogeneous ideal of k[x1, . . . , xn]. Then the initial ideal in≺(I) of I is in≺(I) := in≺(f )f ∈I.

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Grobner Bases

Definition

Let I be a homogeneous ideal of k[x1, . . . , xn]. A Gr¨

• bner basis for I, also

known as a standard basis for I, is a set of elements f1, . . . , fs ∈ I such that in≺(I) = in≺(f1), . . . , in≺(fs).

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The S-pair Criterion

Definition

Let k[x1, . . . , xn] be endowed with a monomial ordering ≺. Then, the S-pair of f and g, denoted S(f , g), is S(f , g) := in≺(f ) GCD(in≺(f ), in≺(g)) · g − in≺(g) GCD(in≺(f ), in≺(g)) · f .

Theorem (Buchberger’s S-pair Criterion)

Let S := k[x1, . . . , xn] be endowed with a monomial ordering ≺, and let f1, . . . , fs ∈ S. These form a Gr¨

• bner basis for the ideal they generate if

and only if for all i, j the remainder of S(fi, fj) upon division by f1, . . . , fs is 0.

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Petri’s Theorem

Theorem (Noether–Enriques–Babbage–Petri)

Let C be a smooth nonhyperelliptic canonical curve of genus at least 3. The ideal IC is generated by quadrics if and only if C is neither trigonal nor a plane quintic.

Corollary

Further, for any nonhyperelliptic canonical curve, its ideal has a Grobner basis given by an explicit collection of quadrics fij, cubics Gk, and a quartic H, where 1 ≤ i < j ≤ g − 2 and 1 ≤ k ≤ g − 3.

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Petri’s Relations

Let xi be dual coordinates to a simple g − 2 secant P1, . . . , Pg−2. Define αi := λixg−1 + µixg so that αi vanishes to order 2 at Pi. The relations are fi,j := xixj −

g−2

• s=1

asij(xg−1, xg)xs − βij(xg−1, xg) Gi := αix2

i − αg−2x2 g−2 +

• 1≤j≤g−2

k+l=2

gi,j,k,lxjxk

g−1xl g +

• k+l=3

gi,k,lxk

g−1xl g

H := xg−2G1 − α1x1f1,g−2.

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Quadratic Relations

Lemma

For 1 ≤ i < j ≤ g − 2, a basis for the quadrics cutting out C is given by fij := xixj −

g−2

• s=1

asij(xg−1, xg)xs − βij(xg−1, xg).

Proof.

dim H0(C, IC(2)) = dim H0(Pg−1, OPg−1) − dim H0(C, OC(2)) = g + 1 2

• − (3g − 3) =

g − 2 2

• by Noether’s theorem. Since xaxb where either b = a, b = g − 1, or b = g

are 3g − 3 independent monomials, the given relations are all quadric relations.

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Schreyer’s Generalization

Recall:

Definition

A Gorenstein canonically embedded curve is Petri-general if it has a simple g − 2 secant.

Theorem (Schreyer)

The collection of quadrics fij, cubics, Gi and a quartic H form a Grobner basis for the ideal they generate if and only if they define a Petri-general canonical curve.

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Petri Scheme Definition

Recall αs = λsxg−1 + µsxg fi,j := xixj −

g−2

• s=1

asijxs − βij Gi := αix2

i − αg−2x2 g−2 +

• 1≤j≤g−2

k+l=2

gi,j,k,lxjxk

g−1xl g +

• k+l=3

gi,k,lxk

g−1xl g

H := xg−2G1 − α1x1f1,g−2

Definition

Let A := k[ρijk, µs, λs, βij, aiij, gi,j,k,l, gi,k,l]λ1···λg−2. Let J be the ideal generated by all S(F1, F2), where F1 and F2 range over all fij or Gk. The Petri Scheme of genus g is Pg := Spec B/J.

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Example Genus 4

Example (Genus 4)

A canonically embedded genus 4 curve is a complete intersection of f12 and G1, so the S-pair criteria is automatically satisfied. The Petri scheme P4 is then simply the open affine subset U = D(λ1, λ2) of A21

k .

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Proof Outline

Theorem (L–, Zureick-Brown)

Genus 6 Petri-general curves are smoothable.

Remark

Little showed that the locus of the Petri scheme with β1,3 = 1 is irreducible, but incorrectly claimed the locus with β1,3 = 1 is empty.

Proof.

1 Show that Petri-general curves with β1,3 = 1 are the union of a (3, 1)

curve and a (7, 3) curve meeting at three collinear points.

2 Show that the genus 1 and genus 3 curves are separately smoothable,

while maintaining the three points of contact.

3 Show that the three points of contact are smoothable. Aaron Landesman and David Zureick-Brown Smoothability of Genus 6 Petri General Curves 17 / 21

Log Petri Scheme

Theorem (Voight, Zureick-Brown)

A log degree δ Gorenstein curve C for δ ≥ 3 has a simple g − 3 + δ secant if and only if its Grobner basis is of a certain form, analogous to Petri’s relations.

Definition (Log Petri Scheme)

For a choice of δ and g define the log Petri scheme as the affine scheme with generators given by the coefficients of the equations in the Grobner basis for log degree δ, genus g curves, with relations imposed by the S-pair criterion.

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Utility of the Log Petri Scheme

In order to show the genus 1 and 3 curves are smoothable while maintaining their log degree 3 embedding, we need the irreducibility of the log Petri scheme.

Theorem (L–, Zureick-Brown)

The log 3, genus 1 and log 3, genus 3 Petri schemes are irreducible. Hence, degree 3 genus 1 and degree 7 genus 3 curves are smoothable, while maintaining a log degree 3 embedding.

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Utility of the Log Petri Scheme

Figure: A figure of a prototypical genus 6 canonical curve with β13 = 1.

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Further Questions

Theorem (Little)

The Petri Scheme is reducible in genus more than 12.

Question

Is the Petri scheme irreducible in genera 7 ≤ g ≤ 12?

Question

When is the log Petri scheme irreducible?

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