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 ⊂ P r is smoothable” – Eisenbud and Harris, 3264 & All That Aaron Landesman and David Zureick-Brown Smoothability of Genus 6 Petri General Curves 2 / 21
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. Aaron Landesman and David Zureick-Brown Smoothability of Genus 6 Petri General Curves 3 / 21
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) → P ( 5 2 ) − 1 be the Let C be a Petri-general curve of genus 6 and let G (2 , 5) ֒ Pl¨ ucker embedding of the Grassmannian. Then, there exist hyperplanes H 1 , H 2 , H 3 , H 4 and a quadric hypersurface Q in P 9 so that C = G (2 , 5) ∩ H 1 ∩ H 2 ∩ H 3 ∩ H 4 ∩ Q if and only if β 1 , 3 = 0 . Aaron Landesman and David Zureick-Brown Smoothability of Genus 6 Petri General Curves 5 / 21
Petri’s Theorem Theorem (Noether–Enriques–Babbage–Petri) Let C be a smooth nonhyperelliptic canonical curve of genus at least 3 . The ideal I C is generated by quadrics if and only if C is neither trigonal nor a plane quintic. Aaron Landesman and David Zureick-Brown Smoothability of Genus 6 Petri General Curves 6 / 21
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! Aaron Landesman and David Zureick-Brown Smoothability of Genus 6 Petri General Curves 7 / 21
Initial Ideals Definition Let ≺ be an ordering on k [ x 1 , . . . , x n ] and let f ∈ k [ x 1 , . . . , x n ] 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 [ x 1 , . . . , x n ]. Then the initial ideal in ≺ ( I ) of I is in ≺ ( I ) := � in ≺ ( f ) � f ∈ I . Aaron Landesman and David Zureick-Brown Smoothability of Genus 6 Petri General Curves 8 / 21
Grobner Bases Definition Let I be a homogeneous ideal of k [ x 1 , . . . , x n ]. A Gr¨ obner basis for I , also known as a standard basis for I , is a set of elements f 1 , . . . , f s ∈ I such that in ≺ ( I ) = � in ≺ ( f 1 ) , . . . , in ≺ ( f s ) � . Aaron Landesman and David Zureick-Brown Smoothability of Genus 6 Petri General Curves 9 / 21
The S-pair Criterion Definition Let k [ x 1 , . . . , x n ] be endowed with a monomial ordering ≺ . Then, the S-pair of f and g , denoted S ( f , g ), is in ≺ ( f ) in ≺ ( g ) S ( f , g ) := GCD(in ≺ ( f ) , in ≺ ( g )) · g − GCD(in ≺ ( f ) , in ≺ ( g )) · f . Theorem (Buchberger’s S-pair Criterion) Let S := k [ x 1 , . . . , x n ] be endowed with a monomial ordering ≺ , and let f 1 , . . . , f s ∈ S. These form a Gr¨ obner basis for the ideal they generate if and only if for all i , j the remainder of S ( f i , f j ) upon division by f 1 , . . . , f s is 0 . Aaron Landesman and David Zureick-Brown Smoothability of Genus 6 Petri General Curves 10 / 21
Petri’s Theorem Theorem (Noether–Enriques–Babbage–Petri) Let C be a smooth nonhyperelliptic canonical curve of genus at least 3 . The ideal I C 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 f ij , cubics G k , and a quartic H, where 1 ≤ i < j ≤ g − 2 and 1 ≤ k ≤ g − 3 . Aaron Landesman and David Zureick-Brown Smoothability of Genus 6 Petri General Curves 11 / 21
Petri’s Relations Let x i be dual coordinates to a simple g − 2 secant P 1 , . . . , P g − 2 . Define α i := λ i x g − 1 + µ i x g so that α i vanishes to order 2 at P i . The relations are g − 2 � f i , j := x i x j − a sij ( x g − 1 , x g ) x s − β ij ( x g − 1 , x g ) s =1 � � G i := α i x 2 i − α g − 2 x 2 g i , j , k , l x j x k g − 1 x l g i , k , l x k g − 1 x l g − 2 + g + g 1 ≤ j ≤ g − 2 k + l =3 k + l =2 H := x g − 2 G 1 − α 1 x 1 f 1 , g − 2 . Aaron Landesman and David Zureick-Brown Smoothability of Genus 6 Petri General Curves 12 / 21
Quadratic Relations Lemma For 1 ≤ i < j ≤ g − 2 , a basis for the quadrics cutting out C is given by g − 2 � f ij := x i x j − a sij ( x g − 1 , x g ) x s − β ij ( x g − 1 , x g ) . s =1 Proof. dim H 0 ( C , I C (2)) = dim H 0 ( P g − 1 , O P g − 1 ) − dim H 0 ( C , O C (2)) � g + 1 � � g − 2 � = − (3 g − 3) = 2 2 by Noether’s theorem. Since x a x b where either b = a , b = g − 1, or b = g are 3 g − 3 independent monomials, the given relations are all quadric relations. Aaron Landesman and David Zureick-Brown Smoothability of Genus 6 Petri General Curves 13 / 21
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 f ij , cubics, G i and a quartic H form a Grobner basis for the ideal they generate if and only if they define a Petri-general canonical curve. Aaron Landesman and David Zureick-Brown Smoothability of Genus 6 Petri General Curves 14 / 21
Petri Scheme Definition Recall α s = λ s x g − 1 + µ s x g g − 2 � f i , j := x i x j − a sij x s − β ij s =1 � � G i := α i x 2 i − α g − 2 x 2 g i , j , k , l x j x k g − 1 x l g i , k , l x k g − 1 x l g − 2 + g + g 1 ≤ j ≤ g − 2 k + l =3 k + l =2 H := x g − 2 G 1 − α 1 x 1 f 1 , g − 2 Definition Let A := k [ ρ ijk , µ s , λ s , β ij , a iij , g i , j , k , l , g i , k , l ] λ 1 ··· λ g − 2 . Let J be the ideal generated by all S ( F 1 , F 2 ), where F 1 and F 2 range over all f ij or G k . The Petri Scheme of genus g is P g := Spec B / J . Aaron Landesman and David Zureick-Brown Smoothability of Genus 6 Petri General Curves 15 / 21
Example Genus 4 Example (Genus 4) A canonically embedded genus 4 curve is a complete intersection of f 12 and G 1 , so the S-pair criteria is automatically satisfied. The Petri scheme P 4 is then simply the open affine subset U = D ( λ 1 , λ 2 ) of A 21 k . Aaron Landesman and David Zureick-Brown Smoothability of Genus 6 Petri General Curves 16 / 21
Proof Outline Theorem (L–, Zureick-Brown) Genus 6 Petri-general curves are smoothable. Remark Little showed that the locus of the Petri scheme with β 1 , 3 � = 1 is irreducible, but incorrectly claimed the locus with β 1 , 3 = 1 is empty. Proof. 1 Show that Petri-general curves with β 1 , 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. Aaron Landesman and David Zureick-Brown Smoothability of Genus 6 Petri General Curves 18 / 21
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. Aaron Landesman and David Zureick-Brown Smoothability of Genus 6 Petri General Curves 19 / 21
Utility of the Log Petri Scheme Figure: A figure of a prototypical genus 6 canonical curve with β 13 = 1. Aaron Landesman and David Zureick-Brown Smoothability of Genus 6 Petri General Curves 20 / 21
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? Aaron Landesman and David Zureick-Brown Smoothability of Genus 6 Petri General Curves 21 / 21
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