Normality of secant varieties
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1 Normality of secant varieties Brooke Ullery Joint Mathematics Meetings January 6, 2016 Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
2 Introduction Let X be a smooth projective variety over C, and L a very ample line bundle so that X P(H 0 (L)) = P r. Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
3 Introduction Let X be a smooth projective variety over C, and L a very ample line bundle so that X P(H 0 (L)) = P r. Definition The secant variety of X, Σ(X, L) = Σ P r is the Zariski closure of the union of lines spanned by two points of X (i.e. secant and tangent lines). Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
4 Bad example X a curve in P 2, deg X 2. Then Σ = P 2. Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
5 Bad example X a curve in P 2, deg X 2. Then Σ = P 2. Problem: Secant lines intersect away from X. Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
6 Bad example X a curve in P 2, deg X 2. Then Σ = P 2. Problem: Secant lines intersect away from X. Solution: Every point in Σ\X will lie on exactly one secant or tangent line L is 3-very ample (i.e. no four points of X lie on a plane). Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
7 Bad example X a curve in P 2, deg X 2. Then Σ = P 2. Problem: Secant lines intersect away from X. Solution: Every point in Σ\X will lie on exactly one secant or tangent line L is 3-very ample (i.e. no four points of X lie on a plane). As long as L is 3-very ample, Σ will be singular precisely along X. Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
8 Bad example X a curve in P 2, deg X 2. Then Σ = P 2. Problem: Secant lines intersect away from X. Solution: Every point in Σ\X will lie on exactly one secant or tangent line L is 3-very ample (i.e. no four points of X lie on a plane). As long as L is 3-very ample, Σ will be singular precisely along X. Intuition/Motivating question As L becomes more positive, the singularities of Σ should improve. How positive does L need to be for Σ to be normal? Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
9 Main results Theorem Let X be a smooth projective variety, L a 3-very ample line bundle on X, and m x the ideal sheaf of x X. If for all x X and i > 0, the map Sym i H 0 (L m 2 x) H 0 (L i m 2i x ) is surjective, then Σ(X, L) is a normal variety. Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
10 Main results Theorem Let X be a smooth projective variety, L a 3-very ample line bundle on X, and m x the ideal sheaf of x X. If for all x X and i > 0, the map Sym i H 0 (L m 2 x) H 0 (L i m 2i x ) is surjective, then Σ(X, L) is a normal variety. *Note that the surjectivity of these maps is the same as bl x L( 2E) being normally generated for all blowups at a point, where E is the exceptional divisor. Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
11 Corollary 1: curves If X is a curve, and deg L 2g + 3, then Σ(X, L) is normal. Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
12 Corollary 1: curves If X is a curve, and deg L 2g + 3, then Σ(X, L) is normal. Corollary 2: canonical curves If X is a curve with Clifford index CliffX 3, then Σ(X, ω X ) is normal. Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
13 Corollary 1: curves If X is a curve, and deg L 2g + 3, then Σ(X, L) is normal. Corollary 2: canonical curves If X is a curve with Clifford index CliffX 3, then Σ(X, ω X ) is normal. Corollary 3: arbitrary dimension If X is a variety of dimension n, A is very ample, B is nef, and L = K X + 2(n + 1)A + B, then Σ(X, L) is normal. Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
14 Recent progress Theorem (Chou, Song 2015) With hypotheses from Corollary 3, 1 Σ(X, L) has Du Bois singularities, and 2 Σ(X, L) has rational singularities H i (O X ) = 0 for i > 0. Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
15 Resolution of singularities construction (Schwarzenberger, Bertram, Vermeire) Setup: X [2] = Hilbert scheme of length two subschemes on X. Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
16 Resolution of singularities construction (Schwarzenberger, Bertram, Vermeire) Setup: X [2] = Hilbert scheme of length two subschemes on X. Let L be a 3-very ample line bundle, and X P(H 0 (L)) = P r the corresponding embedding. Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
17 Resolution of singularities construction (Schwarzenberger, Bertram, Vermeire) Setup: X [2] = Hilbert scheme of length two subschemes on X. Let L be a 3-very ample line bundle, and X P(H 0 (L)) = P r the corresponding embedding. Recall: X [2] is smooth, and its universal subscheme is the incidence variety Φ = {(x, Z) X X [2] : x Z} = bl (X X ). Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
18 Resolution of singularities construction (Schwarzenberger, Bertram, Vermeire) Setup: X [2] = Hilbert scheme of length two subschemes on X. Let L be a 3-very ample line bundle, and X P(H 0 (L)) = P r the corresponding embedding. Recall: X [2] is smooth, and its universal subscheme is the incidence variety Φ = {(x, Z) X X [2] : x Z} = bl (X X ). Define the two projections Φ. q σ X X [2] Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
19 Φ q σ X X [2] Define E L = σ q L. Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
20 Φ q σ X X [2] Define E L = σ q L. E L is a rank two vector bundle, and the fiber over Z X [2] is H 0 (X, L Z ). Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
21 Φ q σ X X [2] Define E L = σ q L. E L is a rank two vector bundle, and the fiber over Z X [2] is H 0 (X, L Z ). The natural evaluation map H 0 (X, L) O X [2] E L induces a morphism f : P(E L ) P(H 0 (L)) = P r, Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
22 Φ q σ X X [2] Define E L = σ q L. E L is a rank two vector bundle, and the fiber over Z X [2] is H 0 (X, L Z ). The natural evaluation map H 0 (X, L) O X [2] E L induces a morphism f : P(E L ) P(H 0 (L)) = P r, which sends ( Z, H 0 (L Z ) Q ) ( H 0 (L) Q ), where Q is some one-dimensional quotient. Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
23 Notice: The image of f is Σ, Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
24 Notice: The image of f is Σ, since the surjections (H 0 (L) Q) in the image are precisely those that factor through H 0 (L Z ) for some Z X [2] ; i.e. they lie on the line determined by Z. Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
25 Notice: The image of f is Σ, since the surjections (H 0 (L) Q) in the image are precisely those that factor through H 0 (L Z ) for some Z X [2] ; i.e. they lie on the line determined by Z. f is injective away from f 1 (X ): Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
26 Notice: The image of f is Σ, since the surjections (H 0 (L) Q) in the image are precisely those that factor through H 0 (L Z ) for some Z X [2] ; i.e. they lie on the line determined by Z. f is injective away from f 1 (X ): The P 1 over each Z maps to the secant line determined by Z, and two secant lines only meet along X. Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
27 Notice: The image of f is Σ, since the surjections (H 0 (L) Q) in the image are precisely those that factor through H 0 (L Z ) for some Z X [2] ; i.e. they lie on the line determined by Z. f is injective away from f 1 (X ): The P 1 over each Z maps to the secant line determined by Z, and two secant lines only meet along X. In fact... Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
28 Notice: The image of f is Σ, since the surjections (H 0 (L) Q) in the image are precisely those that factor through H 0 (L Z ) for some Z X [2] ; i.e. they lie on the line determined by Z. f is injective away from f 1 (X ): The P 1 over each Z maps to the secant line determined by Z, and two secant lines only meet along X. In fact... Lemma (Bertram (curves), Vermeire (higher dim)) Let t : P(E L ) Σ be f with its target restricted. Then t is an isomorphism away from t 1 (X ). In particular, t is a resolution of singularities. Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
29 Geometry of the resolution Pre-image of X? t 1 (X ) = { ( Z, H 0 (L Z ) H 0 (L x ) x Z} = Φ. (i.e. a choice of a subscheme and a point of that subscheme) Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
30 Geometry of the resolution Pre-image of X? t 1 (X ) = { ( Z, H 0 (L Z ) H 0 (L x ) x Z} = Φ. (i.e. a choice of a subscheme and a point of that subscheme) Pre-image of x? t 1 (x) = bl x X. (i.e. all subschemes containing x) Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
31 Geometry of the resolution Pre-image of X? t 1 (X ) = { ( Z, H 0 (L Z ) H 0 (L x ) x Z} = Φ. (i.e. a choice of a subscheme and a point of that subscheme) Pre-image of x? t 1 (x) = bl x X. (i.e. all subschemes containing x) To summarize: bl x X {x} Φ q X P(E L ) t Σ(X, L) f P r Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
32 Strategy for proof of main theorem: Show that t O P(EL ) = O Σ by checking on the completion at x X (i.e. at the singular points). Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
33 Strategy for proof of main theorem: Show that t O P(EL ) = O Σ by checking on the completion at x X (i.e. at the singular points). Conjecture If X is a smooth projective curve of genus g, and L a line bundle such that deg L 2g + 2k + 1, then Σ k (X, L) is normal. Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, / 11
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