2 Clarifies terminology

Let $$k$$ be an algebraically closed field. Let $$X/k$$ be a smooth projective variety. For a suitable embedding in $$\mathbb{P}^{n}$$ we can form a Lefschetz pencil $$\widetilde{X} \to D = \mathbb{P}^{1}$$.

: In response to Jason Starr's comment: I assume that every singular fibre of a Lefschetz pencil has a single ordinary double point (and is otherwise nonsingular). [/Edit]

Question: Can we say anything about the number of critical points of this Lefschetz pencil?

Can we give lower/upper bounds, for example involving the dimension/Betti numbers of $$X$$ and/or $$\widetilde{X}$$?

Asking Google gives some results for symplectic manifolds. I could not find anything related to algebraic varieties.

Notation: Let us fix the notation $$j \colon U \to D$$ for the smooth locus of $$f$$, and $$i \colon S \to D$$ the complement of $$U$$ in $$D$$. (So $$S$$ is the subset of $$D$$ with singular fibres.) Furthermore, $$d$$ is the dimension of $$X/k$$, hence also of $$\widetilde{X}/k$$. It is customary to write $$n$$ for the dimension of the fibres, so $$d = n+1$$. Let us write $$q$$ for the number of critical points, so $$q = \#S(k)$$. Finally fix a prime number $$\ell$$, invertible in $$k$$.

Motivation/baby case: (Please take in mind that I am a beginner with perverse sheaves, so the following might be totally wrong.) If the vanishing cycles are zero (a special case, implying $$d$$ is even) the number of critical points, $$q$$, has to be less then $$\dim \mathrm{H}^{d}(\widetilde{X}, \mathbb{Q}_{\ell})$$. I think this can be proven using the Leray spectral sequence for perverse sheaves (so that we have $$\mathrm{E}_{2}$$-degeneration). One can prove that $${}^{p}\mathrm{R}^{d}f_{*}\mathbb{Q}_{\ell} = (\mathrm{R}^{d-1}f_{*}\mathrm{Q}_{\ell})[1] \oplus i_{*}\mathbb{Q}_{\ell}(-d/2)$$. The critical points then contribute to the dimension of $$\mathrm{E}_{2}^{0,d} = \mathrm{H}^{0}(D, {}^{p}\mathrm{R}^{d}f_{*}\mathbb{Q}_{\ell})$$. Using the $$\mathrm{E}_{2}$$-degeneration, we see that $$\mathrm{E}_{2}^{0,d}$$ is a direct summand of $$\mathrm{H}^{d}(\widetilde{X}, \mathbb{Q}_{\ell})$$, proving that $$q$$ is less than the $$d$$-th Betti number.

Probably this has been investigated before, in particular in the case that the vanishing cycles are not zero. If so, I would be very happy with a reference to the literature.

Let $$k$$ be an algebraically closed field. Let $$X/k$$ be a smooth projective variety. For a suitable embedding in $$\mathbb{P}^{n}$$ we can form a Lefschetz pencil $$\widetilde{X} \to D = \mathbb{P}^{1}$$.

Question: Can we say anything about the number of critical points of this Lefschetz pencil?

Can we give lower/upper bounds, for example involving the dimension/Betti numbers of $$X$$ and/or $$\widetilde{X}$$?

Asking Google gives some results for symplectic manifolds. I could not find anything related to algebraic varieties.

Notation: Let us fix the notation $$j \colon U \to D$$ for the smooth locus of $$f$$, and $$i \colon S \to D$$ the complement of $$U$$ in $$D$$. (So $$S$$ is the subset of $$D$$ with singular fibres.) Furthermore, $$d$$ is the dimension of $$X/k$$, hence also of $$\widetilde{X}/k$$. It is customary to write $$n$$ for the dimension of the fibres, so $$d = n+1$$. Let us write $$q$$ for the number of critical points, so $$q = \#S(k)$$. Finally fix a prime number $$\ell$$, invertible in $$k$$.

Motivation/baby case: (Please take in mind that I am a beginner with perverse sheaves, so the following might be totally wrong.) If the vanishing cycles are zero (a special case, implying $$d$$ is even) the number of critical points, $$q$$, has to be less then $$\dim \mathrm{H}^{d}(\widetilde{X}, \mathbb{Q}_{\ell})$$. I think this can be proven using the Leray spectral sequence for perverse sheaves (so that we have $$\mathrm{E}_{2}$$-degeneration). One can prove that $${}^{p}\mathrm{R}^{d}f_{*}\mathbb{Q}_{\ell} = (\mathrm{R}^{d-1}f_{*}\mathrm{Q}_{\ell})[1] \oplus i_{*}\mathbb{Q}_{\ell}(-d/2)$$. The critical points then contribute to the dimension of $$\mathrm{E}_{2}^{0,d} = \mathrm{H}^{0}(D, {}^{p}\mathrm{R}^{d}f_{*}\mathbb{Q}_{\ell})$$. Using the $$\mathrm{E}_{2}$$-degeneration, we see that $$\mathrm{E}_{2}^{0,d}$$ is a direct summand of $$\mathrm{H}^{d}(\widetilde{X}, \mathbb{Q}_{\ell})$$, proving that $$q$$ is less than the $$d$$-th Betti number.

Probably this has been investigated before, in particular in the case that the vanishing cycles are not zero. If so, I would be very happy with a reference to the literature.

Let $$k$$ be an algebraically closed field. Let $$X/k$$ be a smooth projective variety. For a suitable embedding in $$\mathbb{P}^{n}$$ we can form a Lefschetz pencil $$\widetilde{X} \to D = \mathbb{P}^{1}$$.

: In response to Jason Starr's comment: I assume that every singular fibre of a Lefschetz pencil has a single ordinary double point (and is otherwise nonsingular). [/Edit]

Question: Can we say anything about the number of critical points of this Lefschetz pencil?

Can we give lower/upper bounds, for example involving the dimension/Betti numbers of $$X$$ and/or $$\widetilde{X}$$?

Asking Google gives some results for symplectic manifolds. I could not find anything related to algebraic varieties.

Notation: Let us fix the notation $$j \colon U \to D$$ for the smooth locus of $$f$$, and $$i \colon S \to D$$ the complement of $$U$$ in $$D$$. (So $$S$$ is the subset of $$D$$ with singular fibres.) Furthermore, $$d$$ is the dimension of $$X/k$$, hence also of $$\widetilde{X}/k$$. It is customary to write $$n$$ for the dimension of the fibres, so $$d = n+1$$. Let us write $$q$$ for the number of critical points, so $$q = \#S(k)$$. Finally fix a prime number $$\ell$$, invertible in $$k$$.

Motivation/baby case: (Please take in mind that I am a beginner with perverse sheaves, so the following might be totally wrong.) If the vanishing cycles are zero (a special case, implying $$d$$ is even) the number of critical points, $$q$$, has to be less then $$\dim \mathrm{H}^{d}(\widetilde{X}, \mathbb{Q}_{\ell})$$. I think this can be proven using the Leray spectral sequence for perverse sheaves (so that we have $$\mathrm{E}_{2}$$-degeneration). One can prove that $${}^{p}\mathrm{R}^{d}f_{*}\mathbb{Q}_{\ell} = (\mathrm{R}^{d-1}f_{*}\mathrm{Q}_{\ell})[1] \oplus i_{*}\mathbb{Q}_{\ell}(-d/2)$$. The critical points then contribute to the dimension of $$\mathrm{E}_{2}^{0,d} = \mathrm{H}^{0}(D, {}^{p}\mathrm{R}^{d}f_{*}\mathbb{Q}_{\ell})$$. Using the $$\mathrm{E}_{2}$$-degeneration, we see that $$\mathrm{E}_{2}^{0,d}$$ is a direct summand of $$\mathrm{H}^{d}(\widetilde{X}, \mathbb{Q}_{\ell})$$, proving that $$q$$ is less than the $$d$$-th Betti number.

Probably this has been investigated before, in particular in the case that the vanishing cycles are not zero. If so, I would be very happy with a reference to the literature.

1

# Bounding the number of critical points in a Lefschetz pencil

Let $$k$$ be an algebraically closed field. Let $$X/k$$ be a smooth projective variety. For a suitable embedding in $$\mathbb{P}^{n}$$ we can form a Lefschetz pencil $$\widetilde{X} \to D = \mathbb{P}^{1}$$.

Question: Can we say anything about the number of critical points of this Lefschetz pencil?

Can we give lower/upper bounds, for example involving the dimension/Betti numbers of $$X$$ and/or $$\widetilde{X}$$?

Asking Google gives some results for symplectic manifolds. I could not find anything related to algebraic varieties.

Notation: Let us fix the notation $$j \colon U \to D$$ for the smooth locus of $$f$$, and $$i \colon S \to D$$ the complement of $$U$$ in $$D$$. (So $$S$$ is the subset of $$D$$ with singular fibres.) Furthermore, $$d$$ is the dimension of $$X/k$$, hence also of $$\widetilde{X}/k$$. It is customary to write $$n$$ for the dimension of the fibres, so $$d = n+1$$. Let us write $$q$$ for the number of critical points, so $$q = \#S(k)$$. Finally fix a prime number $$\ell$$, invertible in $$k$$.

Motivation/baby case: (Please take in mind that I am a beginner with perverse sheaves, so the following might be totally wrong.) If the vanishing cycles are zero (a special case, implying $$d$$ is even) the number of critical points, $$q$$, has to be less then $$\dim \mathrm{H}^{d}(\widetilde{X}, \mathbb{Q}_{\ell})$$. I think this can be proven using the Leray spectral sequence for perverse sheaves (so that we have $$\mathrm{E}_{2}$$-degeneration). One can prove that $${}^{p}\mathrm{R}^{d}f_{*}\mathbb{Q}_{\ell} = (\mathrm{R}^{d-1}f_{*}\mathrm{Q}_{\ell})[1] \oplus i_{*}\mathbb{Q}_{\ell}(-d/2)$$. The critical points then contribute to the dimension of $$\mathrm{E}_{2}^{0,d} = \mathrm{H}^{0}(D, {}^{p}\mathrm{R}^{d}f_{*}\mathbb{Q}_{\ell})$$. Using the $$\mathrm{E}_{2}$$-degeneration, we see that $$\mathrm{E}_{2}^{0,d}$$ is a direct summand of $$\mathrm{H}^{d}(\widetilde{X}, \mathbb{Q}_{\ell})$$, proving that $$q$$ is less than the $$d$$-th Betti number.

Probably this has been investigated before, in particular in the case that the vanishing cycles are not zero. If so, I would be very happy with a reference to the literature.