2 Clarifies terminology
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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 $\widetilde{X} \to D = \mathbb{P}^{1}$.

[Edit]: 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 $\widetilde{X} \to D = \mathbb{P}^{1}$.

[Edit]: 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
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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.